Introduction
For ∣ q ∣ < 1 |q| < 1 ∣ q ∣ < 1
ϑ 2 ( q ) = ∑ k ∈ Z q ( k + 1 / 2 ) 2 , ϑ 3 ( q ) = ∑ k ∈ Z q k 2 , ϑ 4 ( q ) = ∑ k ∈ Z ( − 1 ) k q k 2 . \vartheta_2(q)=\sum_{k\in\mathbb{Z}}q^{(k+1/2)^2},\qquad\vartheta_3(q)=\sum_{k\in\mathbb{Z}}q^{k^2},\qquad\vartheta_4(q)=\sum_{k\in\mathbb{Z}}(-1)^kq^{k^2}. ϑ 2 ( q ) = k ∈ Z ∑ q ( k + 1/2 ) 2 , ϑ 3 ( q ) = k ∈ Z ∑ q k 2 , ϑ 4 ( q ) = k ∈ Z ∑ ( − 1 ) k q k 2 . Jacobi [12 p. 90, Eq. (14)] proved the fundamental quartic identity
ϑ 3 ( q ) 4 = ϑ 2 ( q ) 4 + ϑ 4 ( q ) 4 . \vartheta_3(q)^4=\vartheta_2(q)^4+\vartheta_4(q)^4. ϑ 3 ( q ) 4 = ϑ 2 ( q ) 4 + ϑ 4 ( q ) 4 . J. M. Borwein and P. B. Borwein [2 ] discovered a cubic counterpart naturally associated with the root lattice A 2 A_2 A 2
Q ( m , n ) = m 2 + m n + n 2 , ω = exp ( 2 π i / 3 ) , Q(m,n)=m^2+mn+n^2,\qquad\omega=\exp(2\pi i/3), Q ( m , n ) = m 2 + mn + n 2 , ω = exp ( 2 π i /3 ) , and define
a ( q ) = ∑ m , n ∈ Z q Q ( m , n ) , a(q)=\sum_{m,n\in\mathbb{Z}}q^{Q(m,n)}, a ( q ) = m , n ∈ Z ∑ q Q ( m , n ) , b ( q ) = ∑ m , n ∈ Z ω m − n q Q ( m , n ) , b(q)=\sum_{m,n\in\mathbb{Z}}\omega^{m-n}q^{Q(m,n)}, b ( q ) = m , n ∈ Z ∑ ω m − n q Q ( m , n ) , c ( q ) = ∑ m , n ∈ Z q Q ( m + 1 / 3 , n + 1 / 3 ) . c(q)=\sum_{m,n\in\mathbb{Z}}q^{Q(m+1/3,n+1/3)}. c ( q ) = m , n ∈ Z ∑ q Q ( m + 1/3 , n + 1/3 ) . The Borweins [2 Eq. (2.3)] proved that
a ( q ) 3 = b ( q ) 3 + c ( q ) 3 (1) a(q)^3=b(q)^3+c(q)^3 \tag*{(1)} a ( q ) 3 = b ( q ) 3 + c ( q ) 3 (1) so the fourth-power relation for the classical one-dimensional theta functions is replaced by a third-power relation attached to A 2 A_2 A 2 [2 , 3 , 9 ] .
Schultz [15 ] introduced nine cubic theta functions in two complex variables, four of which play the role of the four classical theta functions. Following Schultz [15 pp. 621–622] , let e ( v ) : = exp ( 2 π i v ) e(v):=\exp(2\pi iv) e ( v ) := exp ( 2 π i v ) τ ∈ H 1 \tau\in\mathfrak{H}_1 τ ∈ H 1 q = e ( τ ) q=e(\tau) q = e ( τ ) u = ( u 1 , u 2 ) T ∈ C 2 \boldsymbol{u}=(u_1,u_2)^T\in\mathbb{C}^2 u = ( u 1 , u 2 ) T ∈ C 2 x = e ( u 1 ) x=e(u_1) x = e ( u 1 ) y = e ( u 2 ) y=e(u_2) y = e ( u 2 )
Θ 1 ( u ; q ) = ∑ m , n ∈ Z ω 1 + m − n q Q ( m + 1 / 3 , n + 1 / 3 ) x m + 1 / 3 y n + 1 / 3 , Θ 2 ( u ; q ) = ∑ m , n ∈ Z q Q ( m + 1 / 3 , n + 1 / 3 ) x m + 1 / 3 y n + 1 / 3 , Θ 3 ( u ; q ) = ∑ m , n ∈ Z q Q ( m , n ) x m y n , Θ 4 ( u ; q ) = ∑ m , n ∈ Z ω m − n q Q ( m , n ) x m y n . (2) \begin{aligned}
\Theta_1(\boldsymbol{u};q)&=\sum_{m,n\in\mathbb{Z}}\omega^{1+m-n}q^{Q(m+1/3,n+1/3)}x^{m+1/3}y^{n+1/3},\\
\Theta_2(\boldsymbol{u};q)&=\sum_{m,n\in\mathbb{Z}}q^{Q(m+1/3,n+1/3)}x^{m+1/3}y^{n+1/3},\\
\Theta_3(\boldsymbol{u};q)&=\sum_{m,n\in\mathbb{Z}}q^{Q(m,n)}x^my^n,\\
\Theta_4(\boldsymbol{u};q)&=\sum_{m,n\in\mathbb{Z}}\omega^{m-n}q^{Q(m,n)}x^my^n. \tag*{(2)}
\end{aligned} Θ 1 ( u ; q ) Θ 2 ( u ; q ) Θ 3 ( u ; q ) Θ 4 ( u ; q ) = m , n ∈ Z ∑ ω 1 + m − n q Q ( m + 1/3 , n + 1/3 ) x m + 1/3 y n + 1/3 , = m , n ∈ Z ∑ q Q ( m + 1/3 , n + 1/3 ) x m + 1/3 y n + 1/3 , = m , n ∈ Z ∑ q Q ( m , n ) x m y n , = m , n ∈ Z ∑ ω m − n q Q ( m , n ) x m y n . (2) The fractional powers are interpreted through the additive variables τ , u 1 , u 2 \tau,u_1,u_2 τ , u 1 , u 2 [15 ] . Schultz [15 Corollary 4.3] proved that
Θ 1 ( u ; q ) 3 + Θ 3 ( u ; q ) 3 = Θ 2 ( u ; q ) 3 + Θ 4 ( u ; q ) 3 (3) \Theta_1(\boldsymbol{u};q)^3+\Theta_3(\boldsymbol{u};q)^3=\Theta_2(\boldsymbol{u};q)^3+\Theta_4(\boldsymbol{u};q)^3 \tag*{(3)} Θ 1 ( u ; q ) 3 + Θ 3 ( u ; q ) 3 = Θ 2 ( u ; q ) 3 + Θ 4 ( u ; q ) 3 (3) The definitions give
Θ 1 ( 0 ; q ) = 0 , Θ 2 ( 0 ; q ) = c ( q ) , Θ 3 ( 0 ; q ) = a ( q ) , Θ 4 ( 0 ; q ) = b ( q ) . \Theta_1(0;q)=0,\qquad\Theta_2(0;q)=c(q),\qquad\Theta_3(0;q)=a(q),\qquad\Theta_4(0;q)=b(q). Θ 1 ( 0 ; q ) = 0 , Θ 2 ( 0 ; q ) = c ( q ) , Θ 3 ( 0 ; q ) = a ( q ) , Θ 4 ( 0 ; q ) = b ( q ) . Thus (1.3) specializes to (1.1). The identity (1.3) was subsequently revisited by Chan, Chan, Liu, and Zudilin [4 ] , who gave two further proofs and several related identities. Zhu [17 Theorem 7.12] gave a computational proof of this identity. Zhu [17 Theorem 7.11] also proved an analogous identity for the D 4 D_4 D 4
Our main result is a general theta-series identity whose matrix specializations yield all the identities above, among others.
To describe the scope, let M M M m m m n ≥ 1 n \ge1 n ≥ 1 a , b ∈ R m , n a,b \in\mathbb{R}^{m,n} a , b ∈ R m , n
Θ M , n [ a , b ] ( τ , z ) = ∑ g ∈ Z m , n e ( 1 2 τ ( g + a ) T M ( g + a ) + ( g + a ) T M ( z + b ) ) , \Theta_{M,n}[a,b](\tau,z)=\sum_{g\in\mathbb{Z}^{m,n}}\operatorname{e}\left(\frac{1}{2}\tau(g+a)^{T}M(g+a)+(g+a)^{T}M(z+b)\right), Θ M , n [ a , b ] ( τ , z ) = g ∈ Z m , n ∑ e ( 2 1 τ ( g + a ) T M ( g + a ) + ( g + a ) T M ( z + b ) ) , for ( τ , z ) ∈ H n × C m , n (\tau,z)\in\mathfrak{H}_{n}\times\mathbb{C}^{m,n} ( τ , z ) ∈ H n × C m , n [18 ] studied theta series of this type from the viewpoint of Jacobi forms of higher degree; see Subsection 2.2 for the basic properties used here.
The main result is Theorem 2.7 ; we first state a simplified version. Let n , m , μ n,m,\mu n , m , μ M 1 , M 2 M_1,M_2 M 1 , M 2 m m m τ ∈ H n \tau\in\mathfrak{H}_{n} τ ∈ H n a i , b i ∈ R m , n a_i,b_i\in\mathbb{R}^{m,n} a i , b i ∈ R m , n z i ∈ C m , n z_i\in\mathbb{C}^{m,n} z i ∈ C m , n i = 1 , 2 , … , μ i=1,2,\ldots,\mu i = 1 , 2 , … , μ A ∈ GL m μ ( Q ) A\in\operatorname{GL}_{m\mu}(\mathbb{Q}) A ∈ GL m μ ( Q ) A T M 1 ( μ ) A = M 2 ( μ ) A^{T}M_1^{(\mu)}A=M_2^{(\mu)} A T M 1 ( μ ) A = M 2 ( μ )
( a 1 ′ T , … , a μ ′ T ) = ( a 1 T , … , a μ T ) A − T , (a_1^{\prime T},\ldots,a_\mu^{\prime T})=(a_1^{T},\ldots,a_\mu^{T})A^{-T}, ( a 1 ′ T , … , a μ ′ T ) = ( a 1 T , … , a μ T ) A − T , ( b 1 ′ T , … , b μ ′ T ) = ( b 1 T , … , b μ T ) A − T , (b_1^{\prime T},\ldots,b_\mu^{\prime T})=(b_1^{T},\ldots,b_\mu^{T})A^{-T}, ( b 1 ′ T , … , b μ ′ T ) = ( b 1 T , … , b μ T ) A − T , ( w 1 T , … , w μ T ) = ( z 1 T , … , z μ T ) A − T . (w_1^{T},\ldots,w_\mu^{T})=(z_1^{T},\ldots,z_\mu^{T})A^{-T}. ( w 1 T , … , w μ T ) = ( z 1 T , … , z μ T ) A − T . Then
∏ i = 1 μ Θ M 1 , n [ a i , b i ] ( τ , z i ) = ∑ α , β c α , β ∏ i = 1 μ Θ M 2 , n [ a i ′ + α i , b i ′ + β i ] ( τ , w i ) , (4) \prod_{i=1}^{\mu}\Theta_{M_1,n}[a_i,b_i](\tau,z_i)=\sum_{\alpha,\beta}c_{\alpha,\beta}\prod_{i=1}^{\mu}\Theta_{M_2,n}[a_i'+\alpha_i,b_i'+\beta_i](\tau,w_i), \tag*{(4)} i = 1 ∏ μ Θ M 1 , n [ a i , b i ] ( τ , z i ) = α , β ∑ c α , β i = 1 ∏ μ Θ M 2 , n [ a i ′ + α i , b i ′ + β i ] ( τ , w i ) , (4) where ( α 1 T , … , α μ T ) : = α T (\alpha_1^{T},\ldots,\alpha_\mu^{T}):=\alpha^{T} ( α 1 T , … , α μ T ) := α T ( β 1 T , … , β μ T ) : = β T (\beta_1^{T},\ldots,\beta_\mu^{T}):=\beta^{T} ( β 1 T , … , β μ T ) := β T α \alpha α β \beta β Q m μ , n \mathbb{Q}^{m\mu,n} Q m μ , n c α , β c_{\alpha,\beta} c α , β Theorem 2.7 for the precise statement.
The proof relies on a lemma of Igusa [10, p. 136, Lemma 1]. Igusa [10, pp. 137 and 139] used this lemma to derive two classical theta relations of Riemann. These relations are also special cases of our main theorem, obtained by specializing M 1 = ( 1 ) M_1=(1) M 1 = ( 1 ) Theorems 2.12 and 2.13 . In addition to these classical formulas, theta relations have been studied extensively; see, for example, [5 , 6 , 8 , 11 ] . Gritsenko and Wang [8 ] , in particular, show that certain finite sums of products of theta series are holomorphic Jacobi forms and then derive theta relations from the theory of Jacobi forms. By contrast, the generic theta relation (1.4) can be proved without using the modular and elliptic transformation laws of Θ M , n [ a , b ] ( τ , z ) \Theta_{M,n}[a,b](\tau,z) Θ M , n [ a , b ] ( τ , z ) q q q n n n M 1 , M 2 , A M_1,M_2,A M 1 , M 2 , A A T M 1 ( μ ) A = M 2 ( μ ) A^{T}M_1^{(\mu)}A=M_2^{(\mu)} A T M 1 ( μ ) A = M 2 ( μ ) n n n
Applications
All families of theta identities below arise by specializing the main formula (1.4). The procedure is as follows:
Choose an appropriate A A A
Specify suitable representative sets for α \alpha α β \beta β
Identify theta series on the right-hand side that are proportional to each other.
Combine the relations linearly into the desired relation.
The first application concerns the A 2 A_2 A 2 F 3 = Z / 3 Z \mathbb{F}_3=\mathbb{Z}/3\mathbb{Z} F 3 = Z /3 Z p , q ∈ F 3 n \mathbf{p},\mathbf{q}\in\mathbb{F}_3^n p , q ∈ F 3 n
Θ p , q ( n ) ( τ , z ) : = Θ M , n [ p ρ , q σ ] ( τ , z ) , θ p , q ( n ) ( τ ) : = Θ p , q ( n ) ( τ , 0 ) , \Theta_{\mathbf{p},\mathbf{q}}^{(n)}(\tau,z):=\Theta_{M,n}[\mathbf{p}\rho,\mathbf{q}\sigma](\tau,z),
\qquad
\theta_{\mathbf{p},\mathbf{q}}^{(n)}(\tau):=\Theta_{\mathbf{p},\mathbf{q}}^{(n)}(\tau,0), Θ p , q ( n ) ( τ , z ) := Θ M , n [ p ρ , q σ ] ( τ , z ) , θ p , q ( n ) ( τ ) := Θ p , q ( n ) ( τ , 0 ) , where M = ( 2 − 1 − 1 2 ) M=\begin{pmatrix}2&-1\\-1&2\end{pmatrix} M = ( 2 − 1 − 1 2 ) A 2 A_2 A 2 ρ = ( 1 / 3 , − 1 / 3 ) T \rho=(1/3,-1/3)^{T} ρ = ( 1/3 , − 1/3 ) T σ = ( 1 / 3 , 1 / 3 ) T \sigma=(1/3,1/3)^{T} σ = ( 1/3 , 1/3 ) T n = 1 n=1 n = 1 [15 ] ; see Example 2.4. Let C = ( 1 1 − 1 2 ) C=\begin{pmatrix}1&1\\-1&2\end{pmatrix} C = ( 1 − 1 1 2 )
3 n Θ p , q ( n ) ( τ , z ) 3 = ∑ u , v ∈ F 3 n ω p ⋅ v + q ⋅ u Θ u , v ( n ) ( τ , C z ) θ u , v ( n ) ( τ ) 2 . (5) 3^n\Theta_{\mathbf{p},\mathbf{q}}^{(n)}(\tau,z)^3=\sum_{\mathbf{u},\mathbf{v}\in\mathbb{F}_3^n}\omega^{\mathbf{p}\cdot\mathbf{v}+\mathbf{q}\cdot\mathbf{u}}\Theta_{\mathbf{u},\mathbf{v}}^{(n)}(\tau,Cz)\theta_{\mathbf{u},\mathbf{v}}^{(n)}(\tau)^2. \tag*{(5)} 3 n Θ p , q ( n ) ( τ , z ) 3 = u , v ∈ F 3 n ∑ ω p ⋅ v + q ⋅ u Θ u , v ( n ) ( τ , C z ) θ u , v ( n ) ( τ ) 2 . (5) This is proved in Theorem 3.1. For n = 1 n=1 n = 1 Theorem 3.4 .
In degree two, we classify the theta constants θ p , q ( 2 ) ( τ ) \theta_{\mathbf{p},\mathbf{q}}^{(2)}(\tau) θ p , q ( 2 ) ( τ ) θ p , q ( 2 ) ( τ ) \theta_{\mathbf{p},\mathbf{q}}^{(2)}(\tau) θ p , q ( 2 ) ( τ ) Theorem 3.7 for the precise statement and proof.
To write one of the above cubic relations in expanded form, let
τ = ( τ 1 τ 0 τ 0 τ 2 ) ∈ H 2 , q = e ( τ 1 ) , s = e ( τ 0 ) , t = e ( τ 2 ) . \tau=\begin{pmatrix}\tau_1&\tau_0\\\tau_0&\tau_2\end{pmatrix}\in\mathfrak{H}_2,\qquad q=e(\tau_1),\qquad s=e(\tau_0),\qquad t=e(\tau_2). τ = ( τ 1 τ 0 τ 0 τ 2 ) ∈ H 2 , q = e ( τ 1 ) , s = e ( τ 0 ) , t = e ( τ 2 ) . Then
( ∑ j , k , ℓ , h ∈ Z q Q ( j , k ) s 2 j ℓ + k ℓ + j h + 2 k h t Q ( ℓ , h ) ) 3 = ( ∑ j , k , ℓ , h ∈ Z ω j − k + ℓ − h q Q ( j , k ) s 2 j ℓ + k ℓ + j h + 2 k h t Q ( ℓ , h ) ) 3 + ( ∑ j , k , ℓ , h ∈ Z + 1 / 3 q Q ( j , k ) s 2 j ℓ + k ℓ + j h + 2 k h t Q ( ℓ , h ) ) 3 + ( ∑ j , k ∈ Z ℓ , h ∈ Z + 1 / 3 ω j − k q Q ( j , k ) s 2 j ℓ + k ℓ + j h + 2 k h t Q ( ℓ , h ) ) 3 + ( ∑ j , k ∈ Z + 1 / 3 ℓ , h ∈ Z ω ℓ − h q Q ( j , k ) s 2 j ℓ + k ℓ + j h + 2 k h t Q ( ℓ , h ) ) 3 . (6) \begin{aligned}
\left(\sum_{j,k,\ell,h\in\mathbb{Z}}q^{Q(j,k)}s^{2j\ell+k\ell+jh+2kht}Q(\ell,h)\right)^3
&=\left(\sum_{j,k,\ell,h\in\mathbb{Z}}\omega^{j-k+\ell-h}q^{Q(j,k)}s^{2j\ell+k\ell+jh+2kht}Q(\ell,h)\right)^3\\
&+\left(\sum_{j,k,\ell,h\in\mathbb{Z}+1/3}q^{Q(j,k)}s^{2j\ell+k\ell+jh+2kht}Q(\ell,h)\right)^3\\
&+\left(\sum_{\substack{j,k\in\mathbb{Z}\\\ell,h\in\mathbb{Z}+1/3}}\omega^{j-k}q^{Q(j,k)}s^{2j\ell+k\ell+jh+2kht}Q(\ell,h)\right)^3\\
&+\left(\sum_{\substack{j,k\in\mathbb{Z}+1/3\\\ell,h\in\mathbb{Z}}}\omega^{\ell-h}q^{Q(j,k)}s^{2j\ell+k\ell+jh+2kht}Q(\ell,h)\right)^3. \tag*{(6)}
\end{aligned} j , k , ℓ , h ∈ Z ∑ q Q ( j , k ) s 2 j ℓ + k ℓ + j h + 2 k h t Q ( ℓ , h ) 3 = j , k , ℓ , h ∈ Z ∑ ω j − k + ℓ − h q Q ( j , k ) s 2 j ℓ + k ℓ + j h + 2 k h t Q ( ℓ , h ) 3 + j , k , ℓ , h ∈ Z + 1/3 ∑ q Q ( j , k ) s 2 j ℓ + k ℓ + j h + 2 k h t Q ( ℓ , h ) 3 + j , k ∈ Z ℓ , h ∈ Z + 1/3 ∑ ω j − k q Q ( j , k ) s 2 j ℓ + k ℓ + j h + 2 k h t Q ( ℓ , h ) 3 + j , k ∈ Z + 1/3 ℓ , h ∈ Z ∑ ω ℓ − h q Q ( j , k ) s 2 j ℓ + k ℓ + j h + 2 k h t Q ( ℓ , h ) 3 . (6) All powers in this formula are understood through the additive variables τ 1 , τ 0 , τ 2 \tau_1,\tau_0,\tau_2 τ 1 , τ 0 , τ 2 t → 0 t\to0 t → 0 a ( q ) 3 a(q)^3 a ( q ) 3 b ( q ) 3 b(q)^3 b ( q ) 3 c ( q ) 3 c(q)^3 c ( q ) 3
The second application is the following square counterpart of (1.5):
3 n Θ p , q ( n ) ( τ , z ) Θ p , − q ( n ) ( 2 τ , 2 z ) = ∑ u , v ∈ F 3 n ω p ⋅ v + q ⋅ u Θ u , v ( n ) ( τ , C z ) θ u , v ( n ) ( 2 τ ) . (7) 3^n\Theta_{\mathbf{p},\mathbf{q}}^{(n)}(\tau,z)\Theta_{\mathbf{p},-\mathbf{q}}^{(n)}(2\tau,2z)=\sum_{\mathbf{u},\mathbf{v}\in\mathbb{F}_3^n}\omega^{\mathbf{p}\cdot\mathbf{v}+\mathbf{q}\cdot\mathbf{u}}\Theta_{\mathbf{u},\mathbf{v}}^{(n)}(\tau,Cz)\theta_{\mathbf{u},\mathbf{v}}^{(n)}(2\tau). \tag*{(7)} 3 n Θ p , q ( n ) ( τ , z ) Θ p , − q ( n ) ( 2 τ , 2 z ) = u , v ∈ F 3 n ∑ ω p ⋅ v + q ⋅ u Θ u , v ( n ) ( τ , C z ) θ u , v ( n ) ( 2 τ ) . (7) See Theorem 4.1 for the precise statement and proof. For u = ( u 1 , u 2 ) T \mathbf{u}=(u_1,u_2)^T u = ( u 1 , u 2 ) T
u ∗ = ( u 1 − u 2 − u 2 ) . \mathbf{u}^*=\begin{pmatrix}u_1-u_2\\-u_2\end{pmatrix}. u ∗ = ( u 1 − u 2 − u 2 ) . An important consequence of {(1.7)} is the new identity
Θ 1 ( u ; q ) Θ 1 ( 2 u ∗ ; q 2 ) + Θ 3 ( u ; q ) Θ 3 ( 2 u ∗ ; q 2 ) = Θ 2 ( u ; q ) Θ 2 ( 2 u ∗ ; q 2 ) + Θ 4 ( u ; q ) Θ 4 ( 2 u ∗ ; q 2 ) , (8) \Theta_1(\mathbf{u};q)\Theta_1(2\mathbf{u}^*;q^2)+\Theta_3(\mathbf{u};q)\Theta_3(2\mathbf{u}^*;q^2)
=\Theta_2(\mathbf{u};q)\Theta_2(2\mathbf{u}^*;q^2)+\Theta_4(\mathbf{u};q)\Theta_4(2\mathbf{u}^*;q^2), \tag*{(8)} Θ 1 ( u ; q ) Θ 1 ( 2 u ∗ ; q 2 ) + Θ 3 ( u ; q ) Θ 3 ( 2 u ∗ ; q 2 ) = Θ 2 ( u ; q ) Θ 2 ( 2 u ∗ ; q 2 ) + Θ 4 ( u ; q ) Θ 4 ( 2 u ∗ ; q 2 ) , (8) where we adopt Schultz’s notation (2) . Theorem 4.3 proves a more general form of this identity. At u = 0 u=0 u = 0
a ( q ) a ( q 2 ) = b ( q ) b ( q 2 ) + c ( q ) c ( q 2 ) . a(q)a(q^2)=b(q)b(q^2)+c(q)c(q^2). a ( q ) a ( q 2 ) = b ( q ) b ( q 2 ) + c ( q ) c ( q 2 ) . Borwein, Borwein, and Garvan [3 Theorem 2.6(i)] proved this identity. To the best of our knowledge, the identity (8) with two independent Jacobi variables is new.
In degree two, the same seven relation patterns recur for the square counterparts, with each cube θ ( τ ) 3 \theta(\tau)^3 θ ( τ ) 3 θ ( τ ) θ ( 2 τ ) \theta(\tau)\theta(2\tau) θ ( τ ) θ ( 2 τ ) Theorem 4.6 .
The third application concerns root lattices of arbitrary rank. Let ℓ ≥ 2 \ell\ge2 ℓ ≥ 2 d = ℓ − 1 d=\ell-1 d = ℓ − 1 ξ = e ( 1 / ℓ ) \xi=e(1/\ell) ξ = e ( 1/ ℓ ) λ , κ ∈ ( Z / ℓ Z ) n \lambda,\kappa\in(\mathbb{Z}/\ell\mathbb{Z})^n λ , κ ∈ ( Z / ℓ Z ) n λ ⋅ κ ≢ 0 ( m o d ℓ ) \lambda\cdot\kappa\not\equiv0\pmod{\ell} λ ⋅ κ ≡ 0 ( mod ℓ ) M ℓ M_\ell M ℓ A d A_d A d (64) , and let r = ( 1 2 ⋯ d ) T r=(1\ \ 2\ \ \cdots\ \ d)^{\mathsf{T}} r = ( 1 2 ⋯ d ) T
∑ a ∈ ( Z / ℓ Z ) n , c ∈ { 0 , 1 } n b 1 , … , b d − 1 ∈ ( Z / 2 ℓ Z ) n ξ λ ⋅ a + κ ⋅ ( b 1 + ⋯ + b d − 1 + c ) Θ M ℓ , n 2 ℓ [ r a ℓ , B ′ 2 ℓ ] ( τ , z ) = 0 , (9) \sum_{\substack{a\in(\mathbb{Z}/\ell\mathbb{Z})^n,\ c\in\{0,1\}^n\\ b_1,\ldots,b_{d-1}\in(\mathbb{Z}/2\ell\mathbb{Z})^n}}
\xi^{\lambda\cdot a+\kappa\cdot(b_1+\cdots+b_{d-1}+c)}
\Theta_{M_{\ell,n}}^{2\ell}\left[\frac{ra}{\ell},\frac{B'}{2\ell}\right](\tau,z)=0, \tag*{(9)} a ∈ ( Z / ℓ Z ) n , c ∈ { 0 , 1 } n b 1 , … , b d − 1 ∈ ( Z /2 ℓ Z ) n ∑ ξ λ ⋅ a + κ ⋅ ( b 1 + ⋯ + b d − 1 + c ) Θ M ℓ , n 2 ℓ [ ℓ r a , 2 ℓ B ′ ] ( τ , z ) = 0 , (9) where B ′ = ( b 1 T , … , b d − 1 T , c T ) T B'=(b_1^{\mathsf{T},\ldots,b_{d-1}^{\mathsf{T}},c^{\mathsf{T}})^{\mathsf{T}}} B ′ = ( b 1 T , … , b d − 1 T , c T ) T ℓ ≥ 3 \ell\ge3 ℓ ≥ 3 κ \kappa κ 2 Z / ℓ Z 2\mathbb{Z}/\ell\mathbb{Z} 2 Z / ℓ Z ℓ \ell ℓ
∑ a , b 1 , … , b d − 1 ∈ ( Z / ℓ Z ) n ξ λ ⋅ a + κ ⋅ ( b 1 + ⋯ + b d − 1 ) Θ M ℓ , n ℓ [ r a ℓ , B ℓ ] ( τ , z ) = 0 , (10) \sum_{a,b_1,\ldots,b_{d-1}\in(\mathbb{Z}/\ell\mathbb{Z})^n}
\xi^{\lambda\cdot a+\kappa\cdot(b_1+\cdots+b_{d-1})}
\Theta_{M_{\ell,n}}^\ell\left[\frac{ra}{\ell},\frac{B}{\ell}\right](\tau,z)=0, \tag*{(10)} a , b 1 , … , b d − 1 ∈ ( Z / ℓ Z ) n ∑ ξ λ ⋅ a + κ ⋅ ( b 1 + ⋯ + b d − 1 ) Θ M ℓ , n ℓ [ ℓ r a , ℓ B ] ( τ , z ) = 0 , (10) where B = ( b 1 T , … , b d − 1 T , 0 T ) T B=(b_1^{\mathsf{T},\ldots,b_{d-1}^{\mathsf{T}},0^{\mathsf{T}})^{\mathsf{T}}} B = ( b 1 T , … , b d − 1 T , 0 T ) T Theorems 6.1 and 5.3 , respectively.
For every ℓ ≥ 3 \ell\ge3 ℓ ≥ 3 (10) extends the third-power relations on A 2 A_2 A 2 ℓ \ell ℓ A ℓ − 1 A_{\ell-1} A ℓ − 1 (9) produces 2 ℓ 2\ell 2 ℓ ℓ = 2 \ell=2 ℓ = 2
The case ℓ = 2 \ell=2 ℓ = 2 [12 p. 90, Eq. (14)] in degree one and gives analogues involving a Jacobi variable and valid in arbitrary Siegel degree. In degree two it yields five independent relations among the fourth powers of the ten even theta constants; comparison with Cléry, van der Geer, and Grushevsky [5 Section 6, Eq. (6.1)] shows that they form a basis of all linear relations among these fourth powers. Moreover, we classify the degree-one theta constants associated with A 3 A_3 A 3 A 4 A_4 A 4 (10) .
The paper is organized as follows. Section 2 collects the required facts about finite abelian groups and commensurable lattices, and introduces the Siegel–Jacobi theta series used throughout. The final subsection of Section 2 states and proves the main product formula. Sections 3 and 4 treat the cubic identities and their square counterparts, respectively. The same-order identities on root lattices are developed in Section 5, and the doubled-order identities, including the quartic specializations, are considered in Section 6.
Notation
Let R R R n , m n,m n , m R n , m R^{n,m} R n , m n × m n\times m n × m R R R GL n ( R ) \operatorname{GL}_n(R) GL n ( R ) n n n R = Q R=\mathbb{Q} R = Q R = R R=\mathbb{R} R = R GL n + ( R ) \operatorname{GL}_n^+ (R) GL n + ( R ) n n n Sym n , m ( R ) \operatorname{Sym}_{n,m}(R) Sym n , m ( R ) n × n n\times n n × n A A A A T A^{\mathsf{T}} A T A − T A^{-\mathsf{T}} A − T
For A = ( a i j ) ∈ R n , m A=(a_{ij})\in R^{n,m} A = ( a ij ) ∈ R n , m B ∈ R s , t B\in R^{s,t} B ∈ R s , t A ⊗ B A\otimes B A ⊗ B n s × m t ns\times mt n s × m t ( a i j ⋅ B ) 1 ≤ i ≤ n , 1 ≤ j ≤ m (a_{ij}\cdot B)_{1\le i\le n,1\le j\le m} ( a ij ⋅ B ) 1 ≤ i ≤ n , 1 ≤ j ≤ m μ \mu μ B ( μ ) : = I μ ⊗ B B^{(\mu)}:=I_\mu\otimes B B ( μ ) := I μ ⊗ B I μ I_\mu I μ μ \mu μ M 1 , M 2 , … , M k M_1,M_2,\ldots,M_k M 1 , M 2 , … , M k
[ M 1 , M 2 , … , M k ] : = ( M 1 T , M 2 T , … , M k T ) T . [M_1,M_2,\ldots,M_k]:=(M_1^{\mathsf{T}},M_2^{\mathsf{T}},\ldots,M_k^{\mathsf{T}})^{\mathsf{T}}. [ M 1 , M 2 , … , M k ] := ( M 1 T , M 2 T , … , M k T ) T . The symbol e i e_i e i i i i
The Siegel upper half-space of degree n n n
H n : = { u + i v : u ∈ M n , n s y m ( R ) , v ∈ G L n p o s ( R ) } . \mathfrak{H}_n := \{u+iv:u\in M_{n,n}^{\mathrm{sym}}(\mathbb{R}),\,v\in\mathrm{GL}_n^{\mathrm{pos}}(\mathbb{R})\}. H n := { u + i v : u ∈ M n , n sym ( R ) , v ∈ GL n pos ( R )} . We use the notation e ( x ) : = exp ( 2 π i x ) e(x):=\exp(2\pi ix) e ( x ) := exp ( 2 π i x ) q r : = exp ( 2 π i r τ ) q^r:=\exp(2\pi ir\tau) q r := exp ( 2 π i r τ ) τ ∈ H 1 \tau\in\mathfrak{H}_1 τ ∈ H 1 r ∈ R r\in\mathbb{R} r ∈ R τ = ( τ 1 τ 0 τ 0 τ 2 ) ∈ H 2 \tau=\begin{pmatrix}\tau_1&\tau_0\\\tau_0&\tau_2\end{pmatrix}\in\mathfrak{H}_2 τ = ( τ 1 τ 0 τ 0 τ 2 ) ∈ H 2 q r : = exp ( 2 π i r τ 1 ) q^r:=\exp(2\pi ir\tau_1) q r := exp ( 2 π i r τ 1 ) s r : = exp ( 2 π i r τ 0 ) s^r:=\exp(2\pi ir\tau_0) s r := exp ( 2 π i r τ 0 ) t r : = exp ( 2 π i r τ 2 ) t^r:=\exp(2\pi ir\tau_2) t r := exp ( 2 π i r τ 2 ) τ ∈ H 2 \tau\in\mathfrak{H}_2 τ ∈ H 2 ℑ ( τ 1 ) > 0 \Im(\tau_1)>0 ℑ ( τ 1 ) > 0 ℑ ( τ 2 ) > 0 \Im(\tau_2)>0 ℑ ( τ 2 ) > 0 ℑ ( τ 1 ) ℑ ( τ 2 ) − ℑ ( τ 0 ) 2 > 0 \Im(\tau_1)\Im(\tau_2)-\Im(\tau_0)^2>0 ℑ ( τ 1 ) ℑ ( τ 2 ) − ℑ ( τ 0 ) 2 > 0
The generic formula
Prerequisites: groups and lattices
Let G G G G ^ \widehat{G} G G G G C × \mathbb{C}^{\times} C × G ≅ G ^ G\cong\widehat{G} G ≅ G G G G Theorem 2.7 relies on the following lemma of Igusa.
Lemma 2.1. Let L L L L 1 L_1 L 1 L 2 L_2 L 2
Φ : L ⟶ C \Phi:L\longrightarrow\mathbb{C} Φ : L ⟶ C is an L 1 L^1 L 1 ∑ g ∈ L ∣ Φ ( g ) ∣ < + ∞ \sum_{g\in L}|\Phi(g)|<+\infty ∑ g ∈ L ∣Φ ( g ) ∣ < + ∞
[ L : L 1 ] ∑ g ∈ L 1 Φ ( g ) = ∑ χ ∈ L / L 1 , g ∈ L / L 2 ( ∑ h ∈ L 2 χ ( g + h ) Φ ( g + h ) ) . [L:L_1]\sum_{g\in L_1}\Phi(g)=\sum_{\chi\in L/L_1,g\in L/L_2}\left(\sum_{h\in L_2}\chi(g+h)\Phi(g+h)\right). [ L : L 1 ] g ∈ L 1 ∑ Φ ( g ) = χ ∈ L / L 1 , g ∈ L / L 2 ∑ ( h ∈ L 2 ∑ χ ( g + h ) Φ ( g + h ) ) . Here [ L : L 1 ] [L:L_1] [ L : L 1 ] L 1 L_1 L 1 L L L ∣ L / L 1 ∣ |L/L_1| ∣ L / L 1 ∣ [10 p. 136, Lemma 1] proved this lemma.
We shall use this lemma when L L L Z \mathbb{Z} Z L L L V V V R \mathbb{R} R Q \mathbb{Q} Q L L L Z \mathbb{Z} Z V V V
Fix a space V V V L 1 L_1 L 1 L 2 L_2 L 2 V V V [ L 1 : L 1 ∩ L 2 ] [L_1:L_1\cap L_2] [ L 1 : L 1 ∩ L 2 ] [ L 2 : L 1 ∩ L 2 ] [L_2:L_1\cap L_2] [ L 2 : L 1 ∩ L 2 ] V V V m , n m,n m , n Z m , n \mathbb{Z}^{m,n} Z m , n m × n m\times n m × n R m , n \mathbb{R}^{m,n} R m , n R m , n \mathbb{R}^{m,n} R m , n Z m , n \mathbb{Z}^{m,n} Z m , n Q m , n \mathbb{Q}^{m,n} Q m , n Q m , n \mathbb{Q}^{m,n} Q m , n m × m m\times m m × m A A A A ~ \widetilde{A} A Z m , n + A ⋅ Z m , n = A ~ − 1 ⋅ Z m , n \mathbb{Z}^{m,n}+A\cdot\mathbb{Z}^{m,n}=\widetilde{A}^{-1}\cdot\mathbb{Z}^{m,n} Z m , n + A ⋅ Z m , n = A − 1 ⋅ Z m , n
Applying Theorem 2.7 requires an explicit choice of A ~ \widetilde{A} A A A A
Lemma 2.2. Let m , n m,n m , n A ∈ G L m ( Q ) A\in\mathrm{GL}_m(\mathbb{Q}) A ∈ GL m ( Q )
(a) There exist B 1 , B 2 ∈ G L m ( Z ) B_1,B_2\in\mathrm{GL}_m(\mathbb{Z}) B 1 , B 2 ∈ GL m ( Z ) B 1 − 1 A B 2 B_1^{-1}AB_2 B 1 − 1 A B 2
(b) If B 1 − 1 A B 2 = diag ( q 1 p 1 , … , q m p m ) B_1^{-1}AB_2=\operatorname{diag}\left(\frac{q_1}{p_1},\ldots,\frac{q_m}{p_m}\right) B 1 − 1 A B 2 = diag ( p 1 q 1 , … , p m q m ) p i , q i p_i,q_i p i , q i gcd ( q i , p i ) = 1 \gcd(q_i,p_i)=1 g cd( q i , p i ) = 1 Z m , n + A ⋅ Z m , n = A ~ − 1 ⋅ Z m , n \mathbb{Z}^{m,n}+A\cdot\mathbb{Z}^{m,n}=\widetilde{A}^{-1}\cdot\mathbb{Z}^{m,n} Z m , n + A ⋅ Z m , n = A − 1 ⋅ Z m , n A ~ = diag ( p 1 , … , p m ) ⋅ B 1 − 1 ∈ Z m , m \widetilde{A}=\operatorname{diag}(p_1,\ldots,p_m)\cdot B_1^{-1}\in\mathbb{Z}^{m,m} A = diag ( p 1 , … , p m ) ⋅ B 1 − 1 ∈ Z m , m ∣ det ( A ~ ) ∣ = ∏ i = 1 m p i |\det(\widetilde{A})|=\prod_{i=1}^{m}p_i ∣ det ( A ) ∣ = ∏ i = 1 m p i
Proof. For (a), after multiplying A A A A ∈ Z m , m A\in\mathbb{Z}^{m,m} A ∈ Z m , m d d d A A A ( m , m ) (m,m) ( m , m ) d d d ( m , m ) (m,m) ( m , m ) ∣ d ∣ |d| ∣ d ∣ D 1 , D 2 ∈ GL m ( Z ) D_1,D_2 \in\operatorname{GL}_m(\mathbb{Z}) D 1 , D 2 ∈ GL m ( Z ) d d d
D 1 − 1 A D 2 = diag ( A 0 , d ) , A 0 ∈ Z m − 1 , m − 1 . D_1^{-1} A D_2=\operatorname{diag}(A_0,d),\qquad A_0\in\mathbb{Z}^{m-1,m-1}. D 1 − 1 A D 2 = diag ( A 0 , d ) , A 0 ∈ Z m − 1 , m − 1 . Induction on m m m
For (b), B j Z m , n = Z m , n B_j\mathbb{Z}^{m,n}=\mathbb{Z}^{m,n} B j Z m , n = Z m , n j = 1 , 2 j=1,2 j = 1 , 2 B j ∈ GL m ( Z ) B_j\in\operatorname{GL}_m(\mathbb{Z}) B j ∈ GL m ( Z ) D = diag ( q 1 p 1 , … , q m p m ) D=\operatorname{diag}\left(\frac{q_1}{p_1},\ldots,\frac{q_m}{p_m}\right) D = diag ( p 1 q 1 , … , p m q m )
Z m , n + A Z m , n = B 1 Z m , n + B 1 B 1 − 1 A B 2 B 2 − 1 Z m , n = B 1 Z m , n + B 1 D Z m , n = B 1 ( Z m , n + D Z m , n ) = A ~ − 1 Z m , n . \begin{aligned}
\mathbb{Z}^{m,n}+A\mathbb{Z}^{m,n}=B_1\mathbb{Z}^{m,n}+B_1B_1^{-1}AB_2B_2^{-1}\mathbb{Z}^{m,n}=B_1\mathbb{Z}^{m,n}+B_1D\mathbb{Z}^{m,n} \\
&=B_1\left(\mathbb{Z}^{m,n}+D\mathbb{Z}^{m,n}\right)=\widetilde{A}^{-1}\mathbb{Z}^{m,n}.
\end{aligned} Z m , n + A Z m , n = B 1 Z m , n + B 1 B 1 − 1 A B 2 B 2 − 1 Z m , n = B 1 Z m , n + B 1 D Z m , n = B 1 ( Z m , n + D Z m , n ) = A − 1 Z m , n . Finally, ∣ det B 1 ∣ = 1 |\det B_1|=1 ∣ det B 1 ∣ = 1 ∣ det A ~ ∣ = ∏ i = 1 m p i |\det\widetilde{A}|=\prod_{i=1}^{m}p_i ∣ det A ∣ = ∏ i = 1 m p i
Remark 2.3. (1) If A A A Lemma 2.2 (a) may be strengthened by requiring each diagonal entry of B 1 − 1 A B 2 B_1^{-1}AB_2 B 1 − 1 A B 2 A A A [14 ] , it can be computed by calling A.smith_form(integral=True); the transformation matrices B 1 B_1 B 1 B 2 B_2 B 2 A.smith_form?.
(2) If an integral matrix A ~ \widetilde{A} A Z m , n + A ⋅ Z m , n = A ~ − 1 ⋅ Z m , n \mathbb{Z}^{m,n}+A\cdot\mathbb{Z}^{m,n}=\widetilde{A}^{-1}\cdot\mathbb{Z}^{m,n} Z m , n + A ⋅ Z m , n = A − 1 ⋅ Z m , n G A ~ G\widetilde{A} G A A ~ \widetilde{A} A G ∈ GL m ( Z ) G\in\operatorname{GL}_m(\mathbb{Z}) G ∈ GL m ( Z ) ( G A ~ ) − 1 ⋅ Z m , n = A ~ − 1 G − 1 ⋅ Z m , n = A ~ − 1 ⋅ Z m , n (G\widetilde{A})^{-1}\cdot\mathbb{Z}^{m,n}=\widetilde{A}^{-1}G^{-1}\cdot\mathbb{Z}^{m,n}=\widetilde{A}^{-1}\cdot\mathbb{Z}^{m,n} ( G A ) − 1 ⋅ Z m , n = A − 1 G − 1 ⋅ Z m , n = A − 1 ⋅ Z m , n A ~ \widetilde{A} A [14 ] , the Hermite normal form of a matrix B B B B.hermite_form().
Jacobi theta series of arbitrary degree
Let n , m n,m n , m M M M m × m m\times m m × m Q \mathbb{Q} Q a , b ∈ R m , n a,b\in\mathbb{R}^{m,n} a , b ∈ R m , n
Θ M , n [ a , b ] ( τ , z ) : = ∑ g ∈ Z m , n e ( tr ( 1 2 τ ( g + a ) T ⋅ M ⋅ ( g + a ) + ( g + a ) T ⋅ M ⋅ ( z + b ) ) ) . (11) \Theta_{M,n}[a,b](\tau,z):=\sum_{g\in\mathbb{Z}^{m,n}}\mathop{\mathrm{e}}\left(\operatorname{tr}\left(\frac{1}{2}\tau(g+a)^{T}\cdot M\cdot(g+a)+(g+a)^{T}\cdot M\cdot(z+b)\right)\right). \tag*{(11)} Θ M , n [ a , b ] ( τ , z ) := g ∈ Z m , n ∑ e ( tr ( 2 1 τ ( g + a ) T ⋅ M ⋅ ( g + a ) + ( g + a ) T ⋅ M ⋅ ( z + b ) ) ) . (11) where ( τ , z ) ∈ H n × C m , n (\tau,z)\in\mathfrak{H}_n\times\mathbb{C}^{m,n} ( τ , z ) ∈ H n × C m , n 1 H n × C m , n \mathfrak{H}_n\times\mathbb{C}^{m,n} H n × C m , n
θ M , n [ a , b ] ( τ ) : = Θ M , n [ a , b ] ( τ , 0 ) . \theta_{M,n}[a,b](\tau):=\Theta_{M,n}[a,b](\tau,0). θ M , n [ a , b ] ( τ ) := Θ M , n [ a , b ] ( τ , 0 ) . For a , b ∈ Q m , n a,b\in\mathbb{Q}^{m,n} a , b ∈ Q m , n [18 ] and Zhou, Wang, and Lu [16 ] . One aim of this paper is to derive polynomial relations among Θ M , n [ a , b ] ( τ , z ) \Theta_{M,n}[a,b](\tau,z) Θ M , n [ a , b ] ( τ , z ) q q q
Example 2.4. Let m = 2 m=2 m = 2 M = ( 2 − 1 − 1 2 ) M=\begin{pmatrix}2&-1\\-1&2\end{pmatrix} M = ( 2 − 1 − 1 2 ) n = 1 n=1 n = 1
Θ M , n [ ( a 1 , a 2 ) , ( b 1 , b 2 ) ] ( τ , ( z 1 , z 2 ) ) = e ( 2 a 1 b 1 − a 1 b 2 − a 2 b 1 + 2 a 2 b 2 ) \Theta_{M,n}[(a_1,a_2),(b_1,b_2)](\tau,(z_1,z_2))=\mathop{\mathrm{e}}(2a_1b_1-a_1b_2-a_2b_1+2a_2b_2) Θ M , n [( a 1 , a 2 ) , ( b 1 , b 2 )] ( τ , ( z 1 , z 2 )) = e ( 2 a 1 b 1 − a 1 b 2 − a 2 b 1 + 2 a 2 b 2 ) Θ M , n [ ( a 1 , a 2 ) , ( b 1 , b 2 ) ] ( τ , ( z 1 , z 2 ) ) = ⋅ ∑ g 1 , g 2 ∈ Z e ( 2 g 1 b 1 − g 1 b 2 − g 2 b 1 + 2 g 2 b 2 ) q ( g 1 + a 1 ) 2 − ( g 1 + a 1 ) ( g 2 + a 2 ) + ( g 2 + a 2 ) 2 ζ 1 g 1 + a 1 ζ 2 g 2 + a 2 , \mathrel{\phantom{\Theta_{M,n}[(a_1,a_2),(b_1,b_2)](\tau,(z_1,z_2))={}}}\cdot\sum_{g_1,g_2\in\mathbb{Z}}\mathop{\mathrm{e}}(2g_1b_1-g_1b_2-g_2b_1+2g_2b_2)q^{(g_1+a_1)^2-(g_1+a_1)(g_2+a_2)+(g_2+a_2)^2}\zeta_1^{g_1+a_1}\zeta_2^{g_2+a_2}, Θ M , n [( a 1 , a 2 ) , ( b 1 , b 2 )] ( τ , ( z 1 , z 2 )) = ⋅ g 1 , g 2 ∈ Z ∑ e ( 2 g 1 b 1 − g 1 b 2 − g 2 b 1 + 2 g 2 b 2 ) q ( g 1 + a 1 ) 2 − ( g 1 + a 1 ) ( g 2 + a 2 ) + ( g 2 + a 2 ) 2 ζ 1 g 1 + a 1 ζ 2 g 2 + a 2 , where ζ 1 = e ( 2 z 1 − z 2 ) \zeta_1=\mathop{\mathrm{e}}(2z_1-z_2) ζ 1 = e ( 2 z 1 − z 2 ) ζ 2 = e ( − z 1 + 2 z 2 ) \zeta_2=\mathop{\mathrm{e}}(-z_1+2z_2) ζ 2 = e ( − z 1 + 2 z 2 )
θ M , n [ ( 0 , 0 ) , ( 0 , 0 ) ] = a ( q ) , \theta_{M,n}[(0,0),(0,0)]=a(q), θ M , n [( 0 , 0 ) , ( 0 , 0 )] = a ( q ) , θ M , n [ ( 0 , 0 ) , ( 1 / 3 , 1 / 3 ) ] = b ( q ) , \theta_{M,n}[(0,0),(1/3,1/3)]=b(q), θ M , n [( 0 , 0 ) , ( 1/3 , 1/3 )] = b ( q ) , θ M , n [ ( 2 / 3 , 1 / 3 ) , ( 0 , 0 ) ] = c ( q ) . \theta_{M,n}[(2/3,1/3),(0,0)]=c(q). θ M , n [( 2/3 , 1/3 ) , ( 0 , 0 )] = c ( q ) . The Borweins [2 ] and Hirschhorn, Garvan, and Borwein [9 ] studied the functions a ( q ) a(q) a ( q ) b ( q ) b(q) b ( q ) c ( q ) c(q) c ( q ) Θ M , n [ a , b ] ( τ , z ) \Theta_{M,n}[a,b](\tau,z) Θ M , n [ a , b ] ( τ , z )
a ∈ { ( 0 , 0 ) , ( 2 / 3 , 1 / 3 ) , ( 1 / 3 , 2 / 3 ) } , (12) a \in\{(0,0),(2/3,1/3),(1/3,2/3)\}, \tag*{(12)} a ∈ {( 0 , 0 ) , ( 2/3 , 1/3 ) , ( 1/3 , 2/3 )} , (12) b ∈ { ( 0 , 0 ) , ( 1 / 3 , 1 / 3 ) , ( 2 / 3 , 2 / 3 ) } , (13) b \in\{(0,0),(1/3,1/3),(2/3,2/3)\}, \tag*{(13)} b ∈ {( 0 , 0 ) , ( 1/3 , 1/3 ) , ( 2/3 , 2/3 )} , (13) correspond to the nine functions Θ i j ( u ∣ τ ) \Theta_{ij}(u|\tau) Θ ij ( u ∣ τ ) [15 p. 630] . See also Zhu [17 § 7.2] and Chan, Chan, Liu, and Zudilin [4 ] . Zhu [17 Eqs. (7.76) and (7.77)] records the representatives in (12) and (13) .
Example 2.5. For m = 1 m=1 m = 1 M = ( 1 ) M=(1) M = ( 1 ) Θ ( 1 ) , n [ 0 , 0 ] \Theta_{(1),n}[0,0] Θ ( 1 ) , n [ 0 , 0 ] [13 ] .
We conclude this subsection by observing that two characteristic pairs may determine the same theta series up to a constant factor.
Lemma 2.6. Suppose that a 1 , b 1 , a 2 , b 2 ∈ R m , n a_1,b_1,a_2,b_2 \in\mathbb{R}^{m,n} a 1 , b 1 , a 2 , b 2 ∈ R m , n a 2 − a 1 ∈ Z m , n a_2-a_1 \in\mathbb{Z}^{m,n} a 2 − a 1 ∈ Z m , n b 2 − b 1 ∈ M − 1 Z m , n b_2-b_1 \in M^{-1}\mathbb{Z}^{m,n} b 2 − b 1 ∈ M − 1 Z m , n
Θ M , n [ a 2 , b 2 ] = e tr ( a 1 T M ( b 2 − b 1 ) ) Θ M , n [ a 1 , b 1 ] . \Theta_{M,n}[a_2,b_2] = e^{\operatorname{tr}(a_1^T M(b_2-b_1))}\Theta_{M,n}[a_1,b_1]. Θ M , n [ a 2 , b 2 ] = e tr ( a 1 T M ( b 2 − b 1 )) Θ M , n [ a 1 , b 1 ] . In particular, if a 1 ∈ 1 N Z m , n a_1 \in\frac{1}{N}\mathbb{Z}^{m,n} a 1 ∈ N 1 Z m , n N N N
Θ M , n [ a 2 , b 2 ] N = Θ M , n [ a 1 , b 1 ] N . \Theta_{M,n}[a_2,b_2]^N = \Theta_{M,n}[a_1,b_1]^N. Θ M , n [ a 2 , b 2 ] N = Θ M , n [ a 1 , b 1 ] N . Proof. Apply the change of variables g = g ′ + a 1 − a 2 g=g'+a_1-a_2 g = g ′ + a 1 − a 2 Θ M , n [ a 2 , b 2 ] \Theta_{M,n}[a_2,b_2] Θ M , n [ a 2 , b 2 ]
The main theorem
We now state and prove the main theorem.
Theorem 2.7. Let n , m , μ n,m,\mu n , m , μ M 1 , M 2 ∈ G L m pos ( Q ) M_1,M_2 \in\mathrm{GL}^{\operatorname{pos}}_m(\mathbb{Q}) M 1 , M 2 ∈ GL m pos ( Q ) τ ∈ H n \tau\in\mathfrak{H}_n τ ∈ H n a i , b i ∈ R m , n a_i,b_i \in\mathbb{R}^{m,n} a i , b i ∈ R m , n z i ∈ C m , n z_i \in\mathbb{C}^{m,n} z i ∈ C m , n i = 1 , 2 , … , μ i=1,2,\ldots,\mu i = 1 , 2 , … , μ A ∈ G L m μ ( Q ) A \in\mathrm{GL}_{m\mu}(\mathbb{Q}) A ∈ GL m μ ( Q ) A T M 1 ( μ ) A = M 2 ( μ ) A^T M_1^{(\mu)} A=M_2^{(\mu)} A T M 1 ( μ ) A = M 2 ( μ ) A ~ \widetilde{A} A m μ m\mu m μ Z m μ , n + A Z m μ , n = A ~ − 1 Z m μ , n \mathbb{Z}^{m\mu,n}+A\mathbb{Z}^{m\mu,n}=\widetilde{A}^{-1}\mathbb{Z}^{m\mu,n} Z m μ , n + A Z m μ , n = A − 1 Z m μ , n
( a 1 ′ T , … , a μ ′ T ) = ( a 1 T , … , a μ T ) A − T , (a_1^{\prime T},\ldots,a_\mu^{\prime T})=(a_1^T,\ldots,a_\mu^T)A^{-T}, ( a 1 ′ T , … , a μ ′ T ) = ( a 1 T , … , a μ T ) A − T , ( b 1 ′ T , … , b μ ′ T ) = ( b 1 T , … , b μ T ) A − T , (b_1^{\prime T},\ldots,b_\mu^{\prime T})=(b_1^T,\ldots,b_\mu^T)A^{-T}, ( b 1 ′ T , … , b μ ′ T ) = ( b 1 T , … , b μ T ) A − T , ( w 1 T , … , w μ T ) = ( z 1 T , … , z μ T ) A − T . (w_1^T,\ldots,w_\mu^T)=(z_1^T,\ldots,z_\mu^T)A^{-T}. ( w 1 T , … , w μ T ) = ( z 1 T , … , z μ T ) A − T . Then
∣ det A ~ ∣ n ⋅ ∏ i = 1 μ Θ M 1 , n [ a i , b i ] ( τ , z i ) = ∑ α , β e − tr ( ∑ i = 1 μ a i ′ T M 2 β i ) ⋅ ∏ i = 1 μ Θ M 2 , n [ a i ′ + α i , b i ′ + β i ] ( τ , w i ) , (14) |\det\widetilde{A}|^n\cdot\prod_{i=1}^{\mu}\Theta_{M_1,n}[a_i,b_i](\tau,z_i)
=
\sum_{\alpha,\beta}e^{-\operatorname{tr}\left(\sum_{i=1}^{\mu}a_i^{\prime T}M_2\beta_i\right)}\cdot\prod_{i=1}^{\mu}\Theta_{M_2,n}[a_i'+\alpha_i,b_i'+\beta_i](\tau,w_i), \tag*{(14)} ∣ det A ∣ n ⋅ i = 1 ∏ μ Θ M 1 , n [ a i , b i ] ( τ , z i ) = α , β ∑ e − tr ( ∑ i = 1 μ a i ′ T M 2 β i ) ⋅ i = 1 ∏ μ Θ M 2 , n [ a i ′ + α i , b i ′ + β i ] ( τ , w i ) , (14) where α = ( α 1 T , … , α μ T ) \alpha=(\alpha_1^T,\ldots,\alpha_\mu^T) α = ( α 1 T , … , α μ T )
( A ~ A ) − 1 Z m μ , n / Z m μ , n , (15) (\widetilde{A}A)^{-1}\mathbb{Z}^{m\mu,n}/\mathbb{Z}^{m\mu,n}, \tag*{(15)} ( A A ) − 1 Z m μ , n / Z m μ , n , (15) and β = ( β 1 T , … , β μ T ) \beta=(\beta_1^T,\ldots,\beta_\mu^T) β = ( β 1 T , … , β μ T )
A − 1 M 1 ( μ ) − 1 Z m μ , n / A − 1 M 1 ( μ ) − 1 A ~ T Z m μ , n . (16) A^{-1}M_1^{(\mu)-1}\mathbb{Z}^{m\mu,n}/A^{-1}M_1^{(\mu)-1}\widetilde{A}^T\mathbb{Z}^{m\mu,n}. \tag*{(16)} A − 1 M 1 ( μ ) − 1 Z m μ , n / A − 1 M 1 ( μ ) − 1 A T Z m μ , n . (16) We first prove a preliminary version. Let n , m , μ , M 1 , M 2 , a i , b i , z i n,m,\mu,M_1,M_2,a_i,b_i,z_i n , m , μ , M 1 , M 2 , a i , b i , z i Theorem 2.7 . For v 1 , … , v μ ∈ C m , n v_1,\ldots,v_\mu\in\mathbb{C}^{m,n} v 1 , … , v μ ∈ C m , n
[ v 1 , … , v μ ] : = ( v 1 T , … , v μ T ) T ∈ C m μ , n (17) [v_1,\ldots,v_\mu]:=(v_1^T,\ldots,v_\mu^T)^T\in\mathbb{C}^{m\mu,n} \tag*{(17)} [ v 1 , … , v μ ] := ( v 1 T , … , v μ T ) T ∈ C m μ , n (17) for convenience. For j = 1 , 2 j=1,2 j = 1 , 2
B j : Q m μ , n × Q m μ , n → Q n , n (18) B_j:\mathbb{Q}^{m\mu,n}\times\mathbb{Q}^{m\mu,n}\to\mathbb{Q}^{n,n} \tag*{(18)} B j : Q m μ , n × Q m μ , n → Q n , n (18) ( [ v 1 , … , v μ ] , [ v 1 ′ , … , v μ ′ ] ) ↦ ∑ i = 1 μ v i T M j v i ′ . ([v_1,\ldots,v_\mu],[v'_1,\ldots,v'_\mu])\mapsto\sum_{i=1}^{\mu}v_i^T M_jv'_i. ([ v 1 , … , v μ ] , [ v 1 ′ , … , v μ ′ ]) ↦ i = 1 ∑ μ v i T M j v i ′ . where v i , v i ′ ∈ Q m , n v_i,v'_i\in\mathbb{Q}^{m,n} v i , v i ′ ∈ Q m , n B j ( v , v ′ ) = v T M j ( μ ) v ′ B_j(v,v')=v^T M_j^{(\mu)}v' B j ( v , v ′ ) = v T M j ( μ ) v ′ σ \sigma σ Q \mathbb{Q} Q Q m μ , n \mathbb{Q}^{m\mu,n} Q m μ , n 2
B 1 ( σ ( v ) , σ ( v ′ ) ) = B 2 ( v , v ′ ) (19) B_1(\sigma(v),\sigma(v'))=B_2(v,v') \tag*{(19)} B 1 ( σ ( v ) , σ ( v ′ )) = B 2 ( v , v ′ ) (19) for every v , v ′ ∈ Q m μ , n v,v'\in\mathbb{Q}^{m\mu,n} v , v ′ ∈ Q m μ , n B j B_j B j C m μ , n × C m μ , n \mathbb{C}^{m\mu,n}\times\mathbb{C}^{m\mu,n} C m μ , n × C m μ , n σ \sigma σ C \mathbb{C} C C m μ , n \mathbb{C}^{m\mu,n} C m μ , n (19) remains valid for v , v ′ ∈ C m μ , n v,v'\in\mathbb{C}^{m\mu,n} v , v ′ ∈ C m μ , n
Define
[ a 1 ′ , … , a μ ′ ] = σ − 1 [ a 1 , … , a μ ] , (20) [a'_1,\ldots,a'_\mu]=\sigma^{-1}[a_1,\ldots,a_\mu], \tag*{(20)} [ a 1 ′ , … , a μ ′ ] = σ − 1 [ a 1 , … , a μ ] , (20) [ b 1 ′ , … , b μ ′ ] = σ − 1 [ b 1 , … , b μ ] , (21) [b'_1,\ldots,b'_\mu]=\sigma^{-1}[b_1,\ldots,b_\mu], \tag*{(21)} [ b 1 ′ , … , b μ ′ ] = σ − 1 [ b 1 , … , b μ ] , (21) [ w 1 , … , w μ ] = σ − 1 [ z 1 , … , z μ ] . (22) [w_1,\ldots,w_\mu]=\sigma^{-1}[z_1,\ldots,z_\mu]. \tag*{(22)} [ w 1 , … , w μ ] = σ − 1 [ z 1 , … , z μ ] . (22) Let
L 1 = Z m μ , n , L 2 = σ ( Z m μ , n ) , L = L 1 + L 2 . (23) L_1=\mathbb{Z}^{m\mu,n},\qquad L_2=\sigma(\mathbb{Z}^{m\mu,n}),\qquad L=L_1+L_2. \tag*{(23)} L 1 = Z m μ , n , L 2 = σ ( Z m μ , n ) , L = L 1 + L 2 . (23) These are pairwise commensurable lattices in Q m μ , n \mathbb{Q}^{m\mu,n} Q m μ , n
Lemma 2.8. If σ \sigma σ
∣ L / L 1 ∣ ⋅ ∏ i = 1 μ Θ M 1 , n [ a i , b i ] ( τ , z i ) = ∑ χ ∈ L / L 1 , g ∈ σ − 1 ( L ) / L 1 χ ( σ ( g ) ) ∑ h 1 , … , h μ ∈ Z m , n χ ( σ [ h 1 , … , h μ ] ) ∏ i = 1 μ e ( tr ( 1 2 τ v i T M 2 v i + v i T M 2 ( w i + b i ′ ) ) ) , (24) \lvert L/L_1\rvert\cdot\prod_{i=1}^{\mu}\Theta_{M_{1,n}}[a_i,b_i](\tau,z_i)
=
\sum_{\chi\in L/L_1,\;g\in\sigma^{-1}(L)/L_1}\chi(\sigma(g))
\sum_{h_1,\ldots,h_\mu\in\mathbb{Z}^{m,n}}\chi(\sigma[h_1,\ldots,h_\mu])
\prod_{i=1}^{\mu}e\left(\operatorname{tr}\left(\frac{1}{2}\tau v_i^T M_2v_i+v_i^T M_2(w_i+b'_i)\right)\right), \tag*{(24)} ∣ L / L 1 ∣ ⋅ i = 1 ∏ μ Θ M 1 , n [ a i , b i ] ( τ , z i ) = χ ∈ L / L 1 , g ∈ σ − 1 ( L ) / L 1 ∑ χ ( σ ( g )) h 1 , … , h μ ∈ Z m , n ∑ χ ( σ [ h 1 , … , h μ ]) i = 1 ∏ μ e ( tr ( 2 1 τ v i T M 2 v i + v i T M 2 ( w i + b i ′ ) ) ) , (24) where v i : = h i + g i + a i ′ v_i:=h_i+g_i+a'_i v i := h i + g i + a i ′ g i ∈ Q m , n g_i\in\mathbb{Q}^{m,n} g i ∈ Q m , n [ g 1 , … , g μ ] : = g [g_1,\ldots,g_\mu]:=g [ g 1 , … , g μ ] := g
Proof. For g = [ g 1 , … , g μ ] ∈ L g=[g_1,\ldots,g_\mu]\in L g = [ g 1 , … , g μ ] ∈ L
Φ ( g ) = ∏ i = 1 μ e ( tr ( 1 2 τ ( g i + a i ) T M 1 ( g i + a i ) + ( g i + a i ) T M 1 ( z i + b i ) ) ) . \Phi(g)=\prod_{i=1}^{\mu}e\left(\operatorname{tr}\left(\frac{1}{2}\tau(g_i+a_i)^T M_1(g_i+a_i)+(g_i+a_i)^T M_1(z_i+b_i)\right)\right). Φ ( g ) = i = 1 ∏ μ e ( tr ( 2 1 τ ( g i + a i ) T M 1 ( g i + a i ) + ( g i + a i ) T M 1 ( z i + b i ) ) ) . Since L / L 1 L/L_1 L / L 1 L 1 L_1 L 1 Φ ∈ L 1 ( L ) \Phi\in L^1(L) Φ ∈ L 1 ( L )
∣ L / L 1 ∣ ∑ g ∈ L 1 Φ ( g ) = ∑ χ ∈ L / L 1 , g ∈ L / L 2 ( ∑ h ∈ L 2 χ ( g + h ) Φ ( g + h ) ) . (25) \lvert L/L_1\rvert\sum_{g\in L_1}\Phi(g)
=
\sum_{\chi\in L/L_1,\;g\in L/L_2}\left(\sum_{h\in L_2}\chi(g+h)\Phi(g+h)\right). \tag*{(25)} ∣ L / L 1 ∣ g ∈ L 1 ∑ Φ ( g ) = χ ∈ L / L 1 , g ∈ L / L 2 ∑ ( h ∈ L 2 ∑ χ ( g + h ) Φ ( g + h ) ) . (25) Here and below, sums indexed by quotient groups are taken over arbitrary complete sets of representatives, and characters are evaluated on the corresponding cosets. The map L / L 2 → σ − 1 ( L ) / L 1 L/L_2 \to\sigma^{-1}(L)/L_1 L / L 2 → σ − 1 ( L ) / L 1 g + L 2 ↦ σ − 1 ( g ) + L 1 g+L_2 \mapsto\sigma^{-1}(g)+L_1 g + L 2 ↦ σ − 1 ( g ) + L 1
∑ χ ∈ L / L 1 ^ , g ∈ L / L 2 ( ∑ h ∈ L 2 χ ( g + h ) Φ ( g + h ) ) = ∑ χ ∈ L / L 1 ^ , g ∈ σ − 1 ( L ) / L 1 χ ( σ ( g ) ) ∑ h 1 , … , h μ ∈ Z m , n χ ( σ [ h 1 , … , h μ ] ) Φ ( σ ( g + h ) ) . (26) \sum_{\substack{\chi\in\widehat{L/L_1},\,g\in L/L_2}}
\left(\sum_{h\in L_2}\chi(g+h)\Phi(g+h)\right)
=
\sum_{\substack{\chi\in\widehat{L/L_1},\,g\in\sigma^{-1}(L)/L_1}}
\chi(\sigma(g))\sum_{h_1,\ldots,h_\mu\in\mathbb{Z}^{m,n}}
\chi(\sigma[h_1,\ldots,h_\mu])\Phi(\sigma(g+h)). \tag*{(26)} χ ∈ L / L 1 , g ∈ L / L 2 ∑ ( h ∈ L 2 ∑ χ ( g + h ) Φ ( g + h ) ) = χ ∈ L / L 1 , g ∈ σ − 1 ( L ) / L 1 ∑ χ ( σ ( g )) h 1 , … , h μ ∈ Z m , n ∑ χ ( σ [ h 1 , … , h μ ]) Φ ( σ ( g + h )) . (26) Here h = [ h 1 , … , h μ ] h=[h_1,\ldots,h_\mu] h = [ h 1 , … , h μ ] (19) –(22) ,
Φ ( σ ( g + h ) ) = ∏ i = 1 μ e ( tr ( 1 2 v i T M 2 v i + v i T M 2 ( w i + b i ′ ) ) ) . \Phi(\sigma(g+h))=\prod_{i=1}^{\mu}e\left(\operatorname{tr}\left(\frac{1}{2}v_i^{T}M_2v_i+v_i^{T}M_2(w_i+b_i^{\prime})\right)\right). Φ ( σ ( g + h )) = i = 1 ∏ μ e ( tr ( 2 1 v i T M 2 v i + v i T M 2 ( w i + b i ′ ) ) ) . Substitution into (26) gives
(2.17) RHS of (25) = RHS of (24) .
Absolute convergence also gives
(2.18) LHS of (25) = LHS of (24) .
Equations (25) , (2.17), and (2.18) now imply (24) . □
Theorem 2.7 follows by applying the preceding lemma to a specific σ \sigma σ
Lemma 2.9. Suppose A A A m μ m\mu m μ A T ⋅ M 1 ( μ ) ⋅ A = M 2 ( μ ) A^{T}\cdot M_1^{(\mu)}\cdot A=M_2^{(\mu)} A T ⋅ M 1 ( μ ) ⋅ A = M 2 ( μ ) σ ( v ) = A v \sigma(v)=Av σ ( v ) = A v (19) , and (24) holds for this σ \sigma σ
Proof. For arbitrary v , v ′ ∈ Q m μ , n v,v'\in\mathbb{Q}^{m\mu,n} v , v ′ ∈ Q m μ , n
B 1 ( σ ( v ) , σ ( v ′ ) ) = ( A v ) T M 1 ( μ ) ( A v ′ ) = v T M 2 ( μ ) v ′ = B 2 ( v , v ′ ) . B_1(\sigma(v),\sigma(v'))=(Av)^TM_1^{(\mu)}(Av')=v^TM_2^{(\mu)}v'=B_2(v,v'). B 1 ( σ ( v ) , σ ( v ′ )) = ( A v ) T M 1 ( μ ) ( A v ′ ) = v T M 2 ( μ ) v ′ = B 2 ( v , v ′ ) . Thus (19) holds, and Lemma 2.8 2.8 2.8 (24) . □
The following lemma gives explicit expressions for χ ( σ ( g ) ) \chi(\sigma(g)) χ ( σ ( g )) χ ( σ ( h ) ) \chi(\sigma(h)) χ ( σ ( h )) (24) .
Lemma 2.10. Under the assumptions and with the notation of Theorem 2.7, the map
M 2 ( μ ) − 1 A ~ − T Z m μ , n / M 2 ( μ ) − 1 Z m μ , n ⟶ A ~ − 1 Z m μ , n / Z m μ , n M_2^{(\mu)-1}\widetilde A^{-T}\mathbb{Z}^{m\mu,n}/M_2^{(\mu)-1}\mathbb{Z}^{m\mu,n}
\longrightarrow
\widetilde A^{-1}\mathbb{Z}^{m\mu,n}/\mathbb{Z}^{m\mu,n} M 2 ( μ ) − 1 A − T Z m μ , n / M 2 ( μ ) − 1 Z m μ , n ⟶ A − 1 Z m μ , n / Z m μ , n b ‾ + M 2 ( μ ) − 1 Z m μ , n ⟼ ( v ‾ + Z m μ , n ↦ e ( tr ( b T M 2 ( μ ) A ~ v ) ) ) (27) \overline b+M_2^{(\mu)-1}\mathbb{Z}^{m\mu,n}
\longmapsto
\left(\overline v+\mathbb{Z}^{m\mu,n}\mapsto e\left(\operatorname{tr}(b^TM_2^{(\mu)}\widetilde A v)\right)\right) \tag*{(27)} b + M 2 ( μ ) − 1 Z m μ , n ⟼ ( v + Z m μ , n ↦ e ( tr ( b T M 2 ( μ ) A v ) ) ) (27) is a group isomorphism.
Proof. Because A ~ \widetilde A A (27) are well-defined finite abelian groups. Write b ‾ : = b + M 2 ( μ ) − 1 Z m μ , n \overline b:=b+M_2^{(\mu)-1}\mathbb{Z}^{m\mu,n} b := b + M 2 ( μ ) − 1 Z m μ , n v ‾ : = v + Z m μ , n \overline v:=v+\mathbb{Z}^{m\mu,n} v := v + Z m μ , n
χ b ‾ ( v ‾ ) : = e ( tr ( b T M 2 ( μ ) A ~ v ) ) . (28) \chi_{\overline b}(\overline v):=e\left(\operatorname{tr}(b^TM_2^{(\mu)}\widetilde A v)\right). \tag*{(28)} χ b ( v ) := e ( tr ( b T M 2 ( μ ) A v ) ) . (28) For b ∈ M 2 ( μ ) − 1 A ~ − T Z m μ , n b\in M_2^{(\mu)-1}\widetilde A^{-T}\mathbb{Z}^{m\mu,n} b ∈ M 2 ( μ ) − 1 A − T Z m μ , n v 1 , v 2 ∈ A ~ − 1 Z m μ , n v_1,v_2\in\widetilde A^{-1}\mathbb{Z}^{m\mu,n} v 1 , v 2 ∈ A − 1 Z m μ , n v ‾ 1 = v ‾ 2 \overline v_1=\overline v_2 v 1 = v 2 χ b ‾ ( v ‾ 1 ) = χ b ‾ ( v ‾ 2 ) \chi_{\overline b}(\overline v_1)=\chi_{\overline b}(\overline v_2) χ b ( v 1 ) = χ b ( v 2 ) b T M 2 ( μ ) A ~ ( v 1 − v 2 ) b^TM_2^{(\mu)}\widetilde A(v_1-v_2) b T M 2 ( μ ) A ( v 1 − v 2 ) χ b ‾ \chi_{\overline b} χ b A ~ − 1 Z m μ , n / Z m μ , n \widetilde A^{-1}\mathbb{Z}^{m\mu,n}/\mathbb{Z}^{m\mu,n} A − 1 Z m μ , n / Z m μ , n b ‾ \overline b b χ b ‾ ( v ‾ 1 + v ‾ 2 ) = χ b ‾ ( v ‾ 1 ) ⋅ χ b ‾ ( v ‾ 2 ) \chi_{\overline b}(\overline v_1+\overline v_2)=\chi_{\overline b}(\overline v_1)\cdot\chi_{\overline b}(\overline v_2) χ b ( v 1 + v 2 ) = χ b ( v 1 ) ⋅ χ b ( v 2 ) χ b ‾ ( 0 ) = 1 \chi_{\overline b}(0)=1 χ b ( 0 ) = 1 χ b ‾ ∈ A ~ − 1 Z m μ , n / Z m μ , n \chi_{\overline b}\in\widetilde A^{-1}\mathbb{Z}^{m\mu,n}/\mathbb{Z}^{m\mu,n} χ b ∈ A − 1 Z m μ , n / Z m μ , n χ b ‾ 1 + b ‾ 2 = χ b ‾ 1 χ b ‾ 2 \chi_{\overline b_1+\overline b_2}=\chi_{\overline b_1}\chi_{\overline b_2} χ b 1 + b 2 = χ b 1 χ b 2 χ 0 ( v ‾ ) = 1 \chi_0(\overline v)=1 χ 0 ( v ) = 1 (27) is a group homomorphism. To prove injectivity, suppose that χ b ‾ ( v ‾ ) = 1 \chi_{\overline b}(\overline v)=1 χ b ( v ) = 1 v ∈ A ~ − 1 Z m μ , n v\in\widetilde A^{-1}\mathbb{Z}^{m\mu,n} v ∈ A − 1 Z m μ , n tr ( b T M 2 ( μ ) x ) ∈ Z \operatorname{tr}(b^TM_2^{(\mu)}x)\in\mathbb{Z} tr ( b T M 2 ( μ ) x ) ∈ Z x ∈ Z m μ , n x\in\mathbb{Z}^{m\mu,n} x ∈ Z m μ , n x x x b T M 2 ( μ ) ∈ Z n , m μ b^TM_2^{(\mu)}\in\mathbb{Z}^{n,m\mu} b T M 2 ( μ ) ∈ Z n , m μ b ∈ M 2 ( μ ) − 1 Z m μ , n b\in M_2^{(\mu)-1}\mathbb{Z}^{m\mu,n} b ∈ M 2 ( μ ) − 1 Z m μ , n b ‾ = 0 \overline b=0 b = 0 ∣ det ( A ~ ) ∣ n |\det(\widetilde A)|^n ∣ det ( A ) ∣ n
We now prove the main theorem.
Proof of Theorem 2.7. By Lemma 2.9, (24) holds for σ ( v ) = A v \sigma(v)=Av σ ( v ) = A v (23) and the definition of A ~ \widetilde{A} A L / L 1 = A ~ − 1 Z m μ , n / Z m μ , n L/L_1=\widetilde{A}^{-1}\mathbb{Z}^{m\mu,n}/\mathbb{Z}^{m\mu,n} L / L 1 = A − 1 Z m μ , n / Z m μ , n ∣ L / L 1 ∣ = ∣ det ( A ~ ) ∣ n |L/L_1|=|\det(\widetilde{A})|^n ∣ L / L 1 ∣ = ∣ det ( A ) ∣ n
L H S o f ( 2.14 ) = ∣ det A ~ ∣ n ⋅ ∏ i = 1 μ Θ M 1 , n [ a i , b i ] ( τ , z i ) . (29) \mathrm{LHS\ of\ }(2.14)=|\det\widetilde{A}|^n\cdot\prod_{i=1}^{\mu}\Theta_{M_{1,n}[a_i,b_i]}(\tau,z_i). \tag*{(29)} LHS of ( 2.14 ) = ∣ det A ∣ n ⋅ i = 1 ∏ μ Θ M 1 , n [ a i , b i ] ( τ , z i ) . (29) Let χ b ‾ \chi_{\overline{b}} χ b (28) . Lemma 2.10 allows us to replace the sum ∑ χ ∈ L / L 1 f ( χ ) \sum_{\chi\in L/L_1}f(\chi) ∑ χ ∈ L / L 1 f ( χ ) ∑ b ‾ ∈ M 2 ( μ ) − 1 A ~ − T Z m μ , n / M 2 ( μ ) − 1 Z m μ , n f ( χ b ‾ ) \sum_{\overline{b}\in M_2^{(\mu)-1}\widetilde{A}^{-T}\mathbb{Z}^{m\mu,n}/M_2^{(\mu)-1}\mathbb{Z}^{m\mu,n}}f(\chi_{\overline{b}}) ∑ b ∈ M 2 ( μ ) − 1 A − T Z m μ , n / M 2 ( μ ) − 1 Z m μ , n f ( χ b )
A − 1 M 1 ( μ ) − 1 Z m μ , n / A − 1 M 1 ( μ ) − 1 A ~ T Z m μ , n ⟶ M 2 ( μ ) − 1 A ~ − T Z m μ , n / M 2 ( μ ) − 1 Z m μ , n A^{-1}M_1^{(\mu)-1}\mathbb{Z}^{m\mu,n}/A^{-1}M_1^{(\mu)-1}\widetilde{A}^{T}\mathbb{Z}^{m\mu,n}\longrightarrow M_2^{(\mu)-1}\widetilde{A}^{-T}\mathbb{Z}^{m\mu,n}/M_2^{(\mu)-1}\mathbb{Z}^{m\mu,n} A − 1 M 1 ( μ ) − 1 Z m μ , n / A − 1 M 1 ( μ ) − 1 A T Z m μ , n ⟶ M 2 ( μ ) − 1 A − T Z m μ , n / M 2 ( μ ) − 1 Z m μ , n β + A − 1 M 1 ( μ ) − 1 A ~ T Z m μ , n ⟼ M 2 ( μ ) − 1 A ~ − T M 1 ( μ ) A β + M 2 ( μ ) − 1 Z m μ , n \beta+A^{-1}M_1^{(\mu)-1}\widetilde{A}^{T}\mathbb{Z}^{m\mu,n}\longmapsto M_2^{(\mu)-1}\widetilde{A}^{-T}M_1^{(\mu)}A\beta+M_2^{(\mu)-1}\mathbb{Z}^{m\mu,n} β + A − 1 M 1 ( μ ) − 1 A T Z m μ , n ⟼ M 2 ( μ ) − 1 A − T M 1 ( μ ) A β + M 2 ( μ ) − 1 Z m μ , n is a group isomorphism, so the sum may be written as
∑ β ∈ A − 1 M 1 ( μ ) − 1 Z m μ , n / A − 1 M 1 ( μ ) − 1 A ~ T Z m μ , n f ( χ M 2 ( μ ) − 1 A ~ − T M 1 ( μ ) A β ) . \sum_{\beta\in A^{-1}M_1^{(\mu)-1}\mathbb{Z}^{m\mu,n}/A^{-1}M_1^{(\mu)-1}\widetilde{A}^{T}\mathbb{Z}^{m\mu,n}}
f\left(\chi_{M_2^{(\mu)-1}\widetilde{A}^{-T}M_1^{(\mu)}A\beta}\right). β ∈ A − 1 M 1 ( μ ) − 1 Z m μ , n / A − 1 M 1 ( μ ) − 1 A T Z m μ , n ∑ f ( χ M 2 ( μ ) − 1 A − T M 1 ( μ ) A β ) . Moreover, σ − 1 ( L ) / L 1 = ( A ~ A ) − 1 Z m μ , n / Z m μ , n \sigma^{-1}(L)/L_1=(\widetilde{A}A)^{-1}\mathbb{Z}^{m\mu,n}/\mathbb{Z}^{m\mu,n} σ − 1 ( L ) / L 1 = ( A A ) − 1 Z m μ , n / Z m μ , n (24) becomes
∑ α , β χ M 2 ( μ ) − 1 A ~ − T M 1 ( μ ) A β ( A α ‾ ) ∑ h ∈ Z m μ , n χ M 2 ( μ ) − 1 A ~ − T M 1 ( μ ) A β ( A h ‾ ) ⋅ ∏ i = 1 μ e ( tr ( 1 2 v i T M 2 v i + v i T M 2 ( w i + b i ′ ) ) ) . (30) \sum_{\alpha,\beta}\chi_{M_2^{(\mu)-1}\widetilde{A}^{-T}M_1^{(\mu)}A\beta}(\overline{A\alpha})
\sum_{h\in\mathbb{Z}^{m\mu,n}}\chi_{M_2^{(\mu)-1}\widetilde{A}^{-T}M_1^{(\mu)}A\beta}(\overline{Ah})
\cdot\prod_{i=1}^{\mu}e\left(\operatorname{tr}\left(\frac{1}{2}v_i^TM_2v_i+v_i^TM_2(w_i+b_i')\right)\right). \tag*{(30)} α , β ∑ χ M 2 ( μ ) − 1 A − T M 1 ( μ ) A β ( A α ) h ∈ Z m μ , n ∑ χ M 2 ( μ ) − 1 A − T M 1 ( μ ) A β ( A h ) ⋅ i = 1 ∏ μ e ( tr ( 2 1 v i T M 2 v i + v i T M 2 ( w i + b i ′ ) ) ) . (30) where v i : = h i + α i + a i ′ v_i:=h_i+\alpha_i+a_i' v i := h i + α i + a i ′ [ h 1 , … , h μ ] = h [h_1,\ldots,h_\mu]=h [ h 1 , … , h μ ] = h [ α 1 , … , α μ ] = α [\alpha_1,\ldots,\alpha_\mu]=\alpha [ α 1 , … , α μ ] = α α \alpha α β \beta β (15) and (16) , respectively. Equation (28) gives
χ M 2 ( μ ) − 1 A ~ − T M 1 ( μ ) A β ( A α ‾ ) = e ( tr ( β T A T M 1 ( μ ) A α ) ) = e ( tr ( β T M 2 ( μ ) α ) ) . \chi_{M_2^{(\mu)-1}\widetilde{A}^{-T}M_1^{(\mu)}A\beta}(\overline{A\alpha})=e\left(\operatorname{tr}(\beta^TA^TM_1^{(\mu)}A\alpha)\right)=e\left(\operatorname{tr}(\beta^TM_2^{(\mu)}\alpha)\right). χ M 2 ( μ ) − 1 A − T M 1 ( μ ) A β ( A α ) = e ( tr ( β T A T M 1 ( μ ) A α ) ) = e ( tr ( β T M 2 ( μ ) α ) ) . The analogous formula holds for χ M 2 ( μ ) − 1 A ~ − T M 1 ( μ ) A β ( A h ‾ ) \chi_{M_2^{(\mu)-1}\widetilde{A}^{-T}M_1^{(\mu)}A\beta}(\overline{Ah}) χ M 2 ( μ ) − 1 A − T M 1 ( μ ) A β ( A h ) (30) shows that the right-hand side of (24) equals
∑ α , β e ( − tr ( β T M 2 ( μ ) a ′ ) ) ∑ h ∈ Z m μ , n e ( tr ( β T M 2 ( μ ) ( h + α + a ′ ) ) ) ⋅ ∏ i = 1 μ e ( tr ( 1 2 v i T M 2 v i + v i T M 2 ( w i + b i ′ ) ) ) . \sum_{\alpha,\beta}e\left(-\operatorname{tr}(\beta^TM_2^{(\mu)}a')\right)
\sum_{h\in\mathbb{Z}^{m\mu,n}}e\left(\operatorname{tr}(\beta^TM_2^{(\mu)}(h+\alpha+a'))\right)
\cdot\prod_{i=1}^{\mu}e\left(\operatorname{tr}\left(\frac{1}{2}v_i^TM_2v_i+v_i^TM_2(w_i+b_i')\right)\right). α , β ∑ e ( − tr ( β T M 2 ( μ ) a ′ ) ) h ∈ Z m μ , n ∑ e ( tr ( β T M 2 ( μ ) ( h + α + a ′ )) ) ⋅ i = 1 ∏ μ e ( tr ( 2 1 v i T M 2 v i + v i T M 2 ( w i + b i ′ ) ) ) . where a ′ = [ a 1 ′ , … , a μ ′ ] a'=[a_1',\ldots,a_\mu'] a ′ = [ a 1 ′ , … , a μ ′ ] Θ M 2 , n \Theta_{M_2,n} Θ M 2 , n
∑ α , β e ( − tr ( ∑ i = 1 μ a i ′ T M 2 β i ) ) ⋅ ∏ i = 1 μ Θ M 2 , n [ a i ′ + α i , b i ′ + β i ] ( τ , w i ) . \sum_{\alpha,\beta}e\left(-\operatorname{tr}\left(\sum_{i=1}^{\mu}a_i'^TM_2\beta_i\right)\right)\cdot\prod_{i=1}^{\mu}\Theta_{M_2,n}[a_i'+\alpha_i,b_i'+\beta_i](\tau,w_i). α , β ∑ e ( − tr ( i = 1 ∑ μ a i ′ T M 2 β i ) ) ⋅ i = 1 ∏ μ Θ M 2 , n [ a i ′ + α i , b i ′ + β i ] ( τ , w i ) . Together with (29) and (24) , this proves the theorem. □
Remark 2.11. (1) For given M 1 M_1 M 1 M 2 M_2 M 2 A A A A ~ \widetilde{A} A n n n m m m m μ m\mu m μ α \alpha α β \beta β A ~ \widetilde{A} A
(2) The quotient group (15) is well-defined because the assumption Z m μ , n + A Z m μ , n = A ~ − 1 Z m μ , n \mathbb{Z}^{m\mu,n}+A\mathbb{Z}^{m\mu,n}=\widetilde{A}^{-1}\mathbb{Z}^{m\mu,n} Z m μ , n + A Z m μ , n = A − 1 Z m μ , n A Z m μ , n ⊆ A ~ − 1 Z m μ , n A\mathbb{Z}^{m\mu,n}\subseteq\widetilde{A}^{-1}\mathbb{Z}^{m\mu,n} A Z m μ , n ⊆ A − 1 Z m μ , n Z m μ , n ⊆ ( A ~ A ) − 1 Z m μ , n \mathbb{Z}^{m\mu,n}\subseteq(\widetilde{A}A)^{-1}\mathbb{Z}^{m\mu,n} Z m μ , n ⊆ ( A A ) − 1 Z m μ , n (14) from the representatives of α \alpha α β \beta β α ‾ = α + Z m μ , n ∈ ( A ~ A ) − 1 Z m μ , n / Z m μ , n \overline{\alpha}=\alpha+\mathbb{Z}^{m\mu,n}\in(\widetilde{A}A)^{-1}\mathbb{Z}^{m\mu,n}/\mathbb{Z}^{m\mu,n} α = α + Z m μ , n ∈ ( A A ) − 1 Z m μ , n / Z m μ , n β ‾ = β + A − 1 M 1 ( μ ) − 1 A ~ T Z m μ , n ∈ A − 1 M 1 ( μ ) − 1 Z m μ , n / A − 1 M 1 ( μ ) − 1 A ~ T Z m μ , n \overline{\beta}=\beta+A^{-1}M_{1}^{(\mu)-1}\widetilde{A}^{T}\mathbb{Z}^{m\mu,n}\in A^{-1}M_{1}^{(\mu)-1}\mathbb{Z}^{m\mu,n}/A^{-1}M_{1}^{(\mu)-1}\widetilde{A}^{T}\mathbb{Z}^{m\mu,n} β = β + A − 1 M 1 ( μ ) − 1 A T Z m μ , n ∈ A − 1 M 1 ( μ ) − 1 Z m μ , n / A − 1 M 1 ( μ ) − 1 A T Z m μ , n
exp ( − tr ( ∑ i = 1 μ a i T M 2 β i ) ) ⋅ ∏ i = 1 μ Θ M 2 , n [ a i ′ + α i , b i ′ + β i ] ( τ , w i ) \exp\left(-\operatorname{tr}\left(\sum_{i=1}^{\mu}a_i^{T}M_2\beta_i\right)\right)\cdot\prod_{i=1}^{\mu}\Theta_{M_2,n}[a_i'+\alpha_i,b_i'+\beta_i](\tau,w_i) exp ( − tr ( i = 1 ∑ μ a i T M 2 β i ) ) ⋅ i = 1 ∏ μ Θ M 2 , n [ a i ′ + α i , b i ′ + β i ] ( τ , w i ) is independent of these choices. To prove this, let α ′ ∈ ( A ~ A ) − 1 Z m μ , n \alpha'\in(\widetilde{A}A)^{-1}\mathbb{Z}^{m\mu,n} α ′ ∈ ( A A ) − 1 Z m μ , n β ′ ∈ A − 1 M 1 ( μ ) − 1 Z m μ , n \beta'\in A^{-1}M_{1}^{(\mu)-1}\mathbb{Z}^{m\mu,n} β ′ ∈ A − 1 M 1 ( μ ) − 1 Z m μ , n α ‾ = α ′ ‾ \overline{\alpha}=\overline{\alpha'} α = α ′ β ‾ = β ′ ‾ \overline{\beta}=\overline{\beta'} β = β ′ Δ α i = α i ′ − α i \Delta\alpha_i=\alpha_i'-\alpha_i Δ α i = α i ′ − α i Δ β i = β i ′ − β i \Delta\beta_i=\beta_i'-\beta_i Δ β i = β i ′ − β i
Δ β = β ′ − β ∈ A − 1 M 1 ( μ ) − 1 A ~ T Z m μ , n = M 2 ( μ ) − 1 ( A ~ A ) T Z m μ , n ⊆ M 2 ( μ ) − 1 Z m μ , n , \Delta\beta=\beta'-\beta\in A^{-1}M_{1}^{(\mu)-1}\widetilde{A}^{T}\mathbb{Z}^{m\mu,n}=M_{2}^{(\mu)-1}(\widetilde{A}A)^{T}\mathbb{Z}^{m\mu,n}\subseteq M_{2}^{(\mu)-1}\mathbb{Z}^{m\mu,n}, Δ β = β ′ − β ∈ A − 1 M 1 ( μ ) − 1 A T Z m μ , n = M 2 ( μ ) − 1 ( A A ) T Z m μ , n ⊆ M 2 ( μ ) − 1 Z m μ , n , and hence Δ β i ∈ M 2 − 1 Z m , n \Delta\beta_i\in M_2^{-1}\mathbb{Z}^{m,n} Δ β i ∈ M 2 − 1 Z m , n Δ α i ∈ Z m , n \Delta\alpha_i\in\mathbb{Z}^{m,n} Δ α i ∈ Z m , n Lemma 2.6 therefore gives
exp ( − tr ( ∑ i = 1 μ a i ′ T M 2 β i ′ ) ) ⋅ ∏ i = 1 μ Θ M 2 , n [ a i ′ + α i ′ , b i ′ + β i ′ ] ( τ , w i ) = exp ( − tr ( ∑ i = 1 μ a i ′ T M 2 β i ′ ) ) exp ( tr ( ∑ i = 1 μ ( a i ′ + α i ) T M 2 Δ β i ) ) ⋅ ∏ i = 1 μ Θ M 2 , n [ a i ′ + α i , b i ′ + β i ] ( τ , w i ) = exp ( tr ( ∑ i = 1 μ α i T M 2 Δ β i ) ) exp ( − tr ( ∑ i = 1 μ a i ′ T M 2 β i ) ) ⋅ ∏ i = 1 μ Θ M 2 , n [ a i ′ + α i , b i ′ + β i ] ( τ , w i ) . \begin{aligned}
\exp\left(-\operatorname{tr}\left(\sum_{i=1}^{\mu}a_i^{\prime T}M_2\beta_i'\right)\right)\cdot\prod_{i=1}^{\mu}\Theta_{M_2,n}[a_i'+\alpha_i',b_i'+\beta_i'](\tau,w_i) \\
&=\exp\left(-\operatorname{tr}\left(\sum_{i=1}^{\mu}a_i^{\prime T}M_2\beta_i'\right)\right)\exp\left(\operatorname{tr}\left(\sum_{i=1}^{\mu}(a_i'+\alpha_i)^{T}M_2\Delta\beta_i\right)\right)\cdot\prod_{i=1}^{\mu}\Theta_{M_2,n}[a_i'+\alpha_i,b_i'+\beta_i](\tau,w_i) \\
&=\exp\left(\operatorname{tr}\left(\sum_{i=1}^{\mu}\alpha_i^{T}M_2\Delta\beta_i\right)\right)\exp\left(-\operatorname{tr}\left(\sum_{i=1}^{\mu}a_i^{\prime T}M_2\beta_i\right)\right)\cdot\prod_{i=1}^{\mu}\Theta_{M_2,n}[a_i'+\alpha_i,b_i'+\beta_i](\tau,w_i).
\end{aligned} exp ( − tr ( i = 1 ∑ μ a i ′ T M 2 β i ′ ) ) ⋅ i = 1 ∏ μ Θ M 2 , n [ a i ′ + α i ′ , b i ′ + β i ′ ] ( τ , w i ) = exp ( − tr ( i = 1 ∑ μ a i ′ T M 2 β i ′ ) ) exp ( tr ( i = 1 ∑ μ ( a i ′ + α i ) T M 2 Δ β i ) ) ⋅ i = 1 ∏ μ Θ M 2 , n [ a i ′ + α i , b i ′ + β i ] ( τ , w i ) = exp ( tr ( i = 1 ∑ μ α i T M 2 Δ β i ) ) exp ( − tr ( i = 1 ∑ μ a i ′ T M 2 β i ) ) ⋅ i = 1 ∏ μ Θ M 2 , n [ a i ′ + α i , b i ′ + β i ] ( τ , w i ) . Finally, α T ∈ Z n , m μ ⋅ ( A ~ A ) − T \alpha^{T}\in\mathbb{Z}^{n,m\mu}\cdot(\widetilde{A}A)^{-T} α T ∈ Z n , m μ ⋅ ( A A ) − T M 2 ( μ ) Δ β ∈ ( A ~ A ) T Z m μ , n M_2^{(\mu)}\Delta\beta\in(\widetilde{A}A)^{T}\mathbb{Z}^{m\mu,n} M 2 ( μ ) Δ β ∈ ( A A ) T Z m μ , n
exp ( tr ( ∑ i = 1 μ α i T M 2 Δ β i ) ) = exp ( tr ( α T M 2 ( μ ) Δ β ) ) = 1. \exp\left(\operatorname{tr}\left(\sum_{i=1}^{\mu}\alpha_i^{T}M_2\Delta\beta_i\right)\right)=\exp\left(\operatorname{tr}(\alpha^{T}M_2^{(\mu)}\Delta\beta)\right)=1. exp ( tr ( i = 1 ∑ μ α i T M 2 Δ β i ) ) = exp ( tr ( α T M 2 ( μ ) Δ β ) ) = 1. Riemann’s bilinear and quadrilinear relations
Specializing Theorem 2.7 to m = 1 m=1 m = 1 M 1 = ( 1 ) M_1=(1) M 1 = ( 1 ) Θ ( 1 ) , n [ 0 , 0 ] ( τ , z ) \Theta_{(1),n}[0,0](\tau,z) Θ ( 1 ) , n [ 0 , 0 ] ( τ , z ) ( τ , z ) ∈ H n × C 1 , n (\tau,z)\in\mathfrak{H}_n\times\mathbb{C}^{1,n} ( τ , z ) ∈ H n × C 1 , n n n n Θ [ a , b ] ( τ , z ) : = Θ ( 1 ) , n [ a , b ] ( τ , z ) \Theta[a,b](\tau,z):=\Theta_{(1),n}[a,b](\tau,z) Θ [ a , b ] ( τ , z ) := Θ ( 1 ) , n [ a , b ] ( τ , z )
Theorem 2.12 (Riemann’s bilinear relation). Let n ≥ 1 n\ge1 n ≥ 1 τ ∈ H n \tau\in\mathfrak{H}_n τ ∈ H n z 1 , z 2 ∈ C 1 , n z_1,z_2\in\mathbb{C}^{1,n} z 1 , z 2 ∈ C 1 , n a 1 , b 1 , a 2 , b 2 ∈ R 1 , n a_1,b_1,a_2,b_2\in\mathbb{R}^{1,n} a 1 , b 1 , a 2 , b 2 ∈ R 1 , n
2 n Θ [ a 1 , b 1 ] ( τ , z 1 ) ⋅ Θ [ a 2 , b 2 ] ( τ , z 2 ) = ∑ b ∈ 1 2 Z 1 , n / Z 1 , n exp ( − 2 b ⋅ a 1 T ) ⋅ Θ [ a 1 + a 2 , b 1 + b 2 2 + b ] ( τ 2 , z 1 + z 2 2 ) ⋅ Θ [ a 1 − a 2 , b 1 − b 2 2 + b ] ( τ 2 , z 1 − z 2 2 ) . 2^n\Theta[a_1,b_1](\tau,z_1)\cdot\Theta[a_2,b_2](\tau,z_2)=\sum_{b\in\frac{1}{2}\mathbb{Z}^{1,n}/\mathbb{Z}^{1,n}}\exp(-2b\cdot a_1^{T})\cdot\Theta\left[a_1+a_2,\frac{b_1+b_2}{2}+b\right]\left(\frac{\tau}{2},\frac{z_1+z_2}{2}\right)\cdot\Theta\left[a_1-a_2,\frac{b_1-b_2}{2}+b\right]\left(\frac{\tau}{2},\frac{z_1-z_2}{2}\right). 2 n Θ [ a 1 , b 1 ] ( τ , z 1 ) ⋅ Θ [ a 2 , b 2 ] ( τ , z 2 ) = b ∈ 2 1 Z 1 , n / Z 1 , n ∑ exp ( − 2 b ⋅ a 1 T ) ⋅ Θ [ a 1 + a 2 , 2 b 1 + b 2 + b ] ( 2 τ , 2 z 1 + z 2 ) ⋅ Θ [ a 1 − a 2 , 2 b 1 − b 2 + b ] ( 2 τ , 2 z 1 − z 2 ) . Proof. In Theorem 2.7 , take m = 1 m=1 m = 1 μ = 2 \mu=2 μ = 2 M 1 = ( 1 ) M_1=(1) M 1 = ( 1 ) M 2 = ( 1 / 2 ) M_2=(1/2) M 2 = ( 1/2 ) A = ( 1 / 2 1 / 2 1 / 2 − 1 / 2 ) A=\begin{pmatrix}1/2&1/2\\1/2&-1/2\end{pmatrix} A = ( 1/2 1/2 1/2 − 1/2 ) A T ⋅ M 1 ( μ ) ⋅ A = M 2 ( μ ) A^{T}\cdot M_1^{(\mu)}\cdot A=M_2^{(\mu)} A T ⋅ M 1 ( μ ) ⋅ A = M 2 ( μ ) Z m μ , n + A Z m μ , n = A ~ − 1 Z m μ , n \mathbb{Z}^{m\mu,n}+A\mathbb{Z}^{m\mu,n}=\widetilde{A}^{-1}\mathbb{Z}^{m\mu,n} Z m μ , n + A Z m μ , n = A − 1 Z m μ , n A ~ = A − 1 \widetilde{A}=A^{-1} A = A − 1 (14) becomes
2 n Θ [ a 1 , b 1 ] ( τ , z 1 ) ⋅ Θ [ a 2 , b 2 ] ( τ , z 2 ) = ∑ β ∈ A − 1 Z 2 , n / A − 2 Z 2 , n ϵ ( − 1 2 ( β 1 ⋅ a 1 T + β 2 ⋅ a 2 T ) ) ⋅ Θ ( 1 / 2 ) , n [ a 1 ′ , b 1 ′ + β 1 ] ( τ , w 1 ) ⋅ Θ ( 1 / 2 ) , n [ a 2 ′ , b 2 ′ + β 2 ] ( τ , w 2 ) , (31) 2^n\Theta[a_1,b_1](\tau,z_1)\cdot\Theta[a_2,b_2](\tau,z_2)=\sum_{\beta\in A^{-1}\mathbb{Z}^{2,n}/A^{-2}\mathbb{Z}^{2,n}}\epsilon\left(-\frac{1}{2}(\beta_1\cdot a_1^T+\beta_2\cdot a_2^T)\right)\cdot\Theta_{(1/2),n}[a'_1,b'_1+\beta_1](\tau,w_1)\cdot\Theta_{(1/2),n}[a'_2,b'_2+\beta_2](\tau,w_2), \tag*{(31)} 2 n Θ [ a 1 , b 1 ] ( τ , z 1 ) ⋅ Θ [ a 2 , b 2 ] ( τ , z 2 ) = β ∈ A − 1 Z 2 , n / A − 2 Z 2 , n ∑ ϵ ( − 2 1 ( β 1 ⋅ a 1 T + β 2 ⋅ a 2 T ) ) ⋅ Θ ( 1/2 ) , n [ a 1 ′ , b 1 ′ + β 1 ] ( τ , w 1 ) ⋅ Θ ( 1/2 ) , n [ a 2 ′ , b 2 ′ + β 2 ] ( τ , w 2 ) , (31) where
( a 1 ′ a 2 ′ ) = A − 1 ( a 1 a 2 ) = ( a 1 + a 2 a 1 − a 2 ) , ( b 1 ′ b 2 ′ ) = ( b 1 + b 2 b 1 − b 2 ) , ( w 1 w 2 ) = ( z 1 + z 2 z 1 − z 2 ) . \begin{pmatrix}a'_1\\a'_2\end{pmatrix}=A^{-1}\begin{pmatrix}a_1\\a_2\end{pmatrix}=\begin{pmatrix}a_1+a_2\\a_1-a_2\end{pmatrix},\qquad
\begin{pmatrix}b'_1\\b'_2\end{pmatrix}=\begin{pmatrix}b_1+b_2\\b_1-b_2\end{pmatrix},\qquad
\begin{pmatrix}w_1\\w_2\end{pmatrix}=\begin{pmatrix}z_1+z_2\\z_1-z_2\end{pmatrix}. ( a 1 ′ a 2 ′ ) = A − 1 ( a 1 a 2 ) = ( a 1 + a 2 a 1 − a 2 ) , ( b 1 ′ b 2 ′ ) = ( b 1 + b 2 b 1 − b 2 ) , ( w 1 w 2 ) = ( z 1 + z 2 z 1 − z 2 ) . The map
1 2 Z 1 , n / Z 1 , n ⟶ A − 1 Z 2 , n / A − 2 Z 2 , n , b + Z 1 , n ⟼ ( 2 b 2 b ) + A − 2 Z 2 , n \frac{1}{2}\mathbb{Z}^{1,n}/\mathbb{Z}^{1,n}\longrightarrow A^{-1}\mathbb{Z}^{2,n}/A^{-2}\mathbb{Z}^{2,n},\qquad b+\mathbb{Z}^{1,n}\longmapsto\begin{pmatrix}2b\\2b\end{pmatrix}+A^{-2}\mathbb{Z}^{2,n} 2 1 Z 1 , n / Z 1 , n ⟶ A − 1 Z 2 , n / A − 2 Z 2 , n , b + Z 1 , n ⟼ ( 2 b 2 b ) + A − 2 Z 2 , n is a group monomorphism. Since ∣ 1 2 Z 1 , n / Z 1 , n ∣ = 2 n = ∣ A − 1 Z 2 , n / A − 2 Z 2 , n ∣ \left|\frac{1}{2}\mathbb{Z}^{1,n}/\mathbb{Z}^{1,n}\right|=2^n=\left|A^{-1}\mathbb{Z}^{2,n}/A^{-2}\mathbb{Z}^{2,n}\right| 2 1 Z 1 , n / Z 1 , n = 2 n = A − 1 Z 2 , n / A − 2 Z 2 , n (31) and use
Θ ( 1 / 2 ) , n [ a 1 ′ , b 1 ′ + 2 b ] ( τ , w 1 ) = Θ [ a 1 + a 2 , ( b 1 + b 2 ) / 2 + b ] ( τ 2 , z 1 + z 2 2 ) , \Theta_{(1/2),n}[a'_1,b'_1+2b](\tau,w_1)=\Theta[a_1+a_2,(b_1+b_2)/2+b]\left(\frac{\tau}{2},\frac{z_1+z_2}{2}\right), Θ ( 1/2 ) , n [ a 1 ′ , b 1 ′ + 2 b ] ( τ , w 1 ) = Θ [ a 1 + a 2 , ( b 1 + b 2 ) /2 + b ] ( 2 τ , 2 z 1 + z 2 ) , Θ ( 1 / 2 ) , n [ a 2 ′ , b 2 ′ + 2 b ] ( τ , w 2 ) = Θ [ a 1 − a 2 , ( b 1 − b 2 ) / 2 + b ] ( τ 2 , z 1 − z 2 2 ) , \Theta_{(1/2),n}[a'_2,b'_2+2b](\tau,w_2)=\Theta[a_1-a_2,(b_1-b_2)/2+b]\left(\frac{\tau}{2},\frac{z_1-z_2}{2}\right), Θ ( 1/2 ) , n [ a 2 ′ , b 2 ′ + 2 b ] ( τ , w 2 ) = Θ [ a 1 − a 2 , ( b 1 − b 2 ) /2 + b ] ( 2 τ , 2 z 1 − z 2 ) , to obtain the stated formula. □
Theorem 2.13 (Riemann’s quadrilinear relation). Let n ≥ 1 n\ge1 n ≥ 1 τ ∈ H n \tau\in\mathfrak{H}_n τ ∈ H n z i ∈ C 1 , n z_i\in\mathbb{C}^{1,n} z i ∈ C 1 , n a i , b i ∈ R 1 , n a_i,b_i\in\mathbb{R}^{1,n} a i , b i ∈ R 1 , n i = 1 , 2 , 3 , 4 i=1,2,3,4 i = 1 , 2 , 3 , 4
2 n ∏ i = 1 4 Θ [ a i , b i ] ( τ , z i ) = ∑ a , b ∈ 1 2 Z 1 , n / Z 1 , n ϵ ( − 2 b ⋅ a 1 T ) ∏ i = 1 4 Θ [ a i ′ + a , b i ′ + b ] ( τ , w i ) , (32) 2^n\prod_{i=1}^{4}\Theta[a_i,b_i](\tau,z_i)=\sum_{a,b\in\frac{1}{2}\mathbb{Z}^{1,n}/\mathbb{Z}^{1,n}}\epsilon(-2b\cdot a_1^T)\prod_{i=1}^{4}\Theta[a'_i+a,b'_i+b](\tau,w_i), \tag*{(32)} 2 n i = 1 ∏ 4 Θ [ a i , b i ] ( τ , z i ) = a , b ∈ 2 1 Z 1 , n / Z 1 , n ∑ ϵ ( − 2 b ⋅ a 1 T ) i = 1 ∏ 4 Θ [ a i ′ + a , b i ′ + b ] ( τ , w i ) , (32) where
( a 1 ′ T , a 2 ′ T , a 3 ′ T , a 4 ′ T ) = ( a 1 T , a 2 T , a 3 T , a 4 T ) U , (a_1^{\prime T},a_2^{\prime T},a_3^{\prime T},a_4^{\prime T})=(a_1^T,a_2^T,a_3^T,a_4^T)U, ( a 1 ′ T , a 2 ′ T , a 3 ′ T , a 4 ′ T ) = ( a 1 T , a 2 T , a 3 T , a 4 T ) U , ( b 1 ′ T , b 2 ′ T , b 3 ′ T , b 4 ′ T ) = ( b 1 T , b 2 T , b 3 T , b 4 T ) U , (b_1^{\prime T},b_2^{\prime T},b_3^{\prime T},b_4^{\prime T})=(b_1^T,b_2^T,b_3^T,b_4^T)U, ( b 1 ′ T , b 2 ′ T , b 3 ′ T , b 4 ′ T ) = ( b 1 T , b 2 T , b 3 T , b 4 T ) U , ( w 1 T , w 2 T , w 3 T , w 4 T ) = ( z 1 T , z 2 T , z 3 T , z 4 T ) U , (w_1^T,w_2^T,w_3^T,w_4^T)=(z_1^T,z_2^T,z_3^T,z_4^T)U, ( w 1 T , w 2 T , w 3 T , w 4 T ) = ( z 1 T , z 2 T , z 3 T , z 4 T ) U , and
U = 1 2 ( 1 1 1 1 1 1 − 1 − 1 1 − 1 1 − 1 1 − 1 − 1 1 ) . U=\frac{1}{2}\begin{pmatrix}
1&1&1&1\\
1&1&-1&-1\\
1&-1&1&-1\\
1&-1&-1&1
\end{pmatrix}. U = 2 1 1 1 1 1 1 1 − 1 − 1 1 − 1 1 − 1 1 − 1 − 1 1 . Proof. The proof is similar to that of the preceding theorem, so we give only a sketch. In Theorem 2.7 , take m = 1 m=1 m = 1 μ = 4 \mu=4 μ = 4 M 1 = M 2 = ( 1 ) M_1=M_2=(1) M 1 = M 2 = ( 1 ) A = U − 1 = U A=U^{-1}=U A = U − 1 = U Lemma 2.2 and Remark 2.3 give
A ~ = ( 1 0 0 1 0 1 0 1 0 0 1 1 0 0 0 2 ) . \widetilde{A}=\begin{pmatrix}
1&0&0&1\\
0&1&0&1\\
0&0&1&1\\
0&0&0&2
\end{pmatrix}. A = 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1 2 . All hypotheses of Theorem 2.7 are therefore satisfied. The representative sets needed to reduce (14) to (32) are given by the isomorphisms
1 2 Z 1 , n / Z 1 , n ⟶ ( A ~ A ) − 1 Z 4 , n / Z 4 , n \frac{1}{2}\mathbb{Z}^{1,n}/\mathbb{Z}^{1,n}\longrightarrow(\widetilde{A}A)^{-1}\mathbb{Z}^{4,n}/\mathbb{Z}^{4,n} 2 1 Z 1 , n / Z 1 , n ⟶ ( A A ) − 1 Z 4 , n / Z 4 , n a + Z 1 , n ⟼ [ a , a , a , a ] + Z 4 , n a+\mathbb{Z}^{1,n}\longmapsto[a,a,a,a]+\mathbb{Z}^{4,n} a + Z 1 , n ⟼ [ a , a , a , a ] + Z 4 , n and
1 2 Z 1 , n / Z 1 , n ⟶ A − 1 Z 4 , n / A − 1 A ~ T Z 4 , n \frac{1}{2}\mathbb{Z}^{1,n}/\mathbb{Z}^{1,n}\longrightarrow A^{-1}\mathbb{Z}^{4,n}/A^{-1}\widetilde{A}^{T}\mathbb{Z}^{4,n} 2 1 Z 1 , n / Z 1 , n ⟶ A − 1 Z 4 , n / A − 1 A T Z 4 , n b + Z 1 , n ⟼ [ b , b , b , b ] + A − 1 A ~ T Z 4 , n . b+\mathbb{Z}^{1,n}\longmapsto[b,b,b,b]+A^{-1}\widetilde{A}^{T}\mathbb{Z}^{4,n}. b + Z 1 , n ⟼ [ b , b , b , b ] + A − 1 A T Z 4 , n . For these representatives, ∑ i = 1 4 a i ′ = 2 a 1 \sum_{i=1}^{4}a_i'=2a_1 ∑ i = 1 4 a i ′ = 2 a 1 (14) is ( − 2 b ⋅ a 1 T ) (-2b\cdot a_1^T) ( − 2 b ⋅ a 1 T ) (14) gives (32) .
Cubic theta identities
The cubic Fourier identity
We begin with a direct specialization of Theorem 2.7 . Throughout this section, we use
M = ( 2 − 1 − 1 2 ) , ρ = 1 3 ( 1 − 1 ) , σ = 1 3 ( 1 1 ) , ω = e ( 1 / 3 ) . (33) M=\begin{pmatrix}2&-1\\-1&2\end{pmatrix},\qquad\rho=\frac{1}{3}\begin{pmatrix}1\\-1\end{pmatrix},\qquad\sigma=\frac{1}{3}\begin{pmatrix}1\\1\end{pmatrix},\qquad\omega=e(1/3). \tag*{(33)} M = ( 2 − 1 − 1 2 ) , ρ = 3 1 ( 1 − 1 ) , σ = 3 1 ( 1 1 ) , ω = e ( 1/3 ) . (33) Recall that F 3 = Z / 3 Z \mathbb{F}_3=\mathbb{Z}/3\mathbb{Z} F 3 = Z /3 Z p , q ∈ F 3 n \mathbf{p},\mathbf{q}\in\mathbb{F}_3^n p , q ∈ F 3 n
Θ p , q ( n ) ( τ , z ) : = Θ M , n [ ρ p , σ q ] ( τ , z ) , θ p , q ( n ) ( τ ) : = Θ p , q ( n ) ( τ , 0 ) . (34) \Theta_{\mathbf{p},\mathbf{q}}^{(n)}(\tau,z):=\Theta_{M,n}[\rho\mathbf{p},\sigma\mathbf{q}](\tau,z),\qquad\theta_{\mathbf{p},\mathbf{q}}^{(n)}(\tau):=\Theta_{\mathbf{p},\mathbf{q}}^{(n)}(\tau,0). \tag*{(34)} Θ p , q ( n ) ( τ , z ) := Θ M , n [ ρ p , σ q ] ( τ , z ) , θ p , q ( n ) ( τ ) := Θ p , q ( n ) ( τ , 0 ) . (34) Lemma 2.6 shows that Θ p , q ( n ) \Theta_{\mathbf{p},\mathbf{q}}^{(n)} Θ p , q ( n ) p \mathbf{p} p q \mathbf{q} q 3 3 3 p ⋅ v = p v T \mathbf{p}\cdot\mathbf{v}=\mathbf{p}\mathbf{v}^{T} p ⋅ v = p v T
Theorem 3.1. Let n ≥ 1 n\geq1 n ≥ 1 τ ∈ H n \tau\in\mathfrak{H}_{n} τ ∈ H n z ∈ C 2 n z\in\mathbb{C}^{2n} z ∈ C 2 n C = ( 1 1 − 1 2 ) C=\begin{pmatrix}1&1\\-1&2\end{pmatrix} C = ( 1 − 1 1 2 ) p , q ∈ F 3 n \mathbf{p},\mathbf{q}\in\mathbb{F}_3^n p , q ∈ F 3 n
3 n Θ p , q ( n ) ( τ , z ) 3 = ∑ u , v ∈ F 3 n ω p ⋅ v + q ⋅ u Θ u , v ( n ) ( τ , C z ) θ u , v ( n ) ( τ ) 2 . (35) 3^n\Theta_{\mathbf{p},\mathbf{q}}^{(n)}(\tau,z)^3=\sum_{\mathbf{u},\mathbf{v}\in\mathbb{F}_3^n}\omega^{\mathbf{p}\cdot\mathbf{v}+\mathbf{q}\cdot\mathbf{u}}\Theta_{\mathbf{u},\mathbf{v}}^{(n)}(\tau,Cz)\theta_{\mathbf{u},\mathbf{v}}^{(n)}(\tau)^2. \tag*{(35)} 3 n Θ p , q ( n ) ( τ , z ) 3 = u , v ∈ F 3 n ∑ ω p ⋅ v + q ⋅ u Θ u , v ( n ) ( τ , C z ) θ u , v ( n ) ( τ ) 2 . (35) In particular,
3 n θ p , q ( n ) ( τ ) 3 = ∑ u , v ∈ F 3 n ω p ⋅ v + q ⋅ u θ u , v ( n ) ( τ ) 3 . (36) 3^n\theta_{\mathbf{p},\mathbf{q}}^{(n)}(\tau)^3=\sum_{\mathbf{u},\mathbf{v}\in\mathbb{F}_3^n}\omega^{\mathbf{p}\cdot\mathbf{v}+\mathbf{q}\cdot\mathbf{u}}\theta_{\mathbf{u},\mathbf{v}}^{(n)}(\tau)^3. \tag*{(36)} 3 n θ p , q ( n ) ( τ ) 3 = u , v ∈ F 3 n ∑ ω p ⋅ v + q ⋅ u θ u , v ( n ) ( τ ) 3 . (36) Proof. We apply Theorem 2.7 with m = 2 m=2 m = 2 μ = 3 \mu=3 μ = 3 M 1 = M 2 = M M_1=M_2=M M 1 = M 2 = M
A = 1 3 ( 2 − 1 2 − 1 2 − 1 1 1 1 1 1 1 2 − 1 − 1 − 1 − 1 2 1 1 1 − 2 − 2 1 2 − 1 − 1 2 − 1 − 1 1 1 − 2 1 1 − 2 ) , A=\frac{1}{3}\begin{pmatrix}
2&-1&2&-1&2&-1\\
1&1&1&1&1&1\\
2&-1&-1&-1&-1&2\\
1&1&1&-2&-2&1\\
2&-1&-1&2&-1&-1\\
1&1&-2&1&1&-2
\end{pmatrix}, A = 3 1 2 1 2 1 2 1 − 1 1 − 1 1 − 1 1 2 1 − 1 1 − 1 − 2 − 1 1 − 1 − 2 2 1 2 1 − 1 − 2 − 1 1 − 1 1 2 1 − 1 − 2 , and
A ~ = ( 1 0 0 0 0 − 2 0 1 0 0 0 − 1 0 0 1 0 0 − 2 0 0 0 1 0 − 1 0 0 0 0 1 − 2 0 0 0 0 0 3 ) . \widetilde{A}=\begin{pmatrix}
1&0&0&0&0&-2\\
0&1&0&0&0&-1\\
0&0&1&0&0&-2\\
0&0&0&1&0&-1\\
0&0&0&0&1&-2\\
0&0&0&0&0&3
\end{pmatrix}. A = 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 − 2 − 1 − 2 − 1 − 2 3 . Direct multiplication gives
A T M ( 3 ) A = M ( 3 ) , det A = 1 , det A ~ = 3. (37) A^{T}M^{(3)}A=M^{(3)},\qquad\det A=1,\qquad\det\widetilde{A}=3. \tag*{(37)} A T M ( 3 ) A = M ( 3 ) , det A = 1 , det A = 3. (37) To verify the lattice condition, let
u 0 = ( 2 , 1 , 2 , 1 , 2 , 1 ) T ∈ Z 6 , 1 . u_0=(2,1,2,1,2,1)^T\in\mathbb{Z}^{6,1}. u 0 = ( 2 , 1 , 2 , 1 , 2 , 1 ) T ∈ Z 6 , 1 . Every column of 3 A 3A 3 A u 0 u_0 u 0 3 Z 6 , 1 3\mathbb{Z}^{6,1} 3 Z 6 , 1 u 0 u_0 u 0
Z 6 , n + A Z 6 , n = Z 6 , n + 1 3 u 0 Z 1 , n . \mathbb{Z}^{6,n}+A\mathbb{Z}^{6,n}=\mathbb{Z}^{6,n}+\frac{1}{3}u_0\mathbb{Z}^{1,n}. Z 6 , n + A Z 6 , n = Z 6 , n + 3 1 u 0 Z 1 , n . The first five columns of A ~ − 1 \widetilde{A}^{-1} A − 1 e 1 , … , e 5 e_1,\ldots,e_5 e 1 , … , e 5 u 0 / 3 u_0/3 u 0 /3 e 6 = u 0 − 2 e 1 − e 2 − 2 e 3 − e 4 − 2 e 5 e_6=u_0-2e_1-e_2-2e_3-e_4-2e_5 e 6 = u 0 − 2 e 1 − e 2 − 2 e 3 − e 4 − 2 e 5 Z 6 , 1 + 1 3 Z u 0 \mathbb{Z}^{6,1}+\frac{1}{3}\mathbb{Z}u_0 Z 6 , 1 + 3 1 Z u 0
Z 6 , n + A Z 6 , n = A ~ − 1 Z 6 , n . (38) \mathbb{Z}^{6,n}+A\mathbb{Z}^{6,n}=\widetilde{A}^{-1}\mathbb{Z}^{6,n}. \tag*{(38)} Z 6 , n + A Z 6 , n = A − 1 Z 6 , n . (38) Thus all the hypotheses of Theorem 2.7 are satisfied.
We specialize the characteristics and Jacobi variables as follows:
z 1 = z 2 = z 3 = z , a 1 = a 2 = a 3 = ρ p , b 1 = b 2 = b 3 = σ q . z_1=z_2=z_3=z,\qquad a_1=a_2=a_3=\rho p,\qquad b_1=b_2=b_3=\sigma q. z 1 = z 2 = z 3 = z , a 1 = a 2 = a 3 = ρp , b 1 = b 2 = b 3 = σ q . The left-hand side of (14) is
∣ det A ~ ∣ ∏ i = 1 3 Θ M , n [ a i , b i ] ( τ , z i ) = 3 n Θ p , q ( n ) ( τ , z ) 3 . (39) \lvert\det\widetilde{A}\rvert\prod_{i=1}^{3}\Theta_{M,n}[a_i,b_i](\tau,z_i)=3^n\Theta_{p,q}^{(n)}(\tau,z)^3. \tag*{(39)} ∣ det A ∣ i = 1 ∏ 3 Θ M , n [ a i , b i ] ( τ , z i ) = 3 n Θ p , q ( n ) ( τ , z ) 3 . (39) It remains to identify the representatives in (15) and (16) . For this purpose, we write
ρ ~ = [ ρ , ρ , ρ ] , σ ~ = [ σ , σ , σ ] . \widetilde{\rho}=[\rho,\rho,\rho],\qquad\widetilde{\sigma}=[\sigma,\sigma,\sigma]. ρ = [ ρ , ρ , ρ ] , σ = [ σ , σ , σ ] . The notation is that of (17) . By (38) , A ~ A \widetilde{A}A A A (37) gives ∣ det ( A ~ A ) ∣ = 3 \lvert\det(\widetilde{A}A)\rvert=3 ∣ det ( A A )∣ = 3 A ρ ~ = e 1 A\widetilde{\rho}=e_1 A ρ = e 1 A ~ A ρ ~ = e 1 \widetilde{A}A\widetilde{\rho}=e_1 A A ρ = e 1 ρ ~ ∉ Z 6 , 1 \widetilde{\rho}\notin\mathbb{Z}^{6,1} ρ ∈ / Z 6 , 1 ρ ~ \widetilde{\rho} ρ ( A ~ A ) − 1 Z 6 , 1 / Z 6 , 1 (\widetilde{A}A)^{-1}\mathbb{Z}^{6,1}/\mathbb{Z}^{6,1} ( A A ) − 1 Z 6 , 1 / Z 6 , 1 3 3 3
{ ρ ~ u : u ∈ F 3 n } \{\widetilde{\rho}u:u\in\mathbb{F}_3^n\} { ρ u : u ∈ F 3 n } is a complete set of representatives for ( A ~ A ) − 1 Z 6 , n / Z 6 , n (\widetilde{A}A)^{-1}\mathbb{Z}^{6,n}/\mathbb{Z}^{6,n} ( A A ) − 1 Z 6 , n / Z 6 , n M ( 3 ) A M^{(3)}A M ( 3 ) A
A − 1 M ( 3 ) − 1 Z 6 , 1 A − 1 M ( 3 ) − 1 A ~ T Z 6 , 1 ⟶ Z 6 , 1 A ~ T Z 6 , 1 . (40) \frac{A^{-1}M^{(3)-1}\mathbb{Z}^{6,1}}{A^{-1}M^{(3)-1}\widetilde{A}^{T}\mathbb{Z}^{6,1}}
\longrightarrow
\frac{\mathbb{Z}^{6,1}}{\widetilde{A}^{T}\mathbb{Z}^{6,1}}. \tag*{(40)} A − 1 M ( 3 ) − 1 A T Z 6 , 1 A − 1 M ( 3 ) − 1 Z 6 , 1 ⟶ A T Z 6 , 1 Z 6 , 1 . (40) A direct calculation gives
M ( 3 ) A σ ~ = e 2 , A ~ − T e 2 = ( 0 1 0 0 1 / 3 ) T ∉ Z 6 , 1 . M^{(3)}A\widetilde{\sigma}=e_2,\qquad\widetilde{A}^{-T}e_2=(0\ \ 1\ \ 0\ \ 0\ \ 1/3)^T\notin\mathbb{Z}^{6,1}. M ( 3 ) A σ = e 2 , A − T e 2 = ( 0 1 0 0 1/3 ) T ∈ / Z 6 , 1 . The quotient on the right of (40) has order 3 3 3 e 2 e_2 e 2
{ σ ~ v : v ∈ F 3 n } \{\widetilde{\sigma}v:v\in\mathbb{F}_3^n\} { σ v : v ∈ F 3 n } is a complete set of representatives for
A − 1 M ( 3 ) − 1 Z 6 , n A − 1 M ( 3 ) − 1 A ~ T Z 6 , n . \frac{A^{-1}M^{(3)-1}\mathbb{Z}^{6,n}}{A^{-1}M^{(3)-1}\widetilde{A}^{T}\mathbb{Z}^{6,n}}. A − 1 M ( 3 ) − 1 A T Z 6 , n A − 1 M ( 3 ) − 1 Z 6 , n . Since both quotients are formed columnwise, the preceding one-column computations establish the asserted systems of representatives for every n n n
For every V ∈ C 2 , n V\in\mathbb{C}^{2,n} V ∈ C 2 , n
A − 1 ( V V V ) = ( C V 0 0 ) , C ρ = ( 2 / 3 − 1 ) , C σ = ( 0 1 / 3 ) , M C σ = ( 1 0 ) . A^{-1}
\begin{pmatrix}
V\\
V\\
V
\end{pmatrix}
=
\begin{pmatrix}
CV\\
0\\
0
\end{pmatrix},
\qquad
C_\rho=
\begin{pmatrix}
2/3\\
-1
\end{pmatrix},
\qquad
C_\sigma=
\begin{pmatrix}
0\\
1/3
\end{pmatrix},
\qquad
MC_\sigma=
\begin{pmatrix}
1\\
0
\end{pmatrix}. A − 1 V V V = C V 0 0 , C ρ = ( 2/3 − 1 ) , C σ = ( 0 1/3 ) , M C σ = ( 1 0 ) . Thus the transformed parameters in Theorem 2.7 are
( a 1 ′ , a 2 ′ , a 3 ′ ) = ( C ρ p , 0 , 0 ) , ( b 1 ′ , b 2 ′ , b 3 ′ ) = ( C σ q , 0 , 0 ) , ( w 1 , w 2 , w 3 ) = ( C z , 0 , 0 ) . (a'_1,a'_2,a'_3)=(C_\rho p,0,0),\qquad
(b'_1,b'_2,b'_3)=(C_\sigma q,0,0),\qquad
(w_1,w_2,w_3)=(Cz,0,0). ( a 1 ′ , a 2 ′ , a 3 ′ ) = ( C ρ p , 0 , 0 ) , ( b 1 ′ , b 2 ′ , b 3 ′ ) = ( C σ q , 0 , 0 ) , ( w 1 , w 2 , w 3 ) = ( C z , 0 , 0 ) . For these representatives, the exterior factor in (14) is
ϵ ( − tr ( ( C ρ p ) T M C σ v ) ) = ω p ⋅ v . \epsilon\left(-\operatorname{tr}\left((C_\rho p)^TMC_\sigma v\right)\right)=\omega^{p\cdot v}. ϵ ( − tr ( ( C ρ p ) T M C σ v ) ) = ω p ⋅ v . By Lemma 2.6,
Θ M , n [ C ρ p + ρ u , C σ q + σ v ] ( τ , C z ) = ϵ ( tr ( ( ρ u ) T M C σ q ) ) Θ M , n [ ρ u , σ v ] ( τ , C z ) = ω q ⋅ u Θ u , v ( n ) ( τ , C z ) . (41) \begin{aligned}
\Theta_{M,n}[C_\rho p+\rho u,C_\sigma q+\sigma v](\tau,Cz)
=
\epsilon\left(\operatorname{tr}\left((\rho u)^TMC_\sigma q\right)\right)
\Theta_{M,n}[\rho u,\sigma v](\tau,Cz) \\
&=\omega^{q\cdot u}\Theta_{u,v}^{(n)}(\tau,Cz). \tag*{(41)}
\end{aligned} Θ M , n [ C ρ p + ρ u , C σ q + σ v ] ( τ , C z ) = ϵ ( tr ( ( ρ u ) T M C σ q ) ) Θ M , n [ ρ u , σ v ] ( τ , C z ) = ω q ⋅ u Θ u , v ( n ) ( τ , C z ) . (41) Therefore,
RHS of x r e f ‘ ( 2.4 ) ‘ = ∑ u , v ∈ F 3 n ω p ⋅ v + q ⋅ u Θ u , v ( n ) ( τ , C z ) θ u , v ( n ) ( τ ) 2 . \operatorname{RHS}\text{ of }{xref}`(2.4)`=\sum_{\mathbf{u},\mathbf{v}\in\mathbb{F}_3^n}\omega^{\mathbf{p}\cdot\mathbf{v}+\mathbf{q}\cdot\mathbf{u}}\Theta_{\mathbf{u},\mathbf{v}}^{(n)}(\tau,Cz)\theta_{\mathbf{u},\mathbf{v}}^{(n)}(\tau)^2. RHS of x r e f ‘ ( 2.4 ) ‘ = u , v ∈ F 3 n ∑ ω p ⋅ v + q ⋅ u Θ u , v ( n ) ( τ , C z ) θ u , v ( n ) ( τ ) 2 . Combining this identity with (39) proves (35) . The specialization z = 0 z=0 z = 0 (36) .
The following lemma supplies the two reductions used below. Its vanishing assertion turns the Fourier transform of (35) into a family of relations, while its sign symmetries reduce the number of theta series occurring in degree two.
Lemma 3.2. Let n ≥ 1 n\ge1 n ≥ 1 p , q ∈ F 3 n \mathbf{p},\mathbf{q}\in\mathbb{F}_3^n p , q ∈ F 3 n
p ⋅ q ≠ 0 ⟹ θ p , q ( n ) = 0 , (42) \mathbf{p}\cdot\mathbf{q}\ne0\quad\Longrightarrow\quad\theta_{\mathbf{p},\mathbf{q}}^{(n)}=0, \tag*{(42)} p ⋅ q = 0 ⟹ θ p , q ( n ) = 0 , (42) and
θ p , q ( n ) = θ − p , q ( n ) = θ p , − q ( n ) = θ − p , − q ( n ) . (43) \theta_{\mathbf{p},\mathbf{q}}^{(n)}=\theta_{-\mathbf{p},\mathbf{q}}^{(n)}=\theta_{\mathbf{p},-\mathbf{q}}^{(n)}=\theta_{-\mathbf{p},-\mathbf{q}}^{(n)}. \tag*{(43)} θ p , q ( n ) = θ − p , q ( n ) = θ p , − q ( n ) = θ − p , − q ( n ) . (43) Proof. The matrices
R = ( 0 − 1 1 − 1 ) , J = ( 0 1 1 0 ) R=\begin{pmatrix}0&-1\\1&-1\end{pmatrix},\qquad J=\begin{pmatrix}0&1\\1&0\end{pmatrix} R = ( 0 1 − 1 − 1 ) , J = ( 0 1 1 0 ) satisfy
R T M R = M , J T M J = M , R^{T}MR=M,\qquad J^{T}MJ=M, R T M R = M , J T M J = M , R ρ − ρ = ( 0 1 ) , M ( R σ − σ ) = ( − 1 0 ) , R\rho-\rho=\begin{pmatrix}0\\1\end{pmatrix},\qquad M(R\sigma-\sigma)=\begin{pmatrix}-1\\0\end{pmatrix}, R ρ − ρ = ( 0 1 ) , M ( R σ − σ ) = ( − 1 0 ) , ρ T M ( R σ − σ ) = − 1 3 . \rho^{T}M(R\sigma-\sigma)=-\frac{1}{3}. ρ T M ( R σ − σ ) = − 3 1 . The change of variables induced by R R R Lemma 2.6 , gives
θ p , q ( n ) = θ M , n [ R ρ p , R σ q ] = ω − p ⋅ q θ p , q ( n ) , \theta_{\mathbf{p},\mathbf{q}}^{(n)}=\theta_{M,n}[R\rho\mathbf{p},R\sigma\mathbf{q}]=\omega^{-\mathbf{p}\cdot\mathbf{q}}\theta_{\mathbf{p},\mathbf{q}}^{(n)}, θ p , q ( n ) = θ M , n [ R ρ p , R σ q ] = ω − p ⋅ q θ p , q ( n ) , which proves (42) . The changes induced by J J J − J -J − J
J ρ = − ρ , J σ = σ , ( − J ) ρ = ρ , ( − J ) σ = − σ , J\rho=-\rho,\qquad J\sigma=\sigma,\qquad(-J)\rho=\rho,\qquad(-J)\sigma=-\sigma, J ρ = − ρ , J σ = σ , ( − J ) ρ = ρ , ( − J ) σ = − σ , give (43) .
Corollary 3.3. Let n ≥ 1 n\ge1 n ≥ 1 τ ∈ H n \tau\in\mathfrak{H}_n τ ∈ H n z ∈ C 2 , n z\in\mathbb{C}^{2,n} z ∈ C 2 , n λ , κ ∈ F 3 n \boldsymbol{\lambda},\boldsymbol{\kappa}\in\mathbb{F}_3^n λ , κ ∈ F 3 n λ ⋅ κ ≠ 0 \boldsymbol{\lambda}\cdot\boldsymbol{\kappa}\ne0 λ ⋅ κ = 0
∑ a , b ∈ F 3 n ω λ ⋅ a − κ ⋅ b Θ a , b ( n ) ( τ , z ) 3 = 0. (44) \sum_{\mathbf{a},\mathbf{b}\in\mathbb{F}_3^n}\omega^{\boldsymbol{\lambda}\cdot\mathbf{a}-\boldsymbol{\kappa}\cdot\mathbf{b}}\Theta_{\mathbf{a},\mathbf{b}}^{(n)}(\tau,z)^3=0. \tag*{(44)} a , b ∈ F 3 n ∑ ω λ ⋅ a − κ ⋅ b Θ a , b ( n ) ( τ , z ) 3 = 0. (44) Proof. We multiply (35) by ω λ ⋅ p − κ ⋅ q \omega^{\boldsymbol{\lambda}\cdot\mathbf{p}-\boldsymbol{\kappa}\cdot\mathbf{q}} ω λ ⋅ p − κ ⋅ q p , q ∈ F 3 n \mathbf{p},\mathbf{q}\in\mathbb{F}_3^n p , q ∈ F 3 n
3 n ∑ p , q ∈ F 3 n ω λ ⋅ p − κ ⋅ q Θ p , q ( n ) ( τ , z ) 3 , 3^n\sum_{\mathbf{p},\mathbf{q}\in\mathbb{F}_3^n}\omega^{\boldsymbol{\lambda}\cdot\mathbf{p}-\boldsymbol{\kappa}\cdot\mathbf{q}}\Theta_{\mathbf{p},\mathbf{q}}^{(n)}(\tau,z)^3, 3 n p , q ∈ F 3 n ∑ ω λ ⋅ p − κ ⋅ q Θ p , q ( n ) ( τ , z ) 3 , whereas its right-hand side is
∑ u , v ∈ F 3 n Θ u , v ( n ) ( τ , C z ) θ u , v ( n ) ( τ ) 2 ( ∑ p ∈ F 3 n ω p ⋅ ( λ + v ) ) ( ∑ q ∈ F 3 n ω q ⋅ ( u − κ ) ) . \sum_{\mathbf{u},\mathbf{v}\in\mathbb{F}_3^n}\Theta_{\mathbf{u},\mathbf{v}}^{(n)}(\tau,Cz)\theta_{\mathbf{u},\mathbf{v}}^{(n)}(\tau)^2\left(\sum_{\mathbf{p}\in\mathbb{F}_3^n}\omega^{\mathbf{p}\cdot(\boldsymbol{\lambda}+\mathbf{v})}\right)\left(\sum_{\mathbf{q}\in\mathbb{F}_3^n}\omega^{\mathbf{q}\cdot(\mathbf{u}-\boldsymbol{\kappa})}\right). u , v ∈ F 3 n ∑ Θ u , v ( n ) ( τ , C z ) θ u , v ( n ) ( τ ) 2 p ∈ F 3 n ∑ ω p ⋅ ( λ + v ) q ∈ F 3 n ∑ ω q ⋅ ( u − κ ) . For x ∈ F 3 n \mathbf{x}\in\mathbb{F}_3^n x ∈ F 3 n
∑ p ∈ F 3 n ω p ⋅ x = { 3 n , x = 0 , 0 , x ≠ 0. \sum_{\mathbf{p}\in\mathbb{F}_3^n}\omega^{\mathbf{p}\cdot\mathbf{x}}=
\begin{cases}
3^n,&\mathbf{x}=0,\\
0,&\mathbf{x}\ne0.
\end{cases} p ∈ F 3 n ∑ ω p ⋅ x = { 3 n , 0 , x = 0 , x = 0. Consequently, only ( u , v ) = ( κ , − λ ) (\mathbf{u},\mathbf{v})=(\boldsymbol{\kappa},-\boldsymbol{\lambda}) ( u , v ) = ( κ , − λ ) 3 n 3^n 3 n
∑ a , b ∈ F 3 n ω λ ⋅ a − κ ⋅ b Θ a , b ( n ) ( τ , z ) 3 = 3 n Θ κ , − λ ( n ) ( τ , C z ) θ κ , − λ ( n ) ( τ ) 2 . \sum_{\mathbf{a},\mathbf{b}\in\mathbb{F}_3^n}\omega^{\boldsymbol{\lambda}\cdot\mathbf{a}-\boldsymbol{\kappa}\cdot\mathbf{b}}\Theta_{\mathbf{a},\mathbf{b}}^{(n)}(\tau,z)^3=3^n\Theta_{\boldsymbol{\kappa},-\boldsymbol{\lambda}}^{(n)}(\tau,Cz)\theta_{\boldsymbol{\kappa},-\boldsymbol{\lambda}}^{(n)}(\tau)^2. a , b ∈ F 3 n ∑ ω λ ⋅ a − κ ⋅ b Θ a , b ( n ) ( τ , z ) 3 = 3 n Θ κ , − λ ( n ) ( τ , C z ) θ κ , − λ ( n ) ( τ ) 2 . The right-hand side vanishes by (42) , since λ ⋅ κ ≠ 0 \boldsymbol{\lambda}\cdot\boldsymbol{\kappa}\ne0 λ ⋅ κ = 0
Schultz's identity
We give an independent proof of Schultz’s identity [15 ] . To formulate the indexed version of the identity, let τ ∈ H 1 \tau\in\mathfrak{H}_1 τ ∈ H 1 q = e ( τ ) q=e(\tau) q = e ( τ ) Q ( x , y ) = x 2 + x y + y 2 Q(x,y)=x^2+xy+y^2 Q ( x , y ) = x 2 + x y + y 2 u = ( u 1 , u 2 ) T ∈ C 2 \mathbf{u}=(u_1,u_2)^T\in\mathbb{C}^2 u = ( u 1 , u 2 ) T ∈ C 2 x = e ( u 1 ) x=e(u_1) x = e ( u 1 ) y = e ( u 2 ) y=e(u_2) y = e ( u 2 ) r , s ∈ F 3 r,s\in\mathbb{F}_3 r , s ∈ F 3 q q q r r r s s s 0 , 1 , 2 0,1,2 0 , 1 , 2
T r , s ( u ∣ τ ) : = ω r s ∑ m , n ∈ Z ω s ( m − n ) q Q ( m + r / 3 , n + r / 3 ) x m + r / 3 y n + r / 3 . (45) T_{r,s}(\mathbf{u}\mid\tau):=\omega^{rs}\sum_{m,n\in\mathbb{Z}}\omega^{s(m-n)}q^{Q(m+r/3,n+r/3)}x^{m+r/3}y^{n+r/3}. \tag*{(45)} T r , s ( u ∣ τ ) := ω r s m , n ∈ Z ∑ ω s ( m − n ) q Q ( m + r /3 , n + r /3 ) x m + r /3 y n + r /3 . (45) The fractional powers in (45) are interpreted through the additive variables τ , u 1 , u 2 \tau,u_1,u_2 τ , u 1 , u 2 q , x , y q,x,y q , x , y
z ( u ) = M − 1 ( u 1 − u 2 ) , z(\mathbf{u})=M^{-1}\begin{pmatrix}u_1\\-u_2\end{pmatrix}, z ( u ) = M − 1 ( u 1 − u 2 ) , the substitution g 1 = m g_1=m g 1 = m g 2 = − n g_2=-n g 2 = − n (11) gives, for all r , s ∈ F 3 r,s\in\mathbb{F}_3 r , s ∈ F 3
T r , s ( u ∣ τ ) = ω r s Θ r , s ( 1 ) ( τ , z ( u ) ) . T_{r,s}(\mathbf{u}\mid\tau)=\omega^{rs}\Theta^{(1)}_{r,s}(\tau,z(\mathbf{u})). T r , s ( u ∣ τ ) = ω r s Θ r , s ( 1 ) ( τ , z ( u )) . Under the identification above, the following family is equivalent to that established by Schultz [15 Corollary 4.3] . Schultz [15 p. 622] proved its r = s = 1 r=s=1 r = s = 1 [4 Theorem 1.1] restated this case. The same authors [4 Sections 2 and 4] gave two further proofs. Zhu [17 Theorem 7.12] gave a computer-assisted proof using the theory of Jacobi forms.
Theorem 3.4 (Schultz). Let τ ∈ H 1 \tau\in\mathfrak{H}_1 τ ∈ H 1 u ∈ C 2 \mathbf{u}\in\mathbb{C}^2 u ∈ C 2 r , s ∈ F 3 r,s\in\mathbb{F}_3 r , s ∈ F 3
T r , s ( u ∣ τ ) 3 + T 0 , 0 ( u ∣ τ ) 3 = T r , 0 ( u ∣ τ ) 3 + T 0 , s ( u ∣ τ ) 3 . (46) T_{r,s}(\mathbf{u}\mid\tau)^3+T_{0,0}(\mathbf{u}\mid\tau)^3=T_{r,0}(\mathbf{u}\mid\tau)^3+T_{0,s}(\mathbf{u}\mid\tau)^3. \tag*{(46)} T r , s ( u ∣ τ ) 3 + T 0 , 0 ( u ∣ τ ) 3 = T r , 0 ( u ∣ τ ) 3 + T 0 , s ( u ∣ τ ) 3 . (46) Proof. Fix r , s ∈ F 3 r,s\in\mathbb{F}_3 r , s ∈ F 3 (35) , take n = 1 n=1 n = 1 z = z ( u ) z=z(\mathbf{u}) z = z ( u ) ( r , s ) (r,s) ( r , s ) ( 0 , 0 ) (0,0) ( 0 , 0 ) ( r , 0 ) (r,0) ( r , 0 ) ( 0 , s ) (0,s) ( 0 , s )
∑ a , b ∈ F 3 ( 1 − ω s a ) ( 1 − ω r b ) Θ a , b ( 1 ) ( τ , C z ( u ) ) θ a , b ( 1 ) ( τ ) 2 . \sum_{a,b\in\mathbb{F}_3}(1-\omega^{sa})(1-\omega^{rb})\Theta^{(1)}_{a,b}(\tau,Cz(\mathbf{u}))\theta^{(1)}_{a,b}(\tau)^2. a , b ∈ F 3 ∑ ( 1 − ω s a ) ( 1 − ω r b ) Θ a , b ( 1 ) ( τ , C z ( u )) θ a , b ( 1 ) ( τ ) 2 . Every summand vanishes. Indeed, if s a = 0 sa=0 s a = 0 r b = 0 rb=0 r b = 0 a , b ≠ 0 a,b\ne0 a , b = 0 a b ≠ 0 ab\ne0 ab = 0 F 3 \mathbb{F}_3 F 3 (42) gives θ a , b ( 1 ) ( τ ) = 0 \theta^{(1)}_{a,b}(\tau)=0 θ a , b ( 1 ) ( τ ) = 0 T r , s 3 = ( Θ r , s ( 1 ) ) 3 T_{r,s}^3=(\Theta^{(1)}_{r,s})^3 T r , s 3 = ( Θ r , s ( 1 ) ) 3 (46) . □
Remark. For the four functions in the Introduction, the definitions in (2) and (45) give
Θ 1 = T 1 , 1 , Θ 2 = T 1 , 0 , Θ 3 = T 0 , 0 , Θ 4 = T 0 , 1 . \Theta_1=T_{1,1},\qquad\Theta_2=T_{1,0},\qquad\Theta_3=T_{0,0},\qquad\Theta_4=T_{0,1}. Θ 1 = T 1 , 1 , Θ 2 = T 1 , 0 , Θ 3 = T 0 , 0 , Θ 4 = T 0 , 1 . Thus the r = s = 1 r=s=1 r = s = 1 (46) is precisely (3) , originally proved by Schultz [15 p. 622] .
At u = 0 \mathbf{u}=0 u = 0
a ( q ) = T 0 , 0 ( 0 ∣ τ ) = ∑ m , n ∈ Z q Q ( m , n ) , b ( q ) = T 0 , 1 ( 0 ∣ τ ) = ∑ m , n ∈ Z ω m − n q Q ( m , n ) , c ( q ) = T 1 , 0 ( 0 ∣ τ ) = ∑ m , n ∈ Z q Q ( m + 1 / 3 , n + 1 / 3 ) . (47) \begin{aligned}
a(q)&=T_{0,0}(0\mid\tau)=\sum_{m,n\in\mathbb{Z}}q^{Q(m,n)},\\
b(q)&=T_{0,1}(0\mid\tau)=\sum_{m,n\in\mathbb{Z}}\omega^{m-n}q^{Q(m,n)},\\
c(q)&=T_{1,0}(0\mid\tau)=\sum_{m,n\in\mathbb{Z}}q^{Q(m+1/3,n+1/3)}. \tag*{(47)}
\end{aligned} a ( q ) b ( q ) c ( q ) = T 0 , 0 ( 0 ∣ τ ) = m , n ∈ Z ∑ q Q ( m , n ) , = T 0 , 1 ( 0 ∣ τ ) = m , n ∈ Z ∑ ω m − n q Q ( m , n ) , = T 1 , 0 ( 0 ∣ τ ) = m , n ∈ Z ∑ q Q ( m + 1/3 , n + 1/3 ) . (47) For r = s = 1 r=s=1 r = s = 1 u = 0 \mathbf{u}=0 u = 0 (42) show that T 1 , 1 ( 0 ∣ τ ) = 0 T_{1,1}(0\mid\tau)=0 T 1 , 1 ( 0 ∣ τ ) = 0 (46) specializes to the cubic identity of Borwein and Borwein [2 Eq. (2.3)] .
Corollary 3.5. For ∣ q ∣ < 1 |q|<1 ∣ q ∣ < 1
a ( q ) 3 = b ( q ) 3 + c ( q ) 3 . (48) a(q)^3=b(q)^3+c(q)^3. \tag*{(48)} a ( q ) 3 = b ( q ) 3 + c ( q ) 3 . (48) Degree two
We derive degree-two analogues of the Borweins’ cubic identity (Corollary 3.5).
Let
τ = ( τ 1 τ 0 τ 0 τ 2 ) ∈ H 2 , q = e ( τ 1 ) , s = e ( τ 0 ) , t = e ( τ 2 ) . \tau=\begin{pmatrix}\tau_1 & \tau_0 \\ \tau_0 & \tau_2\end{pmatrix}\in\mathfrak{H}_2,\qquad q=e(\tau_1),\qquad s=e(\tau_0),\qquad t=e(\tau_2). τ = ( τ 1 τ 0 τ 0 τ 2 ) ∈ H 2 , q = e ( τ 1 ) , s = e ( τ 0 ) , t = e ( τ 2 ) . For a = ( a 1 , a 2 ) a=(a_1,a_2) a = ( a 1 , a 2 ) b = ( b 1 , b 2 ) b=(b_1,b_2) b = ( b 1 , b 2 ) F 3 2 \mathbb{F}_3^2 F 3 2 (34) gives the following expansion, obtained by writing the two columns of g + ρ a g+\rho a g + ρ a ( j , − k ) T (j,-k)^{T} ( j , − k ) T ( r , − h ) T (r,-h)^{T} ( r , − h ) T
θ a , b ( 2 ) ( τ ) = ∑ j , k ∈ Z + a 1 / 3 r , h ∈ Z + a 2 / 3 ω b 1 ( j − k ) + b 2 ( r − h ) q Q ( j , k ) s ( 2 j + k ) r + ( j + 2 k ) h t Q ( r , h ) . (49) \theta_{a,b}^{(2)}(\tau)=\sum_{\substack{j,k\in\mathbb{Z}+a_1/3\\r,h\in\mathbb{Z}+a_2/3}}\omega^{b_1(j-k)+b_2(r-h)}q^{Q(j,k)}s^{(2j+k)r+(j+2k)h}t^{Q(r,h)}. \tag*{(49)} θ a , b ( 2 ) ( τ ) = j , k ∈ Z + a 1 /3 r , h ∈ Z + a 2 /3 ∑ ω b 1 ( j − k ) + b 2 ( r − h ) q Q ( j , k ) s ( 2 j + k ) r + ( j + 2 k ) h t Q ( r , h ) . (49) We use the notation
0 = ( 0 , 0 ) , e 1 = ( 1 , 0 ) , e 2 = ( 0 , 1 ) , d + = ( 1 , 1 ) , d − = ( 1 , − 1 ) = ( 1 , 2 ) . \mathbf{0}=(0,0),\qquad e_1=(1,0),\qquad e_2=(0,1),\qquad d_+=(1,1),\qquad d_-=(1,-1)=(1,2). 0 = ( 0 , 0 ) , e 1 = ( 1 , 0 ) , e 2 = ( 0 , 1 ) , d + = ( 1 , 1 ) , d − = ( 1 , − 1 ) = ( 1 , 2 ) . The pairs ( a , b ) (a,b) ( a , b ) a ⋅ b = 0 a\cdot b=0 a ⋅ b = 0 (43) , are represented by
I = { ( 0 , 0 ) , ( 0 , e 1 ) , ( 0 , e 2 ) , ( 0 , d + ) , ( 0 , d − ) , ( e 1 , 0 ) , ( e 2 , 0 ) , ( d + , 0 ) , ( d − , 0 ) , ( e 1 , e 2 ) , ( e 2 , e 1 ) , ( d + , d − ) , ( d − , d + ) } . (50) \mathcal{I}=\{(0,0),(0,e_1),(0,e_2),(0,d_+),(0,d_-),
(e_1,0),(e_2,0),(d_+,0),(d_-,0),(e_1,e_2),(e_2,e_1),(d_+,d_-),(d_-,d_+)\}. \tag*{(50)} I = {( 0 , 0 ) , ( 0 , e 1 ) , ( 0 , e 2 ) , ( 0 , d + ) , ( 0 , d − ) , ( e 1 , 0 ) , ( e 2 , 0 ) , ( d + , 0 ) , ( d − , 0 ) , ( e 1 , e 2 ) , ( e 2 , e 1 ) , ( d + , d − ) , ( d − , d + )} . (50) The first five representatives in I \mathcal{I} I a = 0 a=0 a = 0 b = 0 b=0 b = 0 a ≠ 0 a\ne0 a = 0 (43) . By Lemma 3.2, every nonzero indexed series is therefore represented by one of the 13 pairs in I \mathcal{I} I (49) shows that these 13 series are nonzero and pairwise distinct. For completeness, let c A , B , C = [ q A s B t C ] c_{A,B,C}=[q^A s^B t^C] c A , B , C = [ q A s B t C ]
( 1 / 3 , − 1 / 3 , 1 / 3 ) , ( 1 / 3 , 1 / 3 , 1 / 3 ) , ( 1 / 3 , 0 , 1 ) , ( 1 , 0 , 1 / 3 ) , ( 0 , 0 , 1 ) , ( 1 , 0 , 0 ) , ( 1 , − 1 , 1 ) . (1/3,-1/3,1/3),\quad(1/3,1/3,1/3),\quad(1/3,0,1),\quad(1,0,1/3),\quad(0,0,1),\quad(1,0,0),\quad(1,-1,1). ( 1/3 , − 1/3 , 1/3 ) , ( 1/3 , 1/3 , 1/3 ) , ( 1/3 , 0 , 1 ) , ( 1 , 0 , 1/3 ) , ( 0 , 0 , 1 ) , ( 1 , 0 , 0 ) , ( 1 , − 1 , 1 ) . Direct extraction from (49) gives the following coefficient vectors, in the order in which the 13 representatives occur in I \mathcal{I} I
( 0 , 0 , 0 , 0 , 6 , 6 , 12 ) , ( 0 , 0 , 0 , 0 , 6 , − 3 , − 6 ) , ( 0 , 0 , 0 , 0 , − 3 , 6 , − 6 ) , (0,0,0,0,6,6,12),\quad(0,0,0,0,6,-3,-6),\quad(0,0,0,0,-3,6,-6), ( 0 , 0 , 0 , 0 , 6 , 6 , 12 ) , ( 0 , 0 , 0 , 0 , 6 , − 3 , − 6 ) , ( 0 , 0 , 0 , 0 , − 3 , 6 , − 6 ) , ( 0 , 0 , 0 , 0 , − 3 , − 3 , − 6 ) , ( 0 , 0 , 0 , 0 , − 3 , − 3 , 12 ) , (0,0,0,0,-3,-3,-6),\quad(0,0,0,0,-3,-3,12), ( 0 , 0 , 0 , 0 , − 3 , − 3 , − 6 ) , ( 0 , 0 , 0 , 0 , − 3 , − 3 , 12 ) , ( 0 , 0 , 6 , 0 , 0 , 0 , 0 ) , ( 0 , 0 , 0 , 6 , 0 , 0 , 0 ) , ( 6 , 0 , 0 , 0 , 0 , 0 , 0 ) , ( 0 , 6 , 0 , 0 , 0 , 0 , 0 ) , (0,0,6,0,0,0,0),\quad(0,0,0,6,0,0,0),\quad(6,0,0,0,0,0,0),\quad(0,6,0,0,0,0,0), ( 0 , 0 , 6 , 0 , 0 , 0 , 0 ) , ( 0 , 0 , 0 , 6 , 0 , 0 , 0 ) , ( 6 , 0 , 0 , 0 , 0 , 0 , 0 ) , ( 0 , 6 , 0 , 0 , 0 , 0 , 0 ) , ( 0 , 0 , − 3 , 0 , 0 , 0 , 0 ) , ( 0 , 0 , 0 , − 3 , 0 , 0 , 0 ) , ( − 3 , 0 , 0 , 0 , 0 , 0 , 0 ) , ( 0 , − 3 , 0 , 0 , 0 , 0 , 0 ) . (0,0,-3,0,0,0,0),\quad(0,0,0,-3,0,0,0),\quad(-3,0,0,0,0,0,0),\quad(0,-3,0,0,0,0,0). ( 0 , 0 , − 3 , 0 , 0 , 0 , 0 ) , ( 0 , 0 , 0 , − 3 , 0 , 0 , 0 ) , ( − 3 , 0 , 0 , 0 , 0 , 0 , 0 ) , ( 0 , − 3 , 0 , 0 , 0 , 0 , 0 ) . These vectors show again that the 13 series are nonzero and pairwise distinct. We index them by ( a , b ) ∈ I (a,b)\in\mathcal{I} ( a , b ) ∈ I θ a , b ( τ ) = θ a , b ( 2 ) ( τ ) \theta_{a,b}(\tau)=\theta_{a,b}^{(2)}(\tau) θ a , b ( τ ) = θ a , b ( 2 ) ( τ )
The following linear-algebraic calculation applies to both the cubic and square identities. For an array X = ( X a , b ) X=(X_{a,b}) X = ( X a , b ) F 3 2 × F 3 2 \mathbb{F}_3^2\times\mathbb{F}_3^2 F 3 2 × F 3 2
S λ , κ ( X ) = ∑ a , b ∈ F 3 2 ω λ ⋅ a − κ ⋅ b X a , b . S_{\lambda,\kappa}(X)=\sum_{a,b\in\mathbb{F}_3^2}\omega^{\lambda\cdot a-\kappa\cdot b}X_{a,b}. S λ , κ ( X ) = a , b ∈ F 3 2 ∑ ω λ ⋅ a − κ ⋅ b X a , b . Lemma 3.6. Suppose that
X a , b = 0 if a ⋅ b ≠ 0 , X a , b = X − a , b = X a , − b = X − a , − b . X_{a,b}=0\quad\text{if }a\cdot b\ne0,\qquad X_{a,b}=X_{-a,b}=X_{a,-b}=X_{-a,-b}. X a , b = 0 if a ⋅ b = 0 , X a , b = X − a , b = X a , − b = X − a , − b . Let
S 1 = S e 2 , e 2 ( X ) , S 2 = S e 2 , d + ( X ) , S 3 = S e 2 , d − ( X ) , S 4 = S e 1 , e 1 ( X ) , S_1=S_{e_2,e_2}(X),\qquad S_2=S_{e_2,d_+}(X),\qquad S_3=S_{e_2,d_-}(X),\qquad S_4=S_{e_1,e_1}(X), S 1 = S e 2 , e 2 ( X ) , S 2 = S e 2 , d + ( X ) , S 3 = S e 2 , d − ( X ) , S 4 = S e 1 , e 1 ( X ) , S 5 = S e 1 , d + ( X ) , S 6 = S d + , e 2 ( X ) , S 7 = S d − , e 2 ( X ) . S_5=S_{e_1,d_+}(X),\qquad S_6=S_{d_+,e_2}(X),\qquad S_7=S_{d_-,e_2}(X). S 5 = S e 1 , d + ( X ) , S 6 = S d + , e 2 ( X ) , S 7 = S d − , e 2 ( X ) . We define R ( X ) = ( R 1 , … , R 7 ) T R(\mathbf{X})=(R_1,\ldots,R_7)^{T} R ( X ) = ( R 1 , … , R 7 ) T
R 1 = X 0 , 0 − X 0 , d + − X d + , 0 − X e 2 , e 1 − X e 1 , e 2 , R 2 = ( X 0 , d − − X d + , 0 ) − ( X 0 , d + − X d − , 0 ) , R 3 = ( X 0 , d − − X d + , 0 ) − ( X 0 , e 1 − X e 2 , 0 ) , R 4 = ( X 0 , d − − X d + , 0 ) − ( X 0 , e 2 − X e 1 , 0 ) , R 5 = ( X 0 , d − − X d + , d − ) − ( X 0 , d + − X d − , d + ) , R 6 = ( X 0 , d − − X d + , d − ) − ( X 0 , e 1 − X e 2 , e 1 ) , R 7 = ( X 0 , d − − X d + , d − ) − ( X 0 , e 2 − X e 1 , e 2 ) . \begin{aligned}
R_1&=X_{0,0}-X_{0,d_+}-X_{d_+,0}-X_{e_2,e_1}-X_{e_1,e_2},\\
R_2&=(X_{0,d_-}-X_{d_+,0})-(X_{0,d_+}-X_{d_-,0}),\\
R_3&=(X_{0,d_-}-X_{d_+,0})-(X_{0,e_1}-X_{e_2,0}),\\
R_4&=(X_{0,d_-}-X_{d_+,0})-(X_{0,e_2}-X_{e_1,0}),\\
R_5&=(X_{0,d_-}-X_{d_+,d_-})-(X_{0,d_+}-X_{d_-,d_+}),\\
R_6&=(X_{0,d_-}-X_{d_+,d_-})-(X_{0,e_1}-X_{e_2,e_1}),\\
R_7&=(X_{0,d_-}-X_{d_+,d_-})-(X_{0,e_2}-X_{e_1,e_2}).
\end{aligned} R 1 R 2 R 3 R 4 R 5 R 6 R 7 = X 0 , 0 − X 0 , d + − X d + , 0 − X e 2 , e 1 − X e 1 , e 2 , = ( X 0 , d − − X d + , 0 ) − ( X 0 , d + − X d − , 0 ) , = ( X 0 , d − − X d + , 0 ) − ( X 0 , e 1 − X e 2 , 0 ) , = ( X 0 , d − − X d + , 0 ) − ( X 0 , e 2 − X e 1 , 0 ) , = ( X 0 , d − − X d + , d − ) − ( X 0 , d + − X d − , d + ) , = ( X 0 , d − − X d + , d − ) − ( X 0 , e 1 − X e 2 , e 1 ) , = ( X 0 , d − − X d + , d − ) − ( X 0 , e 2 − X e 1 , e 2 ) . Then
3 R ( X ) = ( 0 1 0 1 0 1 0 0 1 − 1 0 0 1 − 1 0 0 0 0 1 0 − 1 1 0 0 − 1 1 0 − 1 0 1 − 1 0 0 0 0 − 1 1 0 0 0 0 0 0 0 0 − 1 1 0 0 ) ( S 1 S 2 S 3 S 4 S 5 S 6 S 7 ) . (51) 3R(\mathbf{X})=
\begin{pmatrix}
0&1&0&1&0&1&0\\
0&1&-1&0&0&1&-1\\
0&0&0&0&1&0&-1\\
1&0&0&-1&1&0&-1\\
0&1&-1&0&0&0&0\\
-1&1&0&0&0&0&0\\
0&0&0&-1&1&0&0
\end{pmatrix}
\begin{pmatrix}
S_1\\S_2\\S_3\\S_4\\S_5\\S_6\\S_7
\end{pmatrix}. \tag*{(51)} 3 R ( X ) = 0 0 0 1 0 − 1 0 1 1 0 0 1 1 0 0 − 1 0 0 − 1 0 0 1 0 0 − 1 0 0 − 1 0 0 1 1 0 0 1 1 1 0 0 0 0 0 0 − 1 − 1 − 1 0 0 0 S 1 S 2 S 3 S 4 S 5 S 6 S 7 . (51) Moreover, the seven components of R ( X ) R(\mathbf{X}) R ( X ) X a , b X_{a,b} X a , b ( a , b ) ∈ I (\mathbf{a},\mathbf{b})\in\mathcal{I} ( a , b ) ∈ I
Proof. In each character sum S λ , κ ( X ) S_{\lambda,\kappa}(\mathbf{X}) S λ , κ ( X ) ( a , b ) (\mathbf{a},\mathbf{b}) ( a , b ) I \mathcal{I} I 1 + 1 = 2 1+1=2 1 + 1 = 2 ω + ω − 1 = − 1 \omega+\omega^{-1}=-1 ω + ω − 1 = − 1
S 1 = X 0 , 0 + 2 X 0 , e 1 − X 0 , e 2 − X 0 , d + − X 0 , d − + 2 X e 1 , 0 − X e 2 , 0 − X d + , 0 − X d − , 0 − 2 X e 1 , e 2 − 2 X e 2 , e 1 + X d + , d − + X d − , d + . \begin{aligned}
S_1={}&X_{0,0}+2X_{0,e_1}-X_{0,e_2}-X_{0,d_+}-X_{0,d_-}\\
&+2X_{e_1,0}-X_{e_2,0}-X_{d_+,0}-X_{d_-,0}\\
&-2X_{e_1,e_2}-2X_{e_2,e_1}+X_{d_+,d_-}+X_{d_-,d_+}.
\end{aligned} S 1 = X 0 , 0 + 2 X 0 , e 1 − X 0 , e 2 − X 0 , d + − X 0 , d − + 2 X e 1 , 0 − X e 2 , 0 − X d + , 0 − X d − , 0 − 2 X e 1 , e 2 − 2 X e 2 , e 1 + X d + , d − + X d − , d + . Expanding the seven indicated character sums as above gives (51) . To prove independence, consider the minor formed by the columns corresponding to X 0 , 0 X_{0,0} X 0 , 0 X 0 , e 1 X_{0,e_1} X 0 , e 1 X 0 , e 2 X_{0,e_2} X 0 , e 2 X 0 , d + X_{0,d_+} X 0 , d + X e 1 , 0 X_{e_1,0} X e 1 , 0 X e 2 , 0 X_{e_2,0} X e 2 , 0 X d + , 0 X_{d_+,0} X d + , 0 R ( X ) R(\mathbf{X}) R ( X )
Theorem 3.7. The cubes of the thirteen theta series (49) , indexed by the set I \mathcal{I} I (50) , satisfy
θ 0 , 0 3 = θ 0 , d + 3 + θ d + , 0 3 + θ e 2 , e 1 3 + θ e 1 , e 2 3 (52) \theta_{0,0}^{3}=\theta_{0,d_+}^{3}+\theta_{d_+,0}^{3}+\theta_{e_2,e_1}^{3}+\theta_{e_1,e_2}^{3} \tag*{(52)} θ 0 , 0 3 = θ 0 , d + 3 + θ d + , 0 3 + θ e 2 , e 1 3 + θ e 1 , e 2 3 (52) θ 0 , d − 3 − θ d + , 0 3 = θ 0 , d + 3 − θ d − , 0 3 = θ 0 , e 1 3 − θ e 2 , 0 3 = θ 0 , e 2 3 − θ e 1 , 0 3 (53) \theta_{0,d_-}^{3}-\theta_{d_+,0}^{3}
=\theta_{0,d_+}^{3}-\theta_{d_-,0}^{3}
=\theta_{0,e_1}^{3}-\theta_{e_2,0}^{3}
=\theta_{0,e_2}^{3}-\theta_{e_1,0}^{3} \tag*{(53)} θ 0 , d − 3 − θ d + , 0 3 = θ 0 , d + 3 − θ d − , 0 3 = θ 0 , e 1 3 − θ e 2 , 0 3 = θ 0 , e 2 3 − θ e 1 , 0 3 (53) and
θ 0 , d − 3 − θ d + , d − 3 = θ 0 , d + 3 − θ d − , d + 3 = θ 0 , e 1 3 − θ e 2 , e 1 3 = θ 0 , e 2 3 − θ e 1 , e 2 3 . (54) \theta_{0,d_-}^{3}-\theta_{d_+,d_-}^{3}
=\theta_{0,d_+}^{3}-\theta_{d_-,d_+}^{3}
=\theta_{0,e_1}^{3}-\theta_{e_2,e_1}^{3}
=\theta_{0,e_2}^{3}-\theta_{e_1,e_2}^{3}. \tag*{(54)} θ 0 , d − 3 − θ d + , d − 3 = θ 0 , d + 3 − θ d − , d + 3 = θ 0 , e 1 3 − θ e 2 , e 1 3 = θ 0 , e 2 3 − θ e 1 , e 2 3 . (54) Proof. We apply Lemma 3.6 to
X a , b = θ a , b ( τ ) 3 . X_{a,b}=\theta_{a,b}(\tau)^3. X a , b = θ a , b ( τ ) 3 . Its hypotheses follow from (42) and (43) . Corollary 3.3, with n = 2 n=2 n = 2 z = 0 z=0 z = 0 (51) vanishes. Hence R ( X ) = 0 R(\mathbf{X})=0 R ( X ) = 0 (52) , (53) , and (54) . □
Square companions
The square Fourier identity
We retain the notation of § 3 . The following square counterpart of Theorem 3.1 is a direct specialization of Theorem 2.7 .
Theorem 4.1. Let n ≥ 1 n \ge1 n ≥ 1 τ ∈ H n \tau\in\mathfrak{H}_n τ ∈ H n z ∈ C 2 , n z \in\mathbb{C}^{2,n} z ∈ C 2 , n C = ( 1 1 − 1 2 ) C = \begin{pmatrix} 1 & 1 \\ -1 & 2 \end{pmatrix} C = ( 1 − 1 1 2 ) p , q ∈ F 3 n \mathbf{p},\mathbf{q} \in\mathbb{F}_3^n p , q ∈ F 3 n
3 n Θ p , q ( n ) ( τ , z ) Θ p , − q ( n ) ( 2 τ , 2 z ) = ∑ u , v ∈ F 3 n ω p ⋅ v + q ⋅ u Θ u , v ( n ) ( τ , C z ) θ u , v ( n ) ( 2 τ ) . (55) 3^n\Theta_{\mathbf{p},\mathbf{q}}^{(n)}(\tau,z)\Theta_{\mathbf{p},-\mathbf{q}}^{(n)}(2\tau,2z)=\sum_{\mathbf{u},\mathbf{v}\in\mathbb{F}_3^n}\omega^{\mathbf{p}\cdot\mathbf{v}+\mathbf{q}\cdot\mathbf{u}}\Theta_{\mathbf{u},\mathbf{v}}^{(n)}(\tau,Cz)\theta_{\mathbf{u},\mathbf{v}}^{(n)}(2\tau). \tag*{(55)} 3 n Θ p , q ( n ) ( τ , z ) Θ p , − q ( n ) ( 2 τ , 2 z ) = u , v ∈ F 3 n ∑ ω p ⋅ v + q ⋅ u Θ u , v ( n ) ( τ , C z ) θ u , v ( n ) ( 2 τ ) . (55) At z = 0 z=0 z = 0 q \mathbf{q} q − q -\mathbf{q} − q
Proof. We apply Theorem 2.7 with m = 4 m=4 m = 4 μ = 1 \mu=1 μ = 1 M 1 = M 2 = M ~ = diag ( M , 2 M ) M_1=M_2=\widetilde{M}=\operatorname{diag}(M,2M) M 1 = M 2 = M = diag ( M , 2 M )
A = 1 3 ( 2 − 1 − 4 2 1 1 − 2 − 2 2 − 1 2 − 1 1 1 1 1 ) , A ~ = ( 1 0 0 − 2 0 1 0 − 1 0 0 1 − 2 0 0 0 3 ) . A=\frac{1}{3}\begin{pmatrix}2&-1&-4&2\\1&1&-2&-2\\2&-1&2&-1\\1&1&1&1\end{pmatrix},\qquad
\widetilde{A}=\begin{pmatrix}1&0&0&-2\\0&1&0&-1\\0&0&1&-2\\0&0&0&3\end{pmatrix}. A = 3 1 2 1 2 1 − 1 1 − 1 1 − 4 − 2 2 1 2 − 2 − 1 1 , A = 1 0 0 0 0 1 0 0 0 0 1 0 − 2 − 1 − 2 3 . Direct multiplication gives
A T M ~ A = M ~ , det A = 1 , det A ~ = 3. A^{T}\widetilde{M}A=\widetilde{M},\qquad\det A=1,\qquad\det\widetilde{A}=3. A T M A = M , det A = 1 , det A = 3. All columns of 3 A 3A 3 A u 0 = ( 2 , 1 , 2 , 1 ) T u_0=(2,1,2,1)^T u 0 = ( 2 , 1 , 2 , 1 ) T 3 Z 4 3\mathbb{Z}^4 3 Z 4 A ~ − 1 \widetilde{A}^{-1} A − 1 e 1 , e 2 , e 3 e_1,e_2,e_3 e 1 , e 2 , e 3 u 0 / 3 u_0/3 u 0 /3 e 4 = u 0 − 2 e 1 − e 2 − 2 e 3 e_4=u_0-2e_1-e_2-2e_3 e 4 = u 0 − 2 e 1 − e 2 − 2 e 3
Z 4 , n + A Z 4 , n = A ~ − 1 Z 4 , n . \mathbb{Z}^{4,n}+A\mathbb{Z}^{4,n}=\widetilde{A}^{-1}\mathbb{Z}^{4,n}. Z 4 , n + A Z 4 , n = A − 1 Z 4 , n . We choose the characteristics and Jacobi variable as
a 1 = [ ρ , ρ ] p , b 1 = [ σ , σ ] q , z 1 = [ z , z ] . a_1=[\rho,\rho]\mathbf{p},\qquad b_1=[\sigma,\sigma]\mathbf{q},\qquad z_1=[z,z]. a 1 = [ ρ , ρ ] p , b 1 = [ σ , σ ] q , z 1 = [ z , z ] . More explicitly,
LHS of (2.4) = ∣ det A ~ ∣ n ⋅ ∏ i = 1 μ Θ M 1 , n [ a i , b i ] ( τ , z i ) = 3 n Θ M , n [ ρ p , σ q ] ( τ , z ) Θ 2 M , n [ ρ p , σ q ] ( τ , z ) (56) \begin{aligned}
\operatorname{LHS\ of\ (2.4)}=|\det\widetilde{A}|^n\cdot\prod_{i=1}^{\mu}\Theta_{M_1,n}[a_i,b_i](\tau,z_i) \\
&=3^n\Theta_{M,n}[\rho\mathbf{p},\sigma\mathbf{q}](\tau,z)\Theta_{2M,n}[\rho\mathbf{p},\sigma\mathbf{q}](\tau,z) \tag*{(56)}
\end{aligned} LHS of (2.4) = ∣ det A ∣ n ⋅ i = 1 ∏ μ Θ M 1 , n [ a i , b i ] ( τ , z i ) = 3 n Θ M , n [ ρ p , σ q ] ( τ , z ) Θ 2 M , n [ ρ p , σ q ] ( τ , z ) (56) =3^n\Theta_{\mathbf{p},\mathbf{q}}^{(n)}(\tau,z)\Theta_{\mathbf{p},-\mathbf{q}}^{(n)}(2\tau,2z),
where the equality $\Theta_{2M,n}[\rho\mathbf{p},\sigma\mathbf{q}](\tau,z)=\Theta_{\mathbf{p},-\mathbf{q}}^{(n)}(2\tau,2z)$ follows from {xref}`Lemma 2.6`. Complete systems of representatives for the two quotients are
\alpha_{\mathbf{u}}:=\begin{pmatrix}\rho\-\rho\end{pmatrix}\mathbf{u},\qquad \beta_{\mathbf{v}}:=\begin{pmatrix}\sigma\-\sigma\end{pmatrix}\mathbf{v},\qquad \mathbf{u},\mathbf{v}\in\mathbb{F}_3^n.
\widetilde{A}\begin{pmatrix}\rho\-\rho\end{pmatrix}=e_1,\qquad \widetilde{M}A\begin{pmatrix}\sigma\-\sigma\end{pmatrix}=e_2.
Multiplication by $\widetilde{M}A$ identifies the second quotient with $\mathbb{Z}^{4,n}/\widetilde{A}^{T}\mathbb{Z}^{4,n}$. The two quotient groups in {xref}`(2.5)` and {xref}`(2.6)` both have order $3^n$; the displayed classes are nonzero because $(\rho^T,-\rho^T)^T\notin\mathbb{Z}^4$ and $\widetilde{A}^{-T}e_2=(0,1,0,1/3)^T\notin\mathbb{Z}^4$. Since the quotients and the displayed parametrizations are formed column by column, the parametrizations are complete for every $n$. Moreover,
a'_1=\begin{pmatrix}C\rho\0\end{pmatrix}\mathbf{p},\qquad b'_1=\begin{pmatrix}C\sigma\0\end{pmatrix}\mathbf{q},\qquad w_1=\begin{pmatrix}Cz\0\end{pmatrix}.
Therefore,
RHS of ( 2.4 ) = ∑ u , v ∈ F 3 n ω − tr ( a 1 ′ T M β v ) Θ M , n [ a 1 ′ + α u , b 1 ′ + β v ] ( τ , w 1 ) = ∑ u , v ∈ F 3 n ω p ⋅ v + q ⋅ u Θ u , v ( n ) ( τ , C z ) θ u , v ( n ) ( 2 τ ) . \begin{aligned}
\operatorname{RHS}\text{ of }(2.4)=\sum_{u,v\in\mathbb{F}_3^n}\omega^{-\operatorname{tr}\left(a_1^{\prime T}M\beta_v\right)}\Theta_{M,n}\left[a_1^\prime+\alpha_u,b_1^\prime+\beta_v\right](\tau,w_1) \\
&=\sum_{u,v\in\mathbb{F}_3^n}\omega^{p\cdot v+q\cdot u}\Theta_{u,v}^{(n)}(\tau,Cz)\theta_{u,v}^{(n)}(2\tau).
\end{aligned} RHS of ( 2.4 ) = u , v ∈ F 3 n ∑ ω − tr ( a 1 ′ T M β v ) Θ M , n [ a 1 ′ + α u , b 1 ′ + β v ] ( τ , w 1 ) = u , v ∈ F 3 n ∑ ω p ⋅ v + q ⋅ u Θ u , v ( n ) ( τ , C z ) θ u , v ( n ) ( 2 τ ) . The second equality follows from Lemma 2.6, (3.9), and (3.11). Combining this identity with (4.2) in (2.4) gives (4.1). The specialization z = 0 z=0 z = 0
The finite Fourier transform of (4.1) yields the square counterpart of Corollary 3.3. This is the form used both for the square Schultz identity and for the degree-two relations below.
Corollary 4.2. Let n ≥ 1 n\geq1 n ≥ 1 τ ∈ H n \tau\in\mathfrak{H}_n τ ∈ H n z ∈ C 2 , n z\in\mathbb{C}^{2,n} z ∈ C 2 , n λ , κ ∈ F 3 n \lambda,\kappa\in\mathbb{F}_3^n λ , κ ∈ F 3 n λ ⋅ κ ≠ 0 \lambda\cdot\kappa\ne0 λ ⋅ κ = 0
∑ a , b ∈ F 3 n ω λ ⋅ a − κ ⋅ b Θ a , b ( n ) ( τ , z ) Θ a , − b ( n ) ( 2 τ , 2 z ) = 0. (57) \sum_{a,b\in\mathbb{F}_3^n}\omega^{\lambda\cdot a-\kappa\cdot b}\Theta_{a,b}^{(n)}(\tau,z)\Theta_{a,-b}^{(n)}(2\tau,2z)=0. \tag*{(57)} a , b ∈ F 3 n ∑ ω λ ⋅ a − κ ⋅ b Θ a , b ( n ) ( τ , z ) Θ a , − b ( n ) ( 2 τ , 2 z ) = 0. (57) Proof. We multiply (4.1) by ω λ ⋅ p − κ ⋅ q \omega^{\lambda\cdot p-\kappa\cdot q} ω λ ⋅ p − κ ⋅ q p , q ∈ F 3 n p,q\in\mathbb{F}_3^n p , q ∈ F 3 n
3 n ∑ p , q ∈ F 3 n ω λ ⋅ p − κ ⋅ q Θ p , q ( n ) ( τ , z ) Θ p , − q ( n ) ( 2 τ , 2 z ) , 3^n\sum_{p,q\in\mathbb{F}_3^n}\omega^{\lambda\cdot p-\kappa\cdot q}\Theta_{p,q}^{(n)}(\tau,z)\Theta_{p,-q}^{(n)}(2\tau,2z), 3 n p , q ∈ F 3 n ∑ ω λ ⋅ p − κ ⋅ q Θ p , q ( n ) ( τ , z ) Θ p , − q ( n ) ( 2 τ , 2 z ) , whereas its right-hand side is
∑ u , v ∈ F 3 n Θ u , v ( n ) ( τ , C z ) θ u , v ( n ) ( 2 τ ) ( ∑ p ∈ F 3 n ω p ⋅ ( λ + v ) ) ( ∑ q ∈ F 3 n ω q ⋅ ( u − κ ) ) . \sum_{u,v\in\mathbb{F}_3^n}\Theta_{u,v}^{(n)}(\tau,Cz)\theta_{u,v}^{(n)}(2\tau)\left(\sum_{p\in\mathbb{F}_3^n}\omega^{p\cdot(\lambda+v)}\right)\left(\sum_{q\in\mathbb{F}_3^n}\omega^{q\cdot(u-\kappa)}\right). u , v ∈ F 3 n ∑ Θ u , v ( n ) ( τ , C z ) θ u , v ( n ) ( 2 τ ) p ∈ F 3 n ∑ ω p ⋅ ( λ + v ) q ∈ F 3 n ∑ ω q ⋅ ( u − κ ) . For x ∈ F 3 n x\in\mathbb{F}_3^n x ∈ F 3 n
∑ p ∈ F 3 n ω p ⋅ x = { 3 n , x = 0 , 0 , x ≠ 0. \sum_{p\in\mathbb{F}_3^n}\omega^{p\cdot x}=
\begin{cases}
3^n,&x=0,\\
0,&x\ne0.
\end{cases} p ∈ F 3 n ∑ ω p ⋅ x = { 3 n , 0 , x = 0 , x = 0. Consequently, only ( u , v ) = ( κ , − λ ) (u,v)=(\kappa,-\lambda) ( u , v ) = ( κ , − λ ) 3 n 3^n 3 n
∑ a , b ∈ F 3 n ω λ ⋅ a − κ ⋅ b Θ a , b ( n ) ( τ , z ) Θ a , − b ( n ) ( 2 τ , 2 z ) = 3 n Θ κ , − λ ( n ) ( τ , C z ) θ κ , − λ ( n ) ( 2 τ ) . \sum_{a,b\in\mathbb{F}_3^n}\omega^{\lambda\cdot a-\kappa\cdot b}\Theta_{a,b}^{(n)}(\tau,z)\Theta_{a,-b}^{(n)}(2\tau,2z)=3^n\Theta_{\kappa,-\lambda}^{(n)}(\tau,Cz)\theta_{\kappa,-\lambda}^{(n)}(2\tau). a , b ∈ F 3 n ∑ ω λ ⋅ a − κ ⋅ b Θ a , b ( n ) ( τ , z ) Θ a , − b ( n ) ( 2 τ , 2 z ) = 3 n Θ κ , − λ ( n ) ( τ , C z ) θ κ , − λ ( n ) ( 2 τ ) . Since κ ⋅ ( − λ ) = − λ ⋅ κ ≠ 0 \kappa\cdot(-\lambda)=-\lambda\cdot\kappa\ne0 κ ⋅ ( − λ ) = − λ ⋅ κ = 0 2 τ 2\tau 2 τ τ \tau τ
The square analogue of Schultz’s identity
We retain the notation of Subsection 3.2. For u = ( u 1 , u 2 ) T u=(u_1,u_2)^T u = ( u 1 , u 2 ) T
u ∗ = ( u 1 − u 2 − u 2 ) . (58) u^*=\begin{pmatrix}u_1-u_2\\-u_2\end{pmatrix}. \tag*{(58)} u ∗ = ( u 1 − u 2 − u 2 ) . (58) Theorem 4.3. Let τ ∈ H 1 \tau\in\mathfrak{H}_1 τ ∈ H 1 u ∈ C 2 u\in\mathbb{C}^2 u ∈ C 2 r , s ∈ F 3 r,s\in\mathbb{F}_3 r , s ∈ F 3
T r , s ( u ∣ τ ) T r , s ( 2 u ∗ ∣ 2 τ ) + T 0 , 0 ( u ∣ τ ) T 0 , 0 ( 2 u ∗ ∣ 2 τ ) = T r , 0 ( u ∣ τ ) T r , 0 ( 2 u ∗ ∣ 2 τ ) + T 0 , s ( u ∣ τ ) T 0 , s ( 2 u ∗ ∣ 2 τ ) . (59) T_{r,s}(u\mid\tau)T_{r,s}(2u^*\mid2\tau)+T_{0,0}(u\mid\tau)T_{0,0}(2u^*\mid2\tau)
=T_{r,0}(u\mid\tau)T_{r,0}(2u^*\mid2\tau)+T_{0,s}(u\mid\tau)T_{0,s}(2u^*\mid2\tau). \tag*{(59)} T r , s ( u ∣ τ ) T r , s ( 2 u ∗ ∣ 2 τ ) + T 0 , 0 ( u ∣ τ ) T 0 , 0 ( 2 u ∗ ∣ 2 τ ) = T r , 0 ( u ∣ τ ) T r , 0 ( 2 u ∗ ∣ 2 τ ) + T 0 , s ( u ∣ τ ) T 0 , s ( 2 u ∗ ∣ 2 τ ) . (59) Proof. For r , s ∈ F 3 r,s\in\mathbb{F}_3 r , s ∈ F 3
m ′ = m , n ′ = − m − n − r m'=m,\qquad n'=-m-n-r m ′ = m , n ′ = − m − n − r in (3.13) gives
T r , s ( u ∗ ∣ τ ) = T r , − s ( u ∣ τ ) . (60) T_{r,s}(u^*\mid\tau)=T_{r,-s}(u\mid\tau). \tag*{(60)} T r , s ( u ∗ ∣ τ ) = T r , − s ( u ∣ τ ) . (60) :::`
Indeed, if a = m + r / 3 a=m+r/3 a = m + r /3 b = n + r / 3 b=n+r/3 b = n + r /3 n ′ + r / 3 = − a − b n'+r/3=-a-b n ′ + r /3 = − a − b
Q ( a , b ) = Q ( a , − a − b ) , a ( u 1 − u 2 ) − b u 2 = a u 1 + ( − a − b ) u 2 . Q(a,b)=Q(a,-a-b),\qquad a(u_1-u_2)-bu_2=au_1+(-a-b)u_2. Q ( a , b ) = Q ( a , − a − b ) , a ( u 1 − u 2 ) − b u 2 = a u 1 + ( − a − b ) u 2 . Moreover,
ω r s + s ( m − n ) = ω − r s − s ( m ′ − n ′ ) , \omega^{rs+s(m-n)}=\omega^{-rs-s(m'-n')}, ω r s + s ( m − n ) = ω − r s − s ( m ′ − n ′ ) , because
[ r s + s ( m − n ) ] − [ − r s − s ( m ′ − n ′ ) ] = 3 s ( m + r ) ∈ 3 Z . [rs+s(m-n)]-[-rs-s(m'-n')]=3s(m+r)\in3\mathbb{Z}. [ r s + s ( m − n )] − [ − r s − s ( m ′ − n ′ )] = 3 s ( m + r ) ∈ 3 Z . This is exactly the character factor in T r , − s \mathcal{T}_{r,-s} T r , − s
In (55) , take n = 1 n=1 n = 1 z = z ( u ) z=z(\mathbf{u}) z = z ( u ) ( r , s ) (r,s) ( r , s ) ( 0 , 0 ) (0,0) ( 0 , 0 ) ( r , 0 ) (r,0) ( r , 0 ) ( 0 , s ) (0,s) ( 0 , s )
∑ a , b ∈ F 3 ( 1 − ω s a ) ( 1 − ω r b ) Θ a , b ( 1 ) ( τ , C z ( u ) ) θ a , b ( 1 ) ( 2 τ ) . \sum_{a,b\in\mathbb{F}_3}(1-\omega^{sa})(1-\omega^{rb})\Theta^{(1)}_{a,b}(\tau,Cz(\mathbf{u}))\theta^{(1)}_{a,b}(2\tau). a , b ∈ F 3 ∑ ( 1 − ω s a ) ( 1 − ω r b ) Θ a , b ( 1 ) ( τ , C z ( u )) θ a , b ( 1 ) ( 2 τ ) . Every summand vanishes: its coefficient is zero if s a = 0 sa=0 s a = 0 r b = 0 rb=0 r b = 0 a , b ≠ 0 a,b\ne0 a , b = 0 (42) applies.
The normalization in (45) cancels between the two factors on the left-hand side of (55) . Thus the instance indexed by ( p , q ) (p,q) ( p , q )
3 T p , q ( u ∣ τ ) T p , − q ( 2 u ∣ 2 τ ) . 3\mathcal{T}_{p,q}(\mathbf{u}\mid\tau)\mathcal{T}_{p,-q}(2\mathbf{u}\mid2\tau). 3 T p , q ( u ∣ τ ) T p , − q ( 2 u ∣ 2 τ ) . By (60) , this equals
3 T p , q ( u ∣ τ ) T p , q ( 2 u ∗ ∣ 2 τ ) . 3\mathcal{T}_{p,q}(\mathbf{u}\mid\tau)\mathcal{T}_{p,q}(2\mathbf{u}^{*}\mid2\tau). 3 T p , q ( u ∣ τ ) T p , q ( 2 u ∗ ∣ 2 τ ) . The indicated linear combination is therefore three times the difference between the two sides of (59) , which proves the result. □
Remark 4.4. Taking r = s = 1 r=s=1 r = s = 1 Theorem 3.4 , formula (59) becomes exactly the single-subscript identity (8) in Schultz’s notation [15 pp. 621–622] .
Borwein, Borwein, and Garvan [3 Theorem 2.6(i)] proved the quadratic modular equation obtained by taking r = s = 1 r=s=1 r = s = 1 u = 0 u=0 u = 0 Theorem 4.3 . The same authors [3 Theorem 2.6(ii)] stated the equivalent form obtained by combining this equation with (48) .
Corollary 4.5. For ∣ q ∣ < 1 |q|<1 ∣ q ∣ < 1
a ( q ) a ( q 2 ) = b ( q ) b ( q 2 ) + c ( q ) c ( q 2 ) . (61) a(q)a(q^2)=b(q)b(q^2)+c(q)c(q^2). \tag*{(61)} a ( q ) a ( q 2 ) = b ( q ) b ( q 2 ) + c ( q ) c ( q 2 ) . (61) Moreover, if a ( q ) a ( q 2 ) ≠ 0 a(q)a(q^2)\ne0 a ( q ) a ( q 2 ) = 0 λ = c ( q ) / a ( q ) \lambda=c(q)/a(q) λ = c ( q ) / a ( q ) ν = c ( q 2 ) / a ( q 2 ) \nu=c(q^2)/a(q^2) ν = c ( q 2 ) / a ( q 2 )
( 1 − λ ν ) 3 = ( 1 − λ 3 ) ( 1 − ν 3 ) . (1-\lambda\nu)^3=(1-\lambda^3)(1-\nu^3). ( 1 − λ ν ) 3 = ( 1 − λ 3 ) ( 1 − ν 3 ) . Proof. For r = s = 1 r=s=1 r = s = 1 u = 0 \mathbf{u}=0 u = 0 u ∗ = 0 \mathbf{u}^{*}=0 u ∗ = 0 T 1 , 1 \mathcal{T}_{1,1} T 1 , 1 Theorem 3.4 and (42) . Formula (61) follows from this fact, (47) , and (59) . The second conclusion follows from (48) and (61) . □
Degree two
We retain the notation of Subsection 3.3. Define
X a , b = θ a , b ( 2 ) ( τ ) θ a , b ( 2 ) ( 2 τ ) . X_{a,b}=\theta^{(2)}_{a,b}(\tau)\theta^{(2)}_{a,b}(2\tau). X a , b = θ a , b ( 2 ) ( τ ) θ a , b ( 2 ) ( 2 τ ) . Specializing Corollary 4.2 to n = 2 n=2 n = 2 z = 0 z=0 z = 0 (43) in the second factor, gives
S λ , κ ( X ) = 0 ( λ ⋅ κ ≠ 0 ) . (62) S_{\lambda,\kappa}(X)=0\qquad(\lambda\cdot\kappa\ne0). \tag*{(62)} S λ , κ ( X ) = 0 ( λ ⋅ κ = 0 ) . (62) As in the preceding degree-two subsection, we suppress the superscript and write θ a , b ( τ ) = θ a , b ( 2 ) ( τ ) \theta_{a,b}(\tau)=\theta^{(2)}_{a,b}(\tau) θ a , b ( τ ) = θ a , b ( 2 ) ( τ )
Theorem 4.6. The seven relations (52) , (53) , and (54) remain valid after the uniform replacement
θ a , b ( τ ) 3 ⟼ θ a , b ( τ ) θ a , b ( 2 τ ) . (63) \theta_{a,b}(\tau)^3\longmapsto\theta_{a,b}(\tau)\theta_{a,b}(2\tau). \tag*{(63)} θ a , b ( τ ) 3 ⟼ θ a , b ( τ ) θ a , b ( 2 τ ) . (63) Proof. The array X X X (42) and (43) . Equation (62) makes the seven character sums in (51) vanish. Hence R ( X ) = 0 R(X)=0 R ( X ) = 0 (63) .
Theta identities of arbitrary order
Corollary 3.3 gives a cubic character relation on the root lattice A 2 A_2 A 2 A ℓ − 1 A_{\ell-1} A ℓ − 1 ℓ ≥ 3 \ell\ge3 ℓ ≥ 3 A ℓ − 1 A_{\ell-1} A ℓ − 1 ℓ \ell ℓ
The general same-order identity
5.1. The general same-order identity. The argument rests on a finite Fourier relation associated with a Coxeter transformation of A ℓ − 1 A_{\ell-1} A ℓ − 1
Fix ℓ ≥ 2 \ell\ge2 ℓ ≥ 2 d = ℓ − 1 d=\ell-1 d = ℓ − 1
M ℓ = ( 2 − 1 − 1 2 − 1 ⋱ ⋱ ⋱ − 1 2 − 1 − 1 2 ) d × d , r = ( 1 2 ⋯ d ) T , ξ = e ( 1 / ℓ ) . (64) M_{\ell}=
\begin{pmatrix}
2 & -1 & & & \\
-1 & 2 & -1 & & \\
& \ddots& \ddots& \ddots& \\
& & -1 & 2 & -1 \\
& & & -1 & 2
\end{pmatrix}_{d\times d},
\qquad
r=(1\ \ 2\ \ \cdots\ \ d)^{T},
\qquad
\xi=e(1/\ell). \tag*{(64)} M ℓ = 2 − 1 − 1 2 ⋱ − 1 ⋱ − 1 ⋱ 2 − 1 − 1 2 d × d , r = ( 1 2 ⋯ d ) T , ξ = e ( 1/ ℓ ) . (64) Write 1 = ( 1 , … , 1 ) T \mathbf{1}=(1,\ldots,1)^{T} 1 = ( 1 , … , 1 ) T d d d L L L
R = ( 0 0 ⋯ 0 − 1 1 0 ⋯ 0 − 1 0 1 ⋯ 0 − 1 ⋮ ⋱ ⋱ ⋮ ⋮ 0 0 ⋯ 1 − 1 ) d × d . (65) R=
\begin{pmatrix}
0 & 0 & \cdots& 0 & -1 \\
1 & 0 & \cdots& 0 & -1 \\
0 & 1 & \cdots& 0 & -1 \\
\vdots& \ddots& \ddots& \vdots& \vdots\\
0 & 0 & \cdots& 1 & -1
\end{pmatrix}_{d\times d}. \tag*{(65)} R = 0 1 0 ⋮ 0 0 0 1 ⋱ 0 ⋯ ⋯ ⋯ ⋱ ⋯ 0 0 0 ⋮ 1 − 1 − 1 − 1 ⋮ − 1 d × d . (65) (If ℓ = 2 \ell=2 ℓ = 2 R = ( − 1 ) R=(-1) R = ( − 1 )
L 1 = r , R T M ℓ R = M ℓ , ∣ det ( R − I d ) ∣ = ℓ , ( R − 1 − I d ) M ℓ − 1 = − L , I d − M ℓ L = R T , M ℓ r = ℓ e d , R − 1 r = r − ℓ e d , 1 T ( R − 1 − I d ) = − ℓ e d T , 1 T ( R − I d ) = − ℓ e d T . (66) \begin{aligned}
L\mathbf{1}&=r,
&
R^{T}M_{\ell}R&=M_{\ell},
&
\lvert\det(R-I_d)\rvert&=\ell,\\
(R^{-1}-I_d)M_{\ell}^{-1}&=-L,
&
I_d-M_{\ell}L&=R^{T},
&
M_{\ell}r&=\ell e_d,
&
R^{-1}r&=r-\ell e_d,\\
\mathbf{1}^{T}(R^{-1}-I_d)&=-\ell e_d^{T},
&
\mathbf{1}^{T}(R-I_d)&=-\ell e_d^{T}. \tag*{(66)}
\end{aligned} L 1 ( R − 1 − I d ) M ℓ − 1 1 T ( R − 1 − I d ) = r , = − L , = − ℓ e d T , R T M ℓ R I d − M ℓ L 1 T ( R − I d ) = M ℓ , = R T , = − ℓ e d T . ∣ det ( R − I d )∣ M ℓ r = ℓ , = ℓ e d , R − 1 r = r − ℓ e d , (66) They follow by direct multiplication. In particular, the integral matrix
E = ℓ ( R − I d ) − 1 , E i j = { i − ℓ , j ≤ i , i , j > i (67) E=\ell(R-I_d)^{-1},
\qquad
E_{ij}=
\begin{cases}
i-\ell, & j\le i,\\
i, & j>i
\end{cases} \tag*{(67)} E = ℓ ( R − I d ) − 1 , E ij = { i − ℓ , i , j ≤ i , j > i (67) satisfies (we identify Z d \mathbb{Z}^{d} Z d Z d , 1 \mathbb{Z}^{d,1} Z d , 1
( R − I d ) Z d = { x ∈ Z d : 1 T x ≡ 0 ( m o d ℓ ) } , E T Z d = ℓ Z d + Z 1. (68) (R-I_d)\mathbb{Z}^{d}=\{x\in\mathbb{Z}^{d}:\mathbf{1}^{T}x\equiv0\pmod{\ell}\},
\qquad
E^{T}\mathbb{Z}^{d}=\ell\mathbb{Z}^{d}+\mathbb{Z}\mathbf{1}. \tag*{(68)} ( R − I d ) Z d = { x ∈ Z d : 1 T x ≡ 0 ( mod ℓ )} , E T Z d = ℓ Z d + Z 1 . (68) Indeed, 1 T ( R − I d ) = − ℓ e d T \mathbf{1}^{T}(R-I_d)=-\ell e_d^{T} 1 T ( R − I d ) = − ℓ e d T (68) . Both lattices have index ℓ \ell ℓ Z d \mathbb{Z}^{d} Z d ∣ det ( R − I d ) ∣ = ℓ \lvert\det(R-I_d)\rvert=\ell ∣ det ( R − I d )∣ = ℓ x ↦ 1 T x ( m o d ℓ ) x\mapsto\mathbf{1}^{T}x\pmod{\ell} x ↦ 1 T x ( mod ℓ ) (67) shows that every row of E E E ℓ Z d + Z 1 \ell\mathbb{Z}^{d}+\mathbb{Z}\mathbf{1} ℓ Z d + Z 1
∣ det E ∣ = ℓ d − 1 and [ Z d : ℓ Z d + Z 1 ] = ℓ d − 1 . \lvert\det E\rvert=\ell^{d-1}
\qquad\text{and}\qquad
[\mathbb{Z}^{d}:\ell\mathbb{Z}^{d}+\mathbb{Z}\mathbf{1}]=\ell^{d-1}. ∣ det E ∣ = ℓ d − 1 and [ Z d : ℓ Z d + Z 1 ] = ℓ d − 1 . For n ≥ 1 n\ge1 n ≥ 1
X n = { x ∈ ( Z / ℓ Z ) d , n : 1 T x = 0 } , Y n = ( Z / ℓ Z ) d , n { 1 η : η ∈ ( Z / ℓ Z ) 1 , n } . (69) \mathcal{X}_{n}=\{x\in(\mathbb{Z}/\ell\mathbb{Z})^{d,n}:\mathbf{1}^{T}x=0\},
\qquad
\mathcal{Y}_{n}=
\frac{(\mathbb{Z}/\ell\mathbb{Z})^{d,n}}
{\{\mathbf{1}\eta:\eta\in(\mathbb{Z}/\ell\mathbb{Z})^{1,n}\}}. \tag*{(69)} X n = { x ∈ ( Z / ℓ Z ) d , n : 1 T x = 0 } , Y n = { 1 η : η ∈ ( Z / ℓ Z ) 1 , n } ( Z / ℓ Z ) d , n . (69) The denominator in Y n \mathcal{Y}_{n} Y n X n \mathcal{X}_{n} X n Y n \mathcal{Y}_{n} Y n ℓ ( d − 1 ) n \ell^{(d-1)n} ℓ ( d − 1 ) n
⟨ x , s ⟩ = ξ tr ( x T s ) . (70) \langle x,s\rangle=\xi^{\operatorname{tr}(x^{T}s)}. \tag*{(70)} ⟨ x , s ⟩ = ξ tr ( x T s ) . (70) for ( x , s ) ∈ X n × Y n (x,s) \in\mathcal{X}_n \times\mathcal{Y}_n ( x , s ) ∈ X n × Y n
Lemma 5.1. The formula (70) gives a well-defined bicharacter on X n × Y n \mathcal{X}_n \times\mathcal{Y}_n X n × Y n x ≠ 0 x \ne0 x = 0 ℓ ≥ 3 \ell\ge3 ℓ ≥ 3 s ∈ Y n s \in\mathcal{Y}_n s ∈ Y n ⟨ x , s ⟩ ≠ 1 \langle x,s\rangle\ne1 ⟨ x , s ⟩ = 1
Proof. For x ∈ X n x \in\mathcal{X}_n x ∈ X n y ∈ ( Z / ℓ Z ) d , n y \in(\mathbb{Z}/\ell\mathbb{Z})^{d,n} y ∈ ( Z / ℓ Z ) d , n ⟨ x , y ⟩ 0 = ξ tr ( x T y ) \langle x,y\rangle_0 = \xi^{\operatorname{tr}(x^T y)} ⟨ x , y ⟩ 0 = ξ tr ( x T y ) s = y + { 1 η : η ∈ ( Z / ℓ Z ) 1 , n } ∈ Y n s=y+\{1\eta:\eta\in(\mathbb{Z}/\ell\mathbb{Z})^{1,n}\}\in\mathcal{Y}_n s = y + { 1 η : η ∈ ( Z / ℓ Z ) 1 , n } ∈ Y n ⟨ x , s ⟩ = ⟨ x , y ⟩ 0 \langle x,s\rangle=\langle x,y\rangle_0 ⟨ x , s ⟩ = ⟨ x , y ⟩ 0 y ′ y' y ′ s s s y − y ′ = 1 η y-y'=1\eta y − y ′ = 1 η η ∈ ( Z / ℓ Z ) 1 , n \eta\in(\mathbb{Z}/\ell\mathbb{Z})^{1,n} η ∈ ( Z / ℓ Z ) 1 , n
⟨ x , y − y ′ ⟩ 0 = ξ tr ( x T ( y − y ′ ) ) = ξ tr ( x T ( 1 η ) ) = ξ tr ( ( x T 1 ) ⋅ η ) = 1 , \langle x,y-y'\rangle_0=\xi^{\operatorname{tr}(x^T(y-y'))}=\xi^{\operatorname{tr}(x^T(1\eta))}=\xi^{\operatorname{tr}((x^T1)\cdot\eta)}=1, ⟨ x , y − y ′ ⟩ 0 = ξ tr ( x T ( y − y ′ )) = ξ tr ( x T ( 1 η )) = ξ tr (( x T 1 ) ⋅ η ) = 1 , since 1 T x = 0 1^T x=0 1 T x = 0 ( Z / ℓ Z ) 1 , n (\mathbb{Z}/\ell\mathbb{Z})^{1,n} ( Z / ℓ Z ) 1 , n X n \mathcal{X}_n X n ⟨ x , s ⟩ \langle x,s\rangle ⟨ x , s ⟩ s s s (70) is well defined on X n × Y n \mathcal{X}_n \times\mathcal{Y}_n X n × Y n y ∈ ( Z / ℓ Z ) d , n y\in(\mathbb{Z}/\ell\mathbb{Z})^{d,n} y ∈ ( Z / ℓ Z ) d , n tr ( x T y ) ≢ 0 ( m o d ℓ ) \operatorname{tr}(x^T y)\not\equiv0\pmod{\ell} tr ( x T y ) ≡ 0 ( mod ℓ ) x ≠ 0 x\ne0 x = 0 ⟨ x , y ⟩ 0 ≠ 1 \langle x,y\rangle_0\ne1 ⟨ x , y ⟩ 0 = 1 ⟨ x , s ⟩ ≠ 1 \langle x,s\rangle\ne1 ⟨ x , s ⟩ = 1 s = y + { 1 η : η ∈ ( Z / ℓ Z ) 1 , n } s=y+\{1\eta:\eta\in(\mathbb{Z}/\ell\mathbb{Z})^{1,n}\} s = y + { 1 η : η ∈ ( Z / ℓ Z ) 1 , n }
We need the following character projection relation.
Lemma 5.2. Suppose that a family F x , s F_{x,s} F x , s x ∈ ( Z / ℓ Z ) d , n x\in(\mathbb{Z}/\ell\mathbb{Z})^{d,n} x ∈ ( Z / ℓ Z ) d , n s ∈ Y n s\in\mathcal{Y}_n s ∈ Y n X n \mathcal{X}_n X n Y n \mathcal{Y}_n Y n (69) , and satisfies
ℓ ( d − 1 ) n F R − 1 x , t = ∑ y ∈ x + X n s ∈ Y n ⟨ y − x , s − t ⟩ F y , s , (71) \ell^{(d-1)n}F_{R^{-1}x,t}=\sum_{\substack{y\in x+\mathcal{X}_n\\s\in\mathcal{Y}_n}}\langle y-x,s-t\rangle F_{y,s}, \tag*{(71)} ℓ ( d − 1 ) n F R − 1 x , t = y ∈ x + X n s ∈ Y n ∑ ⟨ y − x , s − t ⟩ F y , s , (71) for every x ∈ ( Z / ℓ Z ) d , n x\in(\mathbb{Z}/\ell\mathbb{Z})^{d,n} x ∈ ( Z / ℓ Z ) d , n t ∈ Y n t\in\mathcal{Y}_n t ∈ Y n λ , κ ∈ ( Z / ℓ Z ) 1 , n \lambda,\kappa\in(\mathbb{Z}/\ell\mathbb{Z})^{1,n} λ , κ ∈ ( Z / ℓ Z ) 1 , n 3
r κ ∈ X n , λ κ T ≠ 0. (72) r\kappa\in\mathcal{X}_n,\qquad\lambda\kappa^T\ne0. \tag*{(72)} r κ ∈ X n , λ κ T = 0. (72) Then
∑ a ∈ ( Z / ℓ Z ) 1 , n t ∈ Y n ξ λ a T ⟨ r κ , t ⟩ F r a , t = 0. (73) \sum_{\substack{a\in(\mathbb{Z}/\ell\mathbb{Z})^{1,n}\\t\in\mathcal{Y}_n}}\xi^{\lambda a^T}\langle r\kappa,t\rangle F_{ra,t}=0. \tag*{(73)} a ∈ ( Z / ℓ Z ) 1 , n t ∈ Y n ∑ ξ λ a T ⟨ r κ , t ⟩ F r a , t = 0. (73) Proof. Since R − 1 r = r − ℓ e d R^{-1}r=r-\ell e_d R − 1 r = r − ℓ e d F R − 1 r a , t = F r a , t F_{R^{-1}ra,t}=F_{ra,t} F R − 1 r a , t = F r a , t (71) , with x = r a x=ra x = r a ξ λ a T ⟨ r κ , t ⟩ \xi^{\lambda a^T}\langle r\kappa,t\rangle ξ λ a T ⟨ r κ , t ⟩ a a a t t t Y n \mathcal{Y}_n Y n
∑ t ∈ Y n ⟨ r κ , t ⟩ ⟨ y − r a , − t ⟩ = ∑ t ∈ Y n ⟨ r κ − ( y − r a ) , t ⟩ = { ℓ ( d − 1 ) n , y − r a = r κ , 0 , otherwise . \sum_{t\in\mathcal{Y}_n}\langle r\kappa,t\rangle\langle y-ra,-t\rangle
=\sum_{t\in\mathcal{Y}_n}\langle r\kappa-(y-ra),t\rangle
=\begin{cases}
\ell^{(d-1)n},&y-ra=r\kappa,\\
0,&\text{otherwise}.
\end{cases} t ∈ Y n ∑ ⟨ r κ , t ⟩ ⟨ y − r a , − t ⟩ = t ∈ Y n ∑ ⟨ r κ − ( y − r a ) , t ⟩ = { ℓ ( d − 1 ) n , 0 , y − r a = r κ , otherwise . The value y = r ( a + κ ) y=r(a+\kappa) y = r ( a + κ ) r a + X n ra+\mathcal{X}_n r a + X n (72) . Writing S S S (73) , the left-hand side of the summed Fourier relation is ℓ ( d − 1 ) n S \ell^{(d-1)n}S ℓ ( d − 1 ) n S
ℓ ( d − 1 ) n ∑ a , s ξ λ a T ⟨ r κ , s ⟩ F r ( a + κ ) , s = ℓ ( d − 1 ) n ξ − λ κ T S . \ell^{(d-1)n}\sum_{a,s}\xi^{\lambda a^T}\langle r\kappa,s\rangle F_{r(a+\kappa),s}
=\ell^{(d-1)n}\xi^{-\lambda\kappa^T}S. ℓ ( d − 1 ) n a , s ∑ ξ λ a T ⟨ r κ , s ⟩ F r ( a + κ ) , s = ℓ ( d − 1 ) n ξ − λ κ T S . Since ξ − λ κ T ≠ 1 \xi^{-\lambda\kappa^T}\ne1 ξ − λ κ T = 1 S = 0 S=0 S = 0
Since 1 T r = ℓ ( ℓ − 1 ) / 2 1^T r=\ell(\ell-1)/2 1 T r = ℓ ( ℓ − 1 ) /2 (72) is automatic when ℓ \ell ℓ ℓ \ell ℓ κ \kappa κ 2 Z / ℓ Z 2\mathbb{Z}/\ell\mathbb{Z} 2 Z / ℓ Z λ \lambda λ λ + ( ℓ / 2 ) ε \lambda+(\ell/2)\varepsilon λ + ( ℓ /2 ) ε ε ∈ { 0 , 1 } 1 , n \varepsilon\in\{0,1\}^{1,n} ε ∈ { 0 , 1 } 1 , n (72) . This observation gives the parity refinement in the following theorem, which is the main result of this section. We identify ( Z / ℓ Z ) 1 , n (\mathbb{Z}/\ell\mathbb{Z})^{1,n} ( Z / ℓ Z ) 1 , n ( Z / ℓ Z ) n (\mathbb{Z}/\ell\mathbb{Z})^n ( Z / ℓ Z ) n λ ⋅ κ = λ κ T \lambda\cdot\kappa=\lambda\kappa^T λ ⋅ κ = λ κ T
Theorem 5.3. Let ℓ ≥ 3 \ell\ge3 ℓ ≥ 3 n ≥ 1 n \ge1 n ≥ 1 τ ∈ H n \tau\in\mathfrak{H}_n τ ∈ H n z ∈ C d , n z \in\mathbb{C}^{d,n} z ∈ C d , n d = ℓ − 1 d=\ell-1 d = ℓ − 1 λ , κ ∈ ( Z / ℓ Z ) n \lambda,\kappa\in(\mathbb{Z}/\ell\mathbb{Z})^n λ , κ ∈ ( Z / ℓ Z ) n λ ⋅ κ ≢ 0 ( m o d ℓ ) \lambda\cdot\kappa\not\equiv0 \pmod{\ell} λ ⋅ κ ≡ 0 ( mod ℓ ) ℓ \ell ℓ κ \kappa κ 2 Z / ℓ Z 2\mathbb{Z}/\ell\mathbb{Z} 2 Z / ℓ Z
∑ a , b 1 , … , b d − 1 ∈ ( Z / ℓ Z ) n ξ λ ⋅ a + κ ⋅ ( b 1 + ⋯ + b d − 1 ) Θ M ℓ , n ℓ [ r a ℓ B ℓ ] ( τ , z ) = 0. (74) \sum_{\substack{a,b_1,\ldots,b_{d-1}\in(\mathbb{Z}/\ell\mathbb{Z})^n}}
\xi^{\lambda\cdot a+\kappa\cdot(b_1+\cdots+b_{d-1})}
\Theta_{M_{\ell,n}}^\ell\left[\begin{matrix} \frac{ra}{\ell}\\ \frac{B}{\ell}\end{matrix}\right](\tau,z)=0. \tag*{(74)} a , b 1 , … , b d − 1 ∈ ( Z / ℓ Z ) n ∑ ξ λ ⋅ a + κ ⋅ ( b 1 + ⋯ + b d − 1 ) Θ M ℓ , n ℓ [ ℓ r a ℓ B ] ( τ , z ) = 0. (74) Here B = [ b 1 , … , b d − 1 , 0 ] B=[b_1,\ldots,b_{d-1},0] B = [ b 1 , … , b d − 1 , 0 ] ℓ \ell ℓ a 0 ∈ ( Z / ℓ Z ) n a_0\in(\mathbb{Z}/\ell\mathbb{Z})^n a 0 ∈ ( Z / ℓ Z ) n
∑ b 1 , … , b d − 1 ∈ ( Z / ℓ Z ) n a ∈ a 0 + ( 2 Z / ℓ Z ) n ξ λ ⋅ a + κ ⋅ ( b 1 + ⋯ + b d − 1 ) Θ M ℓ , n ℓ [ r a ℓ B ℓ ] ( τ , z ) = 0. (75) \sum_{\substack{b_1,\ldots,b_{d-1}\in(\mathbb{Z}/\ell\mathbb{Z})^n\\a\in a_0+(2\mathbb{Z}/\ell\mathbb{Z})^n}}
\xi^{\lambda\cdot a+\kappa\cdot(b_1+\cdots+b_{d-1})}
\Theta_{M_{\ell,n}}^\ell\left[\begin{matrix} \frac{ra}{\ell}\\ \frac{B}{\ell}\end{matrix}\right](\tau,z)=0. \tag*{(75)} b 1 , … , b d − 1 ∈ ( Z / ℓ Z ) n a ∈ a 0 + ( 2 Z / ℓ Z ) n ∑ ξ λ ⋅ a + κ ⋅ ( b 1 + ⋯ + b d − 1 ) Θ M ℓ , n ℓ [ ℓ r a ℓ B ] ( τ , z ) = 0. (75) Proof. Throughout the proof, we use the notation introduced at the beginning of this subsection and the properties in (66) . For x ∈ ( Z / ℓ Z ) d , n x\in(\mathbb{Z}/\ell\mathbb{Z})^{d,n} x ∈ ( Z / ℓ Z ) d , n s ∈ Y n s\in\mathcal{Y}_n s ∈ Y n
γ s = − L s ℓ , F x , s = Θ M ℓ , n ℓ [ x ℓ γ s ] ( τ , z ) . \gamma_s=-\frac{Ls}{\ell},\qquad
F_{x,s}=\Theta_{M_{\ell,n}}^\ell\left[\begin{matrix}\frac{x}{\ell}\\ \gamma_s\end{matrix}\right](\tau,z). γ s = − ℓ L s , F x , s = Θ M ℓ , n ℓ [ ℓ x γ s ] ( τ , z ) . Changing an integral lift of x x x ℓ h \ell h ℓ h h h h s ↦ s + ℓ k s\mapsto s+\ell k s ↦ s + ℓ k γ s \gamma_s γ s − L k -Lk − L k Lemma 2.6 , the resulting factor before taking the ℓ \ell ℓ
ξ − tr ( x T M ℓ L k ) . \xi^{-\operatorname{tr}(x^T M_\ell Lk)}. ξ − tr ( x T M ℓ L k ) . Replacing s s s s + 1 η s+\mathbf{1}\eta s + 1 η γ s \gamma_s γ s − r η / ℓ = M ℓ − 1 ( − e d η ) -r\eta/\ell=M_\ell^{-1}(-e_d\eta) − r η / ℓ = M ℓ − 1 ( − e d η )
ξ − tr ( x T e d η ) . \xi^{-\operatorname{tr}(x^T e_d\eta)}. ξ − tr ( x T e d η ) . Both factors disappear after taking the ℓ \ell ℓ F x , s F_{x,s} F x , s x x x s s s
We apply Theorem 2.7 with
m = d , μ = ℓ , M 1 = M 2 = M ℓ , A = I d ℓ + 1 ℓ J ℓ ⊗ ( R − I d ) , m=d,\qquad\mu=\ell,\qquad M_1=M_2=M_\ell,\qquad
A=I_{d\ell}+\frac{1}{\ell}J_\ell\otimes(R-I_d), m = d , μ = ℓ , M 1 = M 2 = M ℓ , A = I d ℓ + ℓ 1 J ℓ ⊗ ( R − I d ) , where J ℓ J_\ell J ℓ ℓ × ℓ \ell\times\ell ℓ × ℓ A T M ℓ ( ℓ ) A = M ℓ ( ℓ ) A^T M_\ell^{(\ell)}A=M_\ell^{(\ell)} A T M ℓ ( ℓ ) A = M ℓ ( ℓ )
The integral matrix A ~ \widetilde{A} A
A ~ ⋅ ( x 1 , … , x ℓ ) T = ( x 1 − x ℓ , … , x ℓ − 1 − x ℓ , x ℓ E T ) T , for all x 1 , … , x ℓ ∈ Z 1 , d . \widetilde{A}\cdot(x_1,\ldots,x_\ell)^T=(x_1-x_\ell,\ldots,x_{\ell-1}-x_\ell,x_\ell E^T)^T,
\qquad\text{for all }x_1,\ldots,x_\ell\in\mathbb{Z}^{1,d}. A ⋅ ( x 1 , … , x ℓ ) T = ( x 1 − x ℓ , … , x ℓ − 1 − x ℓ , x ℓ E T ) T , for all x 1 , … , x ℓ ∈ Z 1 , d . Since E − 1 = ( R − I d ) / ℓ E^{-1}=(R-I_d)/\ell E − 1 = ( R − I d ) / ℓ Z d ℓ , n \mathbb{Z}^{d\ell,n} Z d ℓ , n
A ~ − 1 Z d ℓ , n = Z d ℓ , n + { [ ( R − I d ) u ℓ , … , ( R − I d ) u ℓ ] : u ∈ Z d , n } = Z d ℓ , n + A Z d ℓ , n . \widetilde{A}^{-1}\mathbb{Z}^{d\ell,n}
=
\mathbb{Z}^{d\ell,n}
+
\left\{
\left[
\frac{(R-I_d)u}{\ell},\ldots,\frac{(R-I_d)u}{\ell}
\right]:
u\in\mathbb{Z}^{d,n}
\right\}
=
\mathbb{Z}^{d\ell,n}+A\mathbb{Z}^{d\ell,n}. A − 1 Z d ℓ , n = Z d ℓ , n + { [ ℓ ( R − I d ) u , … , ℓ ( R − I d ) u ] : u ∈ Z d , n } = Z d ℓ , n + A Z d ℓ , n . All the hypotheses of Theorem 2.7 are therefore satisfied, and ∣ det A ~ ∣ = ∣ det E ∣ = ℓ d − 1 |\det\widetilde{A}|=|\det E|=\ell^{d-1} ∣ det A ∣ = ∣ det E ∣ = ℓ d − 1
The two quotient groups in Theorem 2.7 are parametrized by X n \mathcal{X}_n X n Y n \mathcal{Y}_n Y n
X n ⟶ ( A ~ A ) − 1 Z d ℓ , n Z d ℓ , n , v ⟼ [ v ℓ , … , v ℓ ] + Z d ℓ , n . \mathcal{X}_n\longrightarrow
\frac{(\widetilde{A}A)^{-1}\mathbb{Z}^{d\ell,n}}{\mathbb{Z}^{d\ell,n}},
\qquad
v\longmapsto
\left[\frac{v}{\ell},\ldots,\frac{v}{\ell}\right]+\mathbb{Z}^{d\ell,n}. X n ⟶ Z d ℓ , n ( A A ) − 1 Z d ℓ , n , v ⟼ [ ℓ v , … , ℓ v ] + Z d ℓ , n . Changing the lift of v v v
A ~ A [ v ℓ , … , v ℓ ] = A ~ [ R v ℓ , … , R v ℓ ] = [ 0 , … , 0 , ( R − I d ) − 1 R v ] , \widetilde{A}A
\left[\frac{v}{\ell},\ldots,\frac{v}{\ell}\right]
=
\widetilde{A}
\left[\frac{Rv}{\ell},\ldots,\frac{Rv}{\ell}\right]
=
[0,\ldots,0,(R-I_d)^{-1}Rv], A A [ ℓ v , … , ℓ v ] = A [ ℓ R v , … , ℓ R v ] = [ 0 , … , 0 , ( R − I d ) − 1 R v ] , which is integral by the first equality in (68) . Thus the map is well defined and is a homomorphism. Its kernel is trivial, and the target has order ∣ det ( A ~ A ) ∣ n = ℓ ( d − 1 ) n = ∣ X n ∣ |\det(\widetilde{A}A)|^n=\ell^{(d-1)n}=|\mathcal{X}_n| ∣ det ( A A ) ∣ n = ℓ ( d − 1 ) n = ∣ X n ∣
For the second quotient, define
β ( u ) = [ γ u , … , γ u , γ u + M ℓ − 1 u ] . \beta(u)=[\gamma_u,\ldots,\gamma_u,\gamma_u+M_\ell^{-1}u]. β ( u ) = [ γ u , … , γ u , γ u + M ℓ − 1 u ] . Then
M ℓ ( ℓ ) A β ( u ) = [ 0 , … , 0 , u ] . M_{\ell}^{(\ell)}A\beta(u)=[0,\ldots,0,u]. M ℓ ( ℓ ) A β ( u ) = [ 0 , … , 0 , u ] . Since [ 0 , … , 0 , u ] ∈ A ~ T Z d , ℓ n [0,\ldots,0,u]\in\widetilde{A}^{T}\mathbb{Z}^{d,\ell n} [ 0 , … , 0 , u ] ∈ A T Z d , ℓ n u ∈ E T Z d , n u\in E^{T}\mathbb{Z}^{d,n} u ∈ E T Z d , n (68) shows that the class of β ( u ) \beta(u) β ( u ) u u u Y n \mathcal{Y}_{n} Y n
Y n ⟶ A − 1 M ℓ ( ℓ ) − 1 Z d , ℓ n A − 1 M ℓ ( ℓ ) − 1 A ~ T Z d , ℓ n , s ⟼ β ( s ) + A − 1 M ℓ ( ℓ ) − 1 A ~ T Z d , ℓ n \mathcal{Y}_{n}\longrightarrow\frac{A^{-1}M_{\ell}^{(\ell)-1}\mathbb{Z}^{d,\ell n}}{A^{-1}M_{\ell}^{(\ell)-1}\widetilde{A}^{T}\mathbb{Z}^{d,\ell n}},\qquad s\longmapsto\beta(s)+A^{-1}M_{\ell}^{(\ell)-1}\widetilde{A}^{T}\mathbb{Z}^{d,\ell n} Y n ⟶ A − 1 M ℓ ( ℓ ) − 1 A T Z d , ℓ n A − 1 M ℓ ( ℓ ) − 1 Z d , ℓ n , s ⟼ β ( s ) + A − 1 M ℓ ( ℓ ) − 1 A T Z d , ℓ n is injective. Its target has order [ Z d , ℓ n : A ~ T Z d , ℓ n ] = ℓ ( d − 1 ) n = ∣ Y n ∣ [\mathbb{Z}^{d,\ell n}:\widetilde{A}^{T}\mathbb{Z}^{d,\ell n}]=\ell^{(d-1)n}=|\mathcal{Y}_{n}| [ Z d , ℓ n : A T Z d , ℓ n ] = ℓ ( d − 1 ) n = ∣ Y n ∣
For fixed x ∈ ( Z / ℓ Z ) d , n x\in(\mathbb{Z}/\ell\mathbb{Z})^{d,n} x ∈ ( Z / ℓ Z ) d , n t ∈ Y n t\in\mathcal{Y}_{n} t ∈ Y n Theorem 2.7 by taking
a i = x ℓ , b i = R γ t , z i = R z ( 1 ≤ i ≤ ℓ ) . a_i=\frac{x}{\ell},\qquad b_i=R\gamma_t,\qquad z_i=Rz\qquad(1\leq i\leq\ell). a i = ℓ x , b i = R γ t , z i = R z ( 1 ≤ i ≤ ℓ ) . Then a i ′ = R − 1 x / ℓ a_i'=R^{-1}x/\ell a i ′ = R − 1 x / ℓ b i ′ = γ t b_i'=\gamma_t b i ′ = γ t w i = z w_i=z w i = z
Θ M ℓ , n [ u , R v ] ( τ , R z ) = Θ M ℓ , n [ R − 1 u , v ] ( τ , z ) \Theta_{M_{\ell,n}}\left[u,Rv\right](\tau,Rz)=\Theta_{M_{\ell,n}}\left[R^{-1}u,v\right](\tau,z) Θ M ℓ , n [ u , R v ] ( τ , R z ) = Θ M ℓ , n [ R − 1 u , v ] ( τ , z ) shows that the product on the left-hand side of (14) is F R − 1 x , t F_{R^{-1}x,t} F R − 1 x , t y = R − 1 x + v y=R^{-1}x+v y = R − 1 x + v s = t + u s=t+u s = t + u (15) and (16) :
[ y − R − 1 x ℓ , … , y − R − 1 x ℓ ] , y ∈ R − 1 x + X n , \left[\frac{y-R^{-1}x}{\ell},\ldots,\frac{y-R^{-1}x}{\ell}\right],\qquad y\in R^{-1}x+\mathcal{X}_{n}, [ ℓ y − R − 1 x , … , ℓ y − R − 1 x ] , y ∈ R − 1 x + X n , and
[ γ u , … , γ u , γ u + M ℓ − 1 u ] , u ∈ Y n . [\gamma_u,\ldots,\gamma_u,\gamma_u+M_{\ell}^{-1}u],\qquad u\in\mathcal{Y}_{n}. [ γ u , … , γ u , γ u + M ℓ − 1 u ] , u ∈ Y n . The identity I d − M ℓ L = R T I_d-M_{\ell}L=R^{T} I d − M ℓ L = R T
M ℓ ( ℓ γ u + M ℓ − 1 u ) = R T u . M_{\ell}(\ell\gamma_u+M_{\ell}^{-1}u)=R^{T}u. M ℓ ( ℓ γ u + M ℓ − 1 u ) = R T u . The exterior factor on the right-hand side of (14) is therefore
ε ( − tr ( ∑ i = 1 ℓ a i ′ T M ℓ β i ) ) = ξ − tr ( x T u ) , \varepsilon\left(-\operatorname{tr}\left(\sum_{i=1}^{\ell}a_i^{\prime T}M_{\ell}\beta_i\right)\right)=\xi^{-\operatorname{tr}(x^{T}u)}, ε ( − tr ( i = 1 ∑ ℓ a i ′ T M ℓ β i ) ) = ξ − tr ( x T u ) , whereas removing M ℓ − 1 u M_{\ell}^{-1}u M ℓ − 1 u ξ tr ( y T u ) \xi^{\operatorname{tr}(y^{T}u)} ξ tr ( y T u ) ( y , s ) (y,s) ( y , s ) (14) is therefore
ξ − tr ( x T u ) ξ tr ( y T u ) Θ M ℓ , n ℓ [ y ℓ , γ s ] ( τ , z ) = ⟨ y − x , s − t ⟩ F y , s . \xi^{-\operatorname{tr}(x^{T}u)}\xi^{\operatorname{tr}(y^{T}u)}\Theta_{M_{\ell,n}}^{\ell}\left[\frac{y}{\ell},\gamma_s\right](\tau,z)=\langle y-x,s-t\rangle F_{y,s}. ξ − tr ( x T u ) ξ tr ( y T u ) Θ M ℓ , n ℓ [ ℓ y , γ s ] ( τ , z ) = ⟨ y − x , s − t ⟩ F y , s . Consequently, Theorem 2.7 gives
ℓ ( d − 1 ) n F R − 1 x , t = ∑ y ∈ R − 1 x + X n s ∈ Y n ⟨ y − x , s − t ⟩ F y , s . \ell^{(d-1)n}F_{R^{-1}x,t}=\sum_{\substack{y\in R^{-1}x+\mathcal{X}_{n}\\s\in\mathcal{Y}_{n}}}\langle y-x,s-t\rangle F_{y,s}. ℓ ( d − 1 ) n F R − 1 x , t = y ∈ R − 1 x + X n s ∈ Y n ∑ ⟨ y − x , s − t ⟩ F y , s . Since 1 T ( R − 1 − I d ) = − ℓ e 1 T 1^{T}(R^{-1}-I_d)=-\ell e_1^{T} 1 T ( R − 1 − I d ) = − ℓ e 1 T ( R − 1 − I d ) x ∈ X n (R^{-1}-I_d)x\in\mathcal{X}_{n} ( R − 1 − I d ) x ∈ X n x + X n x+\mathcal{X}_{n} x + X n
ℓ ( d − 1 ) n F R − 1 x , t = ∑ y ∈ x + X n s ∈ Y n ⟨ y − x , s − t ⟩ F y , s . (76) \ell^{(d-1)n}F_{R^{-1}x,t}=\sum_{\substack{y\in x+\mathcal{X}_{n}\\s\in\mathcal{Y}_{n}}}\langle y-x,s-t\rangle F_{y,s}. \tag*{(76)} ℓ ( d − 1 ) n F R − 1 x , t = y ∈ x + X n s ∈ Y n ∑ ⟨ y − x , s − t ⟩ F y , s . (76) which is exactly the relation required in (71) . The two conditions in (72) are also satisfied: the first by the argument in the paragraph preceding Theorem 5.3 , and the second by assumption. Lemma 5.2 therefore yields (73) for F x , s = Θ M ℓ , n ℓ [ x ℓ , γ s ] ( τ , z ) F_{x,s}=\Theta_{M_{\ell,n}}^{\ell}\left[\frac{x}{\ell},\gamma_s\right](\tau,z) F x , s = Θ M ℓ , n ℓ [ ℓ x , γ s ] ( τ , z )
It remains to express the character projection in the stated coordinates. Replacing s s s s + 1 η s+1\eta s + 1 η B = − L s B=-Ls B = − L s − r η -r\eta − r η − d η -d\eta − d η d ≡ − 1 d\equiv-1 d ≡ − 1 ℓ \ell ℓ s ∈ Y n s \in\mathcal{Y}_n s ∈ Y n B B B
γ s = B ℓ , r T s = − r T L − 1 B = 1 T B ( m o d ℓ ) . (77) \gamma_s=\frac{B}{\ell},\qquad r^T s=-r^T L^{-1}B=\mathbf{1}^T B\pmod{\ell}. \tag*{(77)} γ s = ℓ B , r T s = − r T L − 1 B = 1 T B ( mod ℓ ) . (77) The result now follows by combining (77) with (73) .
When ℓ \ell ℓ λ + ( ℓ / 2 ) ε \lambda+(\ell/2)\varepsilon λ + ( ℓ /2 ) ε ε ∈ { 0 , 1 } n \varepsilon\in\{0,1\}^n ε ∈ { 0 , 1 } n κ \kappa κ (74) by S λ , κ S_{\lambda,\kappa} S λ , κ S λ + ( ℓ / 2 ) ε , κ = 0 S_{\lambda+(\ell/2)\varepsilon,\kappa}=0 S λ + ( ℓ /2 ) ε , κ = 0 ε ∈ { 0 , 1 } n \varepsilon\in\{0,1\}^n ε ∈ { 0 , 1 } n
0 = ∑ ε ∈ ( Z / 2 Z ) n a ∈ ( Z / ℓ Z ) ( − 1 ) − a 0 ⋅ ε S λ + ( ℓ / 2 ) ε , κ = ∑ ε ∈ ( Z / 2 Z ) n a ∈ ( Z / ℓ Z ) ( − 1 ) ( a − a 0 ) ⋅ ε ∑ b 1 , … , b d − 1 ∈ ( Z / ℓ Z ) ξ λ ⋅ a + κ ⋅ ( b 1 + ⋯ + b d − 1 ) Θ M ℓ , n ℓ ( τ , z ) = 2 n ∑ a − a 0 ∈ ( 2 Z / ℓ Z ) n ∑ b 1 , … , b d − 1 ∈ ( Z / ℓ Z ) ξ λ ⋅ a + κ ⋅ ( b 1 + ⋯ + b d − 1 ) Θ M ℓ , n ℓ ( τ , z ) , \begin{aligned}
0=\sum_{\substack{\varepsilon\in(\mathbb{Z}/2\mathbb{Z})^n\\ a\in(\mathbb{Z}/\ell\mathbb{Z})}}(-1)^{-a_0\cdot\varepsilon}S_{\lambda+(\ell/2)\varepsilon,\kappa} \\
&=\sum_{\substack{\varepsilon\in(\mathbb{Z}/2\mathbb{Z})^n\\ a\in(\mathbb{Z}/\ell\mathbb{Z})}}(-1)^{(a-a_0)\cdot\varepsilon}\sum_{b_1,\ldots,b_{d-1}\in(\mathbb{Z}/\ell\mathbb{Z})}\xi^{\lambda\cdot a+\kappa\cdot(b_1+\cdots+b_{d-1})}\Theta^\ell_{M_{\ell,n}}\left(\tau,z\right) \\
&=2^n\sum_{a-a_0\in(2\mathbb{Z}/\ell\mathbb{Z})^n}\sum_{b_1,\ldots,b_{d-1}\in(\mathbb{Z}/\ell\mathbb{Z})}\xi^{\lambda\cdot a+\kappa\cdot(b_1+\cdots+b_{d-1})}\Theta^\ell_{M_{\ell,n}}\left(\tau,z\right),
\end{aligned} 0 = ε ∈ ( Z /2 Z ) n a ∈ ( Z / ℓ Z ) ∑ ( − 1 ) − a 0 ⋅ ε S λ + ( ℓ /2 ) ε , κ = ε ∈ ( Z /2 Z ) n a ∈ ( Z / ℓ Z ) ∑ ( − 1 ) ( a − a 0 ) ⋅ ε b 1 , … , b d − 1 ∈ ( Z / ℓ Z ) ∑ ξ λ ⋅ a + κ ⋅ ( b 1 + ⋯ + b d − 1 ) Θ M ℓ , n ℓ ( τ , z ) = 2 n a − a 0 ∈ ( 2 Z / ℓ Z ) n ∑ b 1 , … , b d − 1 ∈ ( Z / ℓ Z ) ∑ ξ λ ⋅ a + κ ⋅ ( b 1 + ⋯ + b d − 1 ) Θ M ℓ , n ℓ ( τ , z ) , which proves the parity refinement.
The specialization ℓ = 3 \ell=3 ℓ = 3
For ℓ = 3 \ell=3 ℓ = 3 (64) agree with the cubic data in (33) : we have d = 2 d=2 d = 2 M ℓ = M M_\ell=M M ℓ = M ξ = ω \xi=\omega ξ = ω r = ( 1 , 2 ) T r=(1,2)^T r = ( 1 , 2 ) T B = [ b , 0 ] B=[b,0] B = [ b , 0 ]
B 3 − σ ( − b ) = M − 1 ( b 0 ) , r a 3 ≡ ρ a ( m o d Z 2 , n ) , \frac{B}{3}-\sigma(-b)=M^{-1}\begin{pmatrix}b\\0\end{pmatrix},\qquad\frac{ra}{3}\equiv\rho a\pmod{\mathbb{Z}^{2,n}}, 3 B − σ ( − b ) = M − 1 ( b 0 ) , 3 r a ≡ ρ a ( mod Z 2 , n ) , Lemma 2.6 gives
Θ M , n [ r a 3 , B 3 ] ( τ , z ) = ω a ⋅ b Θ a , − b ( n ) ( τ , z ) . (78) \Theta_{M,n}\left[\frac{ra}{3},\frac{B}{3}\right](\tau,z)=\omega^{a\cdot b}\Theta^{(n)}_{a,-b}(\tau,z). \tag*{(78)} Θ M , n [ 3 r a , 3 B ] ( τ , z ) = ω a ⋅ b Θ a , − b ( n ) ( τ , z ) . (78) The phase in (78) disappears after cubing. Consequently, (74) becomes
∑ a , b ∈ F 3 n ω λ ⋅ a + κ ⋅ b Θ a , − b ( n ) ( τ , z ) 3 = 0. \sum_{a,b\in\mathbb{F}_3^n}\omega^{\lambda\cdot a+\kappa\cdot b}\Theta^{(n)}_{a,-b}(\tau,z)^3=0. a , b ∈ F 3 n ∑ ω λ ⋅ a + κ ⋅ b Θ a , − b ( n ) ( τ , z ) 3 = 0. The change of variables b ↦ − b b\mapsto-b b ↦ − b (44) . Thus Theorem 5.3 specializes exactly to Corollary 3.3, which explains in what sense it extends the cubic relation from A 2 A_2 A 2 A ℓ − 1 A_{\ell-1} A ℓ − 1
The specialization ℓ = 4 \ell=4 ℓ = 4
The case ℓ = 4 \ell=4 ℓ = 4 A 3 A_3 A 3 τ ∈ H 1 \tau\in\mathfrak{H}_1 τ ∈ H 1 q = e ( τ ) q=e(\tau) q = e ( τ ) x = ( x 1 , x 2 , x 3 ) T ∈ R 3 x=(x_1,x_2,x_3)^T\in\mathbb{R}^3 x = ( x 1 , x 2 , x 3 ) T ∈ R 3
Q 4 ( x ) = x 1 2 + x 2 2 + x 3 2 − x 1 x 2 − x 2 x 3 . Q_4(x)=x_1^2+x_2^2+x_3^2-x_1x_2-x_2x_3. Q 4 ( x ) = x 1 2 + x 2 2 + x 3 2 − x 1 x 2 − x 2 x 3 . For 0 ≤ a ≤ 3 0\leq a\leq3 0 ≤ a ≤ 3 0 ≤ j ≤ 2 0\leq j\leq2 0 ≤ j ≤ 2
U a , j ( q ) = ∑ x ∈ a 4 ( 1 , 2 , 3 ) T + Z 3 i j ( − x 1 + 2 x 2 − x 3 ) q Q 4 ( x ) . (79) U_{a,j}(q)=\sum_{x\in\frac{a}{4}(1,2,3)^T+\mathbb{Z}^3}i^{j(-x_1+2x_2-x_3)}q^{Q_4(x)}. \tag*{(79)} U a , j ( q ) = x ∈ 4 a ( 1 , 2 , 3 ) T + Z 3 ∑ i j ( − x 1 + 2 x 2 − x 3 ) q Q 4 ( x ) . (79) Lemma 5.4. Write R 4 = Z / 4 Z R_4=\mathbb{Z}/4\mathbb{Z} R 4 = Z /4 Z 0 ≤ a ≤ 3 0\leq a\leq3 0 ≤ a ≤ 3 b 1 , b 2 ∈ R 4 b_1,b_2\in R_4 b 1 , b 2 ∈ R 4 b = ( b 1 , b 2 , 0 ) T b=(b_1,b_2,0)^T b = ( b 1 , b 2 , 0 ) T
U a , b ( q ) : = Θ M 4 , 1 [ a ( 1 , 2 , 3 ) T 4 , b 4 ] ( τ , 0 ) U_{a,b}(q):=\Theta_{M_4,1}\left[\frac{a(1,2,3)^T}{4},\frac{b}{4}\right](\tau,0) U a , b ( q ) := Θ M 4 , 1 [ 4 a ( 1 , 2 , 3 ) T , 4 b ] ( τ , 0 ) = ∑ x ∈ a 4 ( 1 , 2 , 3 ) T + Z 3 i ( 2 b 1 − b 2 ) x 1 + ( − b 1 + 2 b 2 ) x 2 − b 2 x 3 q Q 4 ( x ) . (80) =\sum_{x\in\frac{a}{4}(1,2,3)^T+\mathbb{Z}^3}i^{(2b_1-b_2)x_1+(-b_1+2b_2)x_2-b_2x_3}q^{Q_4(x)}. \tag*{(80)} = x ∈ 4 a ( 1 , 2 , 3 ) T + Z 3 ∑ i ( 2 b 1 − b 2 ) x 1 + ( − b 1 + 2 b 2 ) x 2 − b 2 x 3 q Q 4 ( x ) . (80) Then U a , ( 0 , j , 0 ) T ( q ) = U a , j ( q ) U_{a,(0,j,0)^T}(q)=U_{a,j}(q) U a , ( 0 , j , 0 ) T ( q ) = U a , j ( q ) 0 ≤ j ≤ 2 0\leq j\leq2 0 ≤ j ≤ 2
U a , b ( q ) 4 = { U a , 0 ( q ) 4 , ( b 1 , b 2 ) = ( 0 , 0 ) , U a , 2 ( q ) 4 , ( b 1 , b 2 ) ∈ { ( 0 , 2 ) , ( 2 , 0 ) , ( 2 , 2 ) } , U a , 1 ( q ) 4 , otherwise . (81) U_{a,b}(q)^4=
\begin{cases}
U_{a,0}(q)^4, & (b_1,b_2)=(0,0),\\
U_{a,2}(q)^4, & (b_1,b_2)\in\{(0,2),(2,0),(2,2)\},\\
U_{a,1}(q)^4, & \text{otherwise}.
\end{cases} \tag*{(81)} U a , b ( q ) 4 = ⎩ ⎨ ⎧ U a , 0 ( q ) 4 , U a , 2 ( q ) 4 , U a , 1 ( q ) 4 , ( b 1 , b 2 ) = ( 0 , 0 ) , ( b 1 , b 2 ) ∈ {( 0 , 2 ) , ( 2 , 0 ) , ( 2 , 2 )} , otherwise . (81) Proof. The second equality in (80) follows from M 4 ( 1 , 2 , 3 ) T = 4 e 3 M_4(1,2,3)^T=4e_3 M 4 ( 1 , 2 , 3 ) T = 4 e 3 b b b U a , ( 0 , j , 0 ) T ( q ) = U a , j ( q ) U_{a,(0,j,0)^T}(q)=U_{a,j}(q) U a , ( 0 , j , 0 ) T ( q ) = U a , j ( q )
( x 1 , x 2 , x 3 ) ⟼ ( − x 1 + x 2 , x 2 , x 3 ) , (x_1,x_2,x_3)\longmapsto(-x_1+x_2,x_2,x_3), ( x 1 , x 2 , x 3 ) ⟼ ( − x 1 + x 2 , x 2 , x 3 ) , ( x 1 , x 2 , x 3 ) ⟼ ( x 1 , x 1 − x 2 + x 3 , x 3 ) , (x_1,x_2,x_3)\longmapsto(x_1,x_1-x_2+x_3,x_3), ( x 1 , x 2 , x 3 ) ⟼ ( x 1 , x 1 − x 2 + x 3 , x 3 ) , ( x 1 , x 2 , x 3 ) ⟼ ( x 1 , x 2 , x 2 − x 3 ) (x_1,x_2,x_3)\longmapsto(x_1,x_2,x_2-x_3) ( x 1 , x 2 , x 3 ) ⟼ ( x 1 , x 2 , x 2 − x 3 ) are bijections of a 4 ( 1 , 2 , 3 ) T + Z 3 \frac{a}{4}(1,2,3)^T+\mathbb{Z}^3 4 a ( 1 , 2 , 3 ) T + Z 3 Q 4 ( x ) Q_4(x) Q 4 ( x )
U a , ( b 1 , b 2 , 0 ) T ( q ) = U a , ( b 2 − b 1 , b 2 , 0 ) T ( q ) , U_{a,(b_1,b_2,0)^T}(q)=U_{a,(b_2-b_1,b_2,0)^T}(q), U a , ( b 1 , b 2 , 0 ) T ( q ) = U a , ( b 2 − b 1 , b 2 , 0 ) T ( q ) , U a , ( b 1 , b 2 , 0 ) T ( q ) = U a , ( b 1 , b 1 − b 2 , 0 ) T ( q ) , U_{a,(b_1,b_2,0)^T}(q)=U_{a,(b_1,b_1-b_2,0)^T}(q), U a , ( b 1 , b 2 , 0 ) T ( q ) = U a , ( b 1 , b 1 − b 2 , 0 ) T ( q ) , U a , ( b 1 , b 2 , 0 ) T ( q ) = i a b 2 U a , ( b 1 + b 2 , − b 2 , 0 ) T ( q ) . U_{a,(b_1,b_2,0)^T}(q)=i^{ab_2}U_{a,(b_1+b_2,-b_2,0)^T}(q). U a , ( b 1 , b 2 , 0 ) T ( q ) = i a b 2 U a , ( b 1 + b 2 , − b 2 , 0 ) T ( q ) . Here all entries are read modulo 4 4 4 ( 0 , 0 ) (0,0) ( 0 , 0 ) ( 0 , 2 ) , ( 2 , 0 ) , ( 2 , 2 ) (0,2),(2,0),(2,2) ( 0 , 2 ) , ( 2 , 0 ) , ( 2 , 2 ) ( 0 , 2 ) (0,2) ( 0 , 2 ) ( 0 , 1 ) (0,1) ( 0 , 1 ) i i i (81) .
Corollary 5.5. The six series defined in (79) , where a ∈ { 0 , 2 } a\in\{0,2\} a ∈ { 0 , 2 } 0 ≤ j ≤ 2 0\leq j\leq2 0 ≤ j ≤ 2
U 0 , 0 ( q ) 4 − 4 U 0 , 1 ( q ) 4 + 3 U 0 , 2 ( q ) 4 = U 2 , 0 ( q ) 4 − 4 U 2 , 1 ( q ) 4 + 3 U 2 , 2 ( q ) 4 . (82) U_{0,0}(q)^4-4U_{0,1}(q)^4+3U_{0,2}(q)^4
=
U_{2,0}(q)^4-4U_{2,1}(q)^4+3U_{2,2}(q)^4. \tag*{(82)} U 0 , 0 ( q ) 4 − 4 U 0 , 1 ( q ) 4 + 3 U 0 , 2 ( q ) 4 = U 2 , 0 ( q ) 4 − 4 U 2 , 1 ( q ) 4 + 3 U 2 , 2 ( q ) 4 . (82) Proof. In (75) , take ℓ = 4 \ell=4 ℓ = 4 n = 1 n=1 n = 1 λ = 1 \lambda=1 λ = 1 κ = 2 \kappa=2 κ = 2 a 0 = 0 a_0=0 a 0 = 0 z = 0 z=0 z = 0 λ ⋅ κ = 2 ≢ 0 ( m o d 4 ) \lambda\cdot\kappa=2\not\equiv0\pmod{4} λ ⋅ κ = 2 ≡ 0 ( mod 4 ) κ ∈ 2 Z / 4 Z \kappa\in2\mathbb{Z}/4\mathbb{Z} κ ∈ 2 Z /4 Z a 0 + 2 Z / 4 Z = { 0 , 2 } a_0+2\mathbb{Z}/4\mathbb{Z}=\{0,2\} a 0 + 2 Z /4 Z = { 0 , 2 } ξ = i \xi=i ξ = i
∑ a ∈ { 0 , 2 } i a ∑ b 1 , b 2 ∈ R 4 ( − 1 ) b 1 + b 2 U a , ( b 1 , b 2 , 0 ) T ( q ) 4 = 0. (83) \sum_{a\in\{0,2\}}i^a\sum_{b_1,b_2\in R_4}(-1)^{b_1+b_2}U_{a,(b_1,b_2,0)^T}(q)^4=0. \tag*{(83)} a ∈ { 0 , 2 } ∑ i a b 1 , b 2 ∈ R 4 ∑ ( − 1 ) b 1 + b 2 U a , ( b 1 , b 2 , 0 ) T ( q ) 4 = 0. (83) The first two cases in (81) have signed weights 1 1 1 3 3 3 ( 1 − 1 + 1 − 1 ) 2 = 0 (1-1+1-1)^2=0 ( 1 − 1 + 1 − 1 ) 2 = 0 − 4 -4 − 4
∑ b 1 , b 2 ∈ R 4 ( − 1 ) b 1 + b 2 U a , ( b 1 , b 2 , 0 ) T ( q ) 4 = U a , 0 ( q ) 4 − 4 U a , 1 ( q ) 4 + 3 U a , 2 ( q ) 4 , \sum_{b_1,b_2\in R_4}(-1)^{b_1+b_2}U_{a,(b_1,b_2,0)^T}(q)^4
=
U_{a,0}(q)^4-4U_{a,1}(q)^4+3U_{a,2}(q)^4, b 1 , b 2 ∈ R 4 ∑ ( − 1 ) b 1 + b 2 U a , ( b 1 , b 2 , 0 ) T ( q ) 4 = U a , 0 ( q ) 4 − 4 U a , 1 ( q ) 4 + 3 U a , 2 ( q ) 4 , for a ∈ { 0 , 2 } a\in\{0,2\} a ∈ { 0 , 2 } (83) and using i 2 = − 1 i^2=-1 i 2 = − 1 (82) .
Remark 5.6. Zhu [17 Theorem 7.13] proved three independent quadratic relations among Θ M 4 , 1 [ a ( 1 , 2 , 3 ) T 4 , b 4 ] ( τ , z ) \Theta_{M_4,1}\left[\frac{a(1,2,3)^T}{4},\frac{b}{4}\right](\tau,z) Θ M 4 , 1 [ 4 a ( 1 , 2 , 3 ) T , 4 b ] ( τ , z ) a ∈ { 0 , 2 } a\in\{0,2\} a ∈ { 0 , 2 }
b ∈ { ( 0 , 0 , 0 ) T , ( 0 , 2 , 0 ) T , ( 2 , 0 , 0 ) T , ( 2 , 2 , 0 ) T } . b\in\{(0,0,0)^T,(0,2,0)^T,(2,0,0)^T,(2,2,0)^T\}. b ∈ {( 0 , 0 , 0 ) T , ( 0 , 2 , 0 ) T , ( 2 , 0 , 0 ) T , ( 2 , 2 , 0 ) T } . These relations cannot be deduced from Theorem 5.3 , although we expect Theorem 2.7 to imply them.
The specialization ℓ = 5 \ell=5 ℓ = 5
The case ℓ = 5 \ell=5 ℓ = 5 A 4 A_4 A 4
R 5 = Z / 5 Z , ξ = e ( 1 / 5 ) , q = e ( τ ) , τ ∈ H 1 . R_5=\mathbb{Z}/5\mathbb{Z},\qquad\xi=e(1/5),\qquad q=e(\tau),\qquad\tau\in\mathfrak{H}_1. R 5 = Z /5 Z , ξ = e ( 1/5 ) , q = e ( τ ) , τ ∈ H 1 . For x = ( x 1 , x 2 , x 3 , x 4 ) T ∈ R 4 x=(x_1,x_2,x_3,x_4)^T\in\mathbb{R}^4 x = ( x 1 , x 2 , x 3 , x 4 ) T ∈ R 4
Q 5 ( x ) = x 1 2 + x 2 2 + x 3 2 + x 4 2 − x 1 x 2 − x 2 x 3 − x 3 x 4 . Q_5(x)=x_1^2+x_2^2+x_3^2+x_4^2-x_1x_2-x_2x_3-x_3x_4. Q 5 ( x ) = x 1 2 + x 2 2 + x 3 2 + x 4 2 − x 1 x 2 − x 2 x 3 − x 3 x 4 . For 0 ≤ a ≤ 4 0 \le a \le4 0 ≤ a ≤ 4
T a , 0 ( q ) = ∑ x ∈ a 5 ( 1 , 2 , 3 , 4 ) T + Z 4 q Q 5 ( x ) , T_{a,0}(q)=\sum_{x\in\frac{a}{5}(1,2,3,4)^T+\mathbb{Z}^4}q^{Q_5(x)}, T a , 0 ( q ) = x ∈ 5 a ( 1 , 2 , 3 , 4 ) T + Z 4 ∑ q Q 5 ( x ) , T a , j ( q ) = ∑ x ∈ a 5 ( 1 , 2 , 3 , 4 ) T + Z 4 ξ ( 1 − j ) x 1 + ( 3 − 2 j ) x 2 − ( 1 + j ) x 3 + ( 2 j − 1 ) x 4 q Q 5 ( x ) ( 1 ≤ j ≤ 5 ) . (84) T_{a,j}(q)=\sum_{x\in\frac{a}{5}(1,2,3,4)^T+\mathbb{Z}^4}\xi^{(1-j)x_1+(3-2j)x_2-(1+j)x_3+(2j-1)x_4}q^{Q_5(x)}
\qquad(1\le j\le5). \tag*{(84)} T a , j ( q ) = x ∈ 5 a ( 1 , 2 , 3 , 4 ) T + Z 4 ∑ ξ ( 1 − j ) x 1 + ( 3 − 2 j ) x 2 − ( 1 + j ) x 3 + ( 2 j − 1 ) x 4 q Q 5 ( x ) ( 1 ≤ j ≤ 5 ) . (84) Lemma 5.7. For 0 ≤ a ≤ 4 0\le a\le4 0 ≤ a ≤ 4 b 1 , b 2 , b 3 ∈ R 5 b_1,b_2,b_3\in R_5 b 1 , b 2 , b 3 ∈ R 5 b = ( b 1 , b 2 , b 3 , 0 ) T b=(b_1,b_2,b_3,0)^T b = ( b 1 , b 2 , b 3 , 0 ) T
A a , b ( q ) : = Θ M 5 , 1 [ a ( 1 , 2 , 3 , 4 ) T 5 b 5 ] ( τ , 0 ) \mathcal{A}_{a,b}(q):=\Theta_{M_5,1}\left[\begin{matrix}\frac{a(1,2,3,4)^T}{5}\\[2pt]\frac{b}{5}\end{matrix}\right](\tau,0) A a , b ( q ) := Θ M 5 , 1 [ 5 a ( 1 , 2 , 3 , 4 ) T 5 b ] ( τ , 0 ) = ∑ x ∈ a 5 ( 1 , 2 , 3 , 4 ) T + Z 4 ξ ( 2 b 1 − b 2 ) x 1 + ( − b 1 + 2 b 2 − b 3 ) x 2 + ( − b 2 + 2 b 3 ) x 3 − b 3 x 4 q Q 5 ( x ) . =\sum_{x\in\frac{a}{5}(1,2,3,4)^T+\mathbb{Z}^4}\xi^{(2b_1-b_2)x_1+(-b_1+2b_2-b_3)x_2+(-b_2+2b_3)x_3-b_3x_4}q^{Q_5(x)}. = x ∈ 5 a ( 1 , 2 , 3 , 4 ) T + Z 4 ∑ ξ ( 2 b 1 − b 2 ) x 1 + ( − b 1 + 2 b 2 − b 3 ) x 2 + ( − b 2 + 2 b 3 ) x 3 − b 3 x 4 q Q 5 ( x ) . Then
T a , 0 ( q ) = A a , ( 0 , 0 , 0 , 0 ) T ( q ) , T_{a,0}(q)=\mathcal{A}_{a,(0,0,0,0)^T}(q), T a , 0 ( q ) = A a , ( 0 , 0 , 0 , 0 ) T ( q ) , T a , j ( q ) = A a , ( 2 − 2 j , 3 − 3 j , 1 − 2 j , 0 ) T ( q ) ( 1 ≤ j ≤ 5 ) , (85) T_{a,j}(q)=\mathcal{A}_{a,(2-2j,3-3j,1-2j,0)^T}(q)
\qquad(1\le j\le5), \tag*{(85)} T a , j ( q ) = A a , ( 2 − 2 j , 3 − 3 j , 1 − 2 j , 0 ) T ( q ) ( 1 ≤ j ≤ 5 ) , (85) and
A a , b ( q ) 5 = { T a , 0 ( q ) 5 , b = ( 0 , 0 , 0 , 0 ) T , T a , j ( q ) 5 , Q 5 ( b ) ≡ j ( m o d 5 ) , 1 ≤ j ≤ 4 , T a , 5 ( q ) 5 , Q 5 ( b ) ≡ 0 ( m o d 5 ) , b ≠ ( 0 , 0 , 0 , 0 ) T . \mathcal{A}_{a,b}(q)^5=
\begin{cases}
T_{a,0}(q)^5, & b=(0,0,0,0)^T,\\
T_{a,j}(q)^5, & Q_5(b)\equiv j\pmod5,\quad1\le j\le4,\\
T_{a,5}(q)^5, & Q_5(b)\equiv0\pmod5,\quad b\ne(0,0,0,0)^T.
\end{cases} A a , b ( q ) 5 = ⎩ ⎨ ⎧ T a , 0 ( q ) 5 , T a , j ( q ) 5 , T a , 5 ( q ) 5 , b = ( 0 , 0 , 0 , 0 ) T , Q 5 ( b ) ≡ j ( mod 5 ) , 1 ≤ j ≤ 4 , Q 5 ( b ) ≡ 0 ( mod 5 ) , b = ( 0 , 0 , 0 , 0 ) T . Proof. The displayed expansion of A a , b \mathcal{A}_{a,b} A a , b M 5 ( 1 , 2 , 3 , 4 ) T = 5 e 4 M_5(1,2,3,4)^T=5e_4 M 5 ( 1 , 2 , 3 , 4 ) T = 5 e 4 b b b
The four changes of summation indices
( x 1 , x 2 , x 3 , x 4 ) ⟼ ( − x 1 + x 2 , x 2 , x 3 , x 4 ) , (x_1,x_2,x_3,x_4)\longmapsto(-x_1+x_2,x_2,x_3,x_4), ( x 1 , x 2 , x 3 , x 4 ) ⟼ ( − x 1 + x 2 , x 2 , x 3 , x 4 ) , ( x 1 , x 2 , x 3 , x 4 ) ⟼ ( x 1 , x 1 − x 2 + x 3 , x 3 , x 4 ) , (x_1,x_2,x_3,x_4)\longmapsto(x_1,x_1-x_2+x_3,x_3,x_4), ( x 1 , x 2 , x 3 , x 4 ) ⟼ ( x 1 , x 1 − x 2 + x 3 , x 3 , x 4 ) , ( x 1 , x 2 , x 3 , x 4 ) ⟼ ( x 1 , x 2 , x 2 − x 3 + x 4 , x 4 ) , (x_1,x_2,x_3,x_4)\longmapsto(x_1,x_2,x_2-x_3+x_4,x_4), ( x 1 , x 2 , x 3 , x 4 ) ⟼ ( x 1 , x 2 , x 2 − x 3 + x 4 , x 4 ) , ( x 1 , x 2 , x 3 , x 4 ) ⟼ ( x 1 , x 2 , x 3 , x 3 − x 4 ) (x_1,x_2,x_3,x_4)\longmapsto(x_1,x_2,x_3,x_3-x_4) ( x 1 , x 2 , x 3 , x 4 ) ⟼ ( x 1 , x 2 , x 3 , x 3 − x 4 ) are bijections of a 5 ( 1 , 2 , 3 , 4 ) T + Z 4 \frac{a}{5}(1,2,3,4)^T+\mathbb{Z}^4 5 a ( 1 , 2 , 3 , 4 ) T + Z 4 Q 5 ( x ) Q_5(x) Q 5 ( x ) q q q
A a , ( b 1 , b 2 , b 3 , 0 ) T = A a , ( b 2 − b 1 , b 2 , b 3 , 0 ) T , \mathcal{A}_{a,(b_1,b_2,b_3,0)^T}=\mathcal{A}_{a,(b_2-b_1,b_2,b_3,0)^T}, A a , ( b 1 , b 2 , b 3 , 0 ) T = A a , ( b 2 − b 1 , b 2 , b 3 , 0 ) T , A a , ( b 1 , b 2 , b 3 , 0 ) T = A a , ( b 1 , b 1 + b 3 − b 2 , b 3 , 0 ) T , \mathcal{A}_{a,(b_1,b_2,b_3,0)^T}=\mathcal{A}_{a,(b_1,b_1+b_3-b_2,b_3,0)^T}, A a , ( b 1 , b 2 , b 3 , 0 ) T = A a , ( b 1 , b 1 + b 3 − b 2 , b 3 , 0 ) T , A a , ( b 1 , b 2 , b 3 , 0 ) T = A a , ( b 1 , b 2 , b 2 − b 3 , 0 ) T , \mathcal{A}_{a,(b_1,b_2,b_3,0)^T}=\mathcal{A}_{a,(b_1,b_2,b_2-b_3,0)^T}, A a , ( b 1 , b 2 , b 3 , 0 ) T = A a , ( b 1 , b 2 , b 2 − b 3 , 0 ) T , A a , ( b 1 , b 2 , b 3 , 0 ) T = ξ a b 3 A a , ( b 1 + b 3 , b 2 + 2 b 3 , − b 3 , 0 ) T . \mathcal{A}_{a,(b_1,b_2,b_3,0)^T}=\xi^{ab_3}\mathcal{A}_{a,(b_1+b_3,b_2+2b_3,-b_3,0)^T}. A a , ( b 1 , b 2 , b 3 , 0 ) T = ξ a b 3 A a , ( b 1 + b 3 , b 2 + 2 b 3 , − b 3 , 0 ) T . Here all entries are read modulo 5. In the last relation, the two exponents differ by 5 b 3 ( x 3 − x 2 ) 5b_3(x_3-x_2) 5 b 3 ( x 3 − x 2 )
x 3 − x 2 ∈ a 5 + Z , 5 b 3 ( x 3 − x 2 ) ∈ a b 3 + 5 Z , x_3-x_2\in\frac{a}{5}+\mathbb{Z},\qquad5b_3(x_3-x_2)\in ab_3+5\mathbb{Z}, x 3 − x 2 ∈ 5 a + Z , 5 b 3 ( x 3 − x 2 ) ∈ a b 3 + 5 Z , the last relation has the stated factor ξ a b 3 \xi^{ab_3} ξ a b 3
We now determine the orbits generated by these transformations. For b = ( b 1 , b 2 , b 3 , 0 ) T b=(b_1,b_2,b_3,0)^T b = ( b 1 , b 2 , b 3 , 0 ) T
c ( b ) = ( c 1 , c 2 , c 3 , c 4 , c 5 ) : = ( b 1 , b 2 − b 1 , b 3 − b 2 , − b 3 , 0 ) ∈ R 5 5 . c(b)=(c_1,c_2,c_3,c_4,c_5):=(b_1,b_2-b_1,b_3-b_2,-b_3,0)\in R_5^5. c ( b ) = ( c 1 , c 2 , c 3 , c 4 , c 5 ) := ( b 1 , b 2 − b 1 , b 3 − b 2 , − b 3 , 0 ) ∈ R 5 5 . Thus c 1 + ⋯ + c 5 = 0 c_1+\cdots+c_5=0 c 1 + ⋯ + c 5 = 0 b b b
b i = c 1 + ⋯ + c i ( 1 ≤ i ≤ 3 ) . b_i=c_1+\cdots+c_i\qquad(1\le i\le3). b i = c 1 + ⋯ + c i ( 1 ≤ i ≤ 3 ) . For the four transformed characteristics above, direct calculation gives
c ( ( b 2 − b 1 , b 2 , b 3 , 0 ) T ) = ( c 2 , c 1 , c 3 , c 4 , c 5 ) , c ( ( b 1 , b 1 + b 3 − b 2 , b 3 , 0 ) T ) = ( c 1 , c 3 , c 2 , c 4 , c 5 ) , c ( ( b 1 , b 2 , b 2 − b 3 , 0 ) T ) = ( c 1 , c 2 , c 4 , c 3 , c 5 ) , c ( ( b 1 + b 3 , b 2 + 2 b 3 , − b 3 , 0 ) T ) = ( c 1 , c 2 , c 3 , c 5 , c 4 ) + b 3 ( 1 , 1 , 1 , 1 , 1 ) . \begin{aligned}
c((b_2-b_1,b_2,b_3,0)^T)&=(c_2,c_1,c_3,c_4,c_5),\\
c((b_1,b_1+b_3-b_2,b_3,0)^T)&=(c_1,c_3,c_2,c_4,c_5),\\
c((b_1,b_2,b_2-b_3,0)^T)&=(c_1,c_2,c_4,c_3,c_5),\\
c((b_1+b_3,b_2+2b_3,-b_3,0)^T)&=(c_1,c_2,c_3,c_5,c_4)+b_3(1,1,1,1,1).
\end{aligned} c (( b 2 − b 1 , b 2 , b 3 , 0 ) T ) c (( b 1 , b 1 + b 3 − b 2 , b 3 , 0 ) T ) c (( b 1 , b 2 , b 2 − b 3 , 0 ) T ) c (( b 1 + b 3 , b 2 + 2 b 3 , − b 3 , 0 ) T ) = ( c 2 , c 1 , c 3 , c 4 , c 5 ) , = ( c 1 , c 3 , c 2 , c 4 , c 5 ) , = ( c 1 , c 2 , c 4 , c 3 , c 5 ) , = ( c 1 , c 2 , c 3 , c 5 , c 4 ) + b 3 ( 1 , 1 , 1 , 1 , 1 ) . Hence the four transformations realize the four adjacent transpositions of the entries of c ( b ) c(b) c ( b ) b b b b ′ b' b ′ π \pi π t ∈ R 5 t\in R_5 t ∈ R 5
c ( b ′ ) = ( c π ( 1 ) , c π ( 2 ) , c π ( 3 ) , c π ( 4 ) , c π ( 5 ) ) + t ( 1 , 1 , 1 , 1 , 1 ) . c(b')=(c_{\pi(1)},c_{\pi(2)},c_{\pi(3)},c_{\pi(4)},c_{\pi(5)})+t(1,1,1,1,1). c ( b ′ ) = ( c π ( 1 ) , c π ( 2 ) , c π ( 3 ) , c π ( 4 ) , c π ( 5 ) ) + t ( 1 , 1 , 1 , 1 , 1 ) . In particular, characteristics in the same orbit have equal fifth powers.
It remains to classify the zero-sum multisets of five elements of R 5 R_5 R 5
∑ i = 1 5 c i 2 = 2 Q 5 ( b ) \sum_{i=1}^{5}c_i^2=2Q_5(b) i = 1 ∑ 5 c i 2 = 2 Q 5 ( b ) shows that Q 5 ( b ) Q_5(b) Q 5 ( b )
∑ i = 1 5 ( c i + t ) 2 = ∑ i = 1 5 c i 2 + 2 t ∑ i = 1 5 c i + 5 t 2 = ∑ i = 1 5 c i 2 \sum_{i=1}^{5}(c_i+t)^2=\sum_{i=1}^{5}c_i^2+2t\sum_{i=1}^{5}c_i+5t^2=\sum_{i=1}^{5}c_i^2 i = 1 ∑ 5 ( c i + t ) 2 = i = 1 ∑ 5 c i 2 + 2 t i = 1 ∑ 5 c i + 5 t 2 = i = 1 ∑ 5 c i 2 in R 5 R_5 R 5
Suppose first that an entry is repeated. After translating a value of maximal multiplicity to zero, the types 4 + 1 and 3 + 2 have the forms
{ 0 , 0 , 0 , 0 , d } , { 0 , 0 , 0 , d , d } . \{0,0,0,0,d\},\qquad\{0,0,0,d,d\}. { 0 , 0 , 0 , 0 , d } , { 0 , 0 , 0 , d , d } . The zero-sum condition gives d = 0 d=0 d = 0 2 d = 0 2d=0 2 d = 0 r , s , u ∈ R 5 r,s,u\in R_5 r , s , u ∈ R 5 v v v
0 = 1 + 2 + 3 + 4 = r + s + u + v = v , 0=1+2+3+4=r+s+u+v=v, 0 = 1 + 2 + 3 + 4 = r + s + u + v = v , a contradiction. The remaining nonconstant possibilities are therefore
{ 0 , 0 , 0 , d , − d } , { 0 , 0 , d , d , − 2 d } , d ∈ R 5 × . \{0,0,0,d,-d\},\qquad\{0,0,d,d,-2d\},\qquad d\in R_5^\times. { 0 , 0 , 0 , d , − d } , { 0 , 0 , d , d , − 2 d } , d ∈ R 5 × . Replacing d d d − d -d − d − d -d − d
{ − d , − d , 0 , 0 , − 3 d } = { 0 , 0 , − d , − d , − 2 ( − d ) } . \{-d,-d,0,0,-3d\}=\{0,0,-d,-d,-2(-d)\}. { − d , − d , 0 , 0 , − 3 d } = { 0 , 0 , − d , − d , − 2 ( − d )} . Thus the cases d = 1 , 2 d=1,2 d = 1 , 2 Q 5 ( b ) Q_5(b) Q 5 ( b )
1 2 ( d 2 + ( − d ) 2 ) = d 2 , 1 2 ( 2 d 2 + ( − 2 d ) 2 ) = 3 d 2 ( m o d 5 ) , \frac{1}{2}\left(d^2+(-d)^2\right)=d^2,\qquad\frac{1}{2}\left(2d^2+(-2d)^2\right)=3d^2\pmod5, 2 1 ( d 2 + ( − d ) 2 ) = d 2 , 2 1 ( 2 d 2 + ( − 2 d ) 2 ) = 3 d 2 ( mod 5 ) , respectively, and hence are 1, 4 and 3, 2. If no entry is repeated, the multiset is R 5 R_5 R 5
2 Q 5 ( b ) = 0 2 + 1 2 + 2 2 + 3 2 + 4 2 ≡ 0 ( m o d 5 ) . 2Q_5(b)=0^2+1^2+2^2+3^2+4^2\equiv0\pmod5. 2 Q 5 ( b ) = 0 2 + 1 2 + 2 2 + 3 2 + 4 2 ≡ 0 ( mod 5 ) . Together with the constant multiset, which corresponds to b = 0 b=0 b = 0
b = 0 , Q 5 ( b ) ≡ j ( m o d 5 ) ( 1 ≤ j ≤ 4 ) , Q 5 ( b ) ≡ 0 ( m o d 5 ) and b ≠ 0. b=0,\qquad Q_5(b)\equiv j\pmod5\ (1\leq j\leq4),\qquad Q_5(b)\equiv0\pmod5\ \text{and}\ b\ne0. b = 0 , Q 5 ( b ) ≡ j ( mod 5 ) ( 1 ≤ j ≤ 4 ) , Q 5 ( b ) ≡ 0 ( mod 5 ) and b = 0. Finally, we identify these six orbits with the series in (84) . For 1 ≤ j ≤ 5 1\leq j\leq5 1 ≤ j ≤ 5
M 5 ( 2 − 2 j , 3 − 3 j , 1 − 2 j , 0 ) T = ( 1 − j , 3 − 2 j , − 1 − j , 2 j − 1 ) T M_5(2-2j,3-3j,1-2j,0)^T=(1-j,3-2j,-1-j,2j-1)^T M 5 ( 2 − 2 j , 3 − 3 j , 1 − 2 j , 0 ) T = ( 1 − j , 3 − 2 j , − 1 − j , 2 j − 1 ) T and
Q 5 ( ( 2 − 2 j , 3 − 3 j , 1 − 2 j , 0 ) T ) = 5 j 2 − 9 j + 5 ≡ j ( m o d 5 ) . Q_{5}\left((2-2j,3-3j,1-2j,0)^{T}\right)=5j^{2}-9j+5\equiv j\pmod{5}. Q 5 ( ( 2 − 2 j , 3 − 3 j , 1 − 2 j , 0 ) T ) = 5 j 2 − 9 j + 5 ≡ j ( mod 5 ) . Replacing the characteristic vector b b b b + 5 m b+5m b + 5 m m ∈ Z 4 m\in\mathbb{Z}^{4} m ∈ Z 4 m 4 = 0 m_{4}=0 m 4 = 0 x = g + a 5 ( 1 , 2 , 3 , 4 ) T x=g+\frac{a}{5}(1,2,3,4)^{T} x = g + 5 a ( 1 , 2 , 3 , 4 ) T g ∈ Z 4 g\in\mathbb{Z}^{4} g ∈ Z 4
x T M 5 m = g T M 5 m + a 5 ( 1 , 2 , 3 , 4 ) T M 5 m = g T M 5 m + a m 4 = g T M 5 m ∈ Z . \begin{aligned}
x^{T}M_{5}m=g^{T}M_{5}m+\frac{a}{5}(1,2,3,4)^{T}M_{5}m \\
&=g^{T}M_{5}m+am_{4}=g^{T}M_{5}m\in\mathbb{Z}.
\end{aligned} x T M 5 m = g T M 5 m + 5 a ( 1 , 2 , 3 , 4 ) T M 5 m = g T M 5 m + a m 4 = g T M 5 m ∈ Z . Consequently, ξ x T M 5 ( b + 5 m ) = ξ x T M 5 b \xi^{x^{T}M_{5}(b+5m)}=\xi^{x^{T}M_{5}b} ξ x T M 5 ( b + 5 m ) = ξ x T M 5 b (84) that
T a , j ( q ) = A a , ( 2 − 2 j , 3 − 3 j , 1 − 2 j , 0 ) T ( q ) ( 1 ≤ j ≤ 5 ) . T_{a,j}(q)=\mathcal{A}_{a,(2-2j,3-3j,1-2j,0)^{T}}(q)\qquad(1\leq j\leq5). T a , j ( q ) = A a , ( 2 − 2 j , 3 − 3 j , 1 − 2 j , 0 ) T ( q ) ( 1 ≤ j ≤ 5 ) . where the characteristic is read modulo 5. Hence T a , j T_{a,j} T a , j Q 5 ( b ) ≡ j ( m o d 5 ) Q_{5}(b)\equiv j\pmod{5} Q 5 ( b ) ≡ j ( mod 5 ) 1 ≤ j ≤ 4 1\leq j\leq4 1 ≤ j ≤ 4 j = 5 j=5 j = 5 ( 2 , 3 , 1 , 0 ) T ≠ 0 (2,3,1,0)^{T}\ne0 ( 2 , 3 , 1 , 0 ) T = 0 Q 5 ( b ) ≡ 0 ( m o d 5 ) Q_{5}(b)\equiv0\pmod{5} Q 5 ( b ) ≡ 0 ( mod 5 ) T a , 0 T_{a,0} T a , 0 b = 0 b=0 b = 0 (85) . The last assertion follows from (85) and the classification of the orbits.
Corollary 5.8. The series defined in (84) satisfy
T 0 , 0 ( q ) 5 − 5 T 0 , 1 ( q ) 5 + 5 T 0 , 2 ( q ) 5 + 5 T 0 , 3 ( q ) 5 − 5 T 0 , 4 ( q ) 5 − T 0 , 5 ( q ) 5 = T 1 , 0 ( q ) 5 − 5 T 1 , 1 ( q ) 5 + 5 T 1 , 2 ( q ) 5 + 5 T 1 , 3 ( q ) 5 − 5 T 1 , 4 ( q ) 5 T_{0,0}(q)^{5}-5T_{0,1}(q)^{5}+5T_{0,2}(q)^{5}+5T_{0,3}(q)^{5}-5T_{0,4}(q)^{5}-T_{0,5}(q)^{5}
=T_{1,0}(q)^{5}-5T_{1,1}(q)^{5}+5T_{1,2}(q)^{5}+5T_{1,3}(q)^{5}-5T_{1,4}(q)^{5} T 0 , 0 ( q ) 5 − 5 T 0 , 1 ( q ) 5 + 5 T 0 , 2 ( q ) 5 + 5 T 0 , 3 ( q ) 5 − 5 T 0 , 4 ( q ) 5 − T 0 , 5 ( q ) 5 = T 1 , 0 ( q ) 5 − 5 T 1 , 1 ( q ) 5 + 5 T 1 , 2 ( q ) 5 + 5 T 1 , 3 ( q ) 5 − 5 T 1 , 4 ( q ) 5 = T 2 , 0 ( q ) 5 − 5 T 2 , 1 ( q ) 5 + 5 T 2 , 2 ( q ) 5 + 5 T 2 , 3 ( q ) 5 − 5 T 2 , 4 ( q ) 5 . =T_{2,0}(q)^{5}-5T_{2,1}(q)^{5}+5T_{2,2}(q)^{5}+5T_{2,3}(q)^{5}-5T_{2,4}(q)^{5}. = T 2 , 0 ( q ) 5 − 5 T 2 , 1 ( q ) 5 + 5 T 2 , 2 ( q ) 5 + 5 T 2 , 3 ( q ) 5 − 5 T 2 , 4 ( q ) 5 . Proof. In (74) , take ℓ = 5 \ell=5 ℓ = 5 n = 1 n=1 n = 1 λ = κ = 1 \lambda=\kappa=1 λ = κ = 1 z = 0 z=0 z = 0 λ ⋅ κ = 1 ≠ 0 \lambda\cdot\kappa=1\ne0 λ ⋅ κ = 1 = 0 R 5 R_{5} R 5
∑ a , b 1 , b 2 , b 3 ∈ R 5 ξ a + b 1 + b 2 + b 3 A a , ( b 1 , b 2 , b 3 , 0 ) T ( q ) 5 = 0. \sum_{a,b_{1},b_{2},b_{3}\in R_{5}}\xi^{a+b_{1}+b_{2}+b_{3}}\mathcal{A}_{a,(b_{1},b_{2},b_{3},0)^{T}}(q)^{5}=0. a , b 1 , b 2 , b 3 ∈ R 5 ∑ ξ a + b 1 + b 2 + b 3 A a , ( b 1 , b 2 , b 3 , 0 ) T ( q ) 5 = 0. Write s = b 1 + b 2 + b 3 s=b_{1}+b_{2}+b_{3} s = b 1 + b 2 + b 3
( b 1 , b 2 , b 3 ) ⟼ ( s , b 2 + 2 b 3 , 2 b 1 + b 2 + 3 b 3 ) (b_{1},b_{2},b_{3})\longmapsto(s,b_{2}+2b_{3},2b_{1}+b_{2}+3b_{3}) ( b 1 , b 2 , b 3 ) ⟼ ( s , b 2 + 2 b 3 , 2 b 1 + b 2 + 3 b 3 ) has determinant 3 3 3 R 5 R_{5} R 5
2 Q 5 ( b ) = ( b 2 + 2 b 3 ) 2 + s ( 2 b 1 + b 2 + 3 b 3 ) . 2Q_{5}(b)=(b_{2}+2b_{3})^{2}+s(2b_{1}+b_{2}+3b_{3}). 2 Q 5 ( b ) = ( b 2 + 2 b 3 ) 2 + s ( 2 b 1 + b 2 + 3 b 3 ) . For fixed s ≠ 0 s\ne0 s = 0 Q 5 ( b ) Q_{5}(b) Q 5 ( b ) s = 0 s=0 s = 0 Q 5 ( b ) ≡ 0 , 1 , 2 , 3 , 4 ( m o d 5 ) Q_{5}(b)\equiv0,1,2,3,4\pmod{5} Q 5 ( b ) ≡ 0 , 1 , 2 , 3 , 4 ( mod 5 ) 5 , 0 , 10 , 10 , 0 5,0,10,10,0 5 , 0 , 10 , 10 , 0 ( b 2 + 2 b 3 ) 2 = 2 Q 5 ( b ) (b_{2}+2b_{3})^{2}=2Q_{5}(b) ( b 2 + 2 b 3 ) 2 = 2 Q 5 ( b ) ∑ s = 1 4 ξ s = − 1 \sum_{s=1}^{4}\xi^{s}=-1 ∑ s = 1 4 ξ s = − 1 b = 0 b=0 b = 0 j = 0 , … , 5 j=0,\ldots,5 j = 0 , … , 5
1 , − 5 , 5 , 5 , − 5 , − 1. 1,\ -5,\ 5,\ 5,\ -5,\ -1. 1 , − 5 , 5 , 5 , − 5 , − 1. Suppressing the common argument q q q
H a : = T a , 0 5 − 5 T a , 1 5 + 5 T a , 2 5 + 5 T a , 3 5 − 5 T a , 4 5 − T a , 5 5 , H_{a}:=T_{a,0}^{5}-5T_{a,1}^{5}+5T_{a,2}^{5}+5T_{a,3}^{5}-5T_{a,4}^{5}-T_{a,5}^{5}, H a := T a , 0 5 − 5 T a , 1 5 + 5 T a , 2 5 + 5 T a , 3 5 − 5 T a , 4 5 − T a , 5 5 , so the preceding identity becomes
∑ a = 0 4 ξ a H a = 0. \sum_{a=0}^{4}\xi^{a}H_{a}=0. a = 0 ∑ 4 ξ a H a = 0. The change of variables x ↦ − x x\mapsto-x x ↦ − x A 5 − a , b = A a , − b \mathcal{A}_{5-a,b}=\mathcal{A}_{a,-b} A 5 − a , b = A a , − b 1 ≤ a ≤ 4 1\leq a\leq4 1 ≤ a ≤ 4 Q 5 ( − b ) = Q 5 ( b ) Q_{5}(-b)=Q_{5}(b) Q 5 ( − b ) = Q 5 ( b ) H 4 = H 1 H_{4}=H_{1} H 4 = H 1 H 3 = H 2 H_{3}=H_{2} H 3 = H 2
H 0 + ( ξ + ξ 4 ) H 1 + ( ξ 2 + ξ 3 ) H 2 = 0. H_{0}+(\xi+\xi^{4})H_{1}+(\xi^{2}+\xi^{3})H_{2}=0. H 0 + ( ξ + ξ 4 ) H 1 + ( ξ 2 + ξ 3 ) H 2 = 0. The coefficientwise Galois automorphism σ \sigma σ Q ( ξ ) \mathbb{Q}(\xi) Q ( ξ ) q q q σ ( ξ ) = ξ 2 \sigma(\xi)=\xi^{2} σ ( ξ ) = ξ 2 A a , b 5 \mathcal{A}^{5}_{a,b} A a , b 5 A a , 2 b 5 \mathcal{A}^{5}_{a,2b} A a , 2 b 5 Q 5 ( 2 b ) ≡ 4 Q 5 ( b ) ( m o d 5 ) Q_{5}(2b)\equiv4Q_{5}(b)\pmod{5} Q 5 ( 2 b ) ≡ 4 Q 5 ( b ) ( mod 5 ) 1 1 1 4 4 4 H a H_a H a σ \sigma σ H a H_a H a σ \sigma σ
( ξ + ξ 4 − ξ 2 − ξ 3 ) ( H 1 − H 2 ) = 0. (\xi+\xi^4-\xi^2-\xi^3)(H_1-H_2)=0. ( ξ + ξ 4 − ξ 2 − ξ 3 ) ( H 1 − H 2 ) = 0. The first factor is nonzero, so H 1 = H 2 H_1=H_2 H 1 = H 2 1 + ξ + ξ 2 + ξ 3 + ξ 4 = 0 1+\xi+\xi^2+\xi^3+\xi^4=0 1 + ξ + ξ 2 + ξ 3 + ξ 4 = 0 H 0 = H 1 H_0=H_1 H 0 = H 1
Finally, applying, respectively, the composition of the first and fourth transformations and that of the second and third transformations from the proof of the lemma to ( 2 , 3 , 1 , 0 ) T (2,3,1,0)^T ( 2 , 3 , 1 , 0 ) T
T a , 5 = ξ a A a , ( 2 , 0 , 4 , 0 ) T , T a , 5 = A a , ( 2 , 0 , 4 , 0 ) T . T_{a,5}=\xi^a A_{a,(2,0,4,0)^T},\qquad T_{a,5}=A_{a,(2,0,4,0)^T}. T a , 5 = ξ a A a , ( 2 , 0 , 4 , 0 ) T , T a , 5 = A a , ( 2 , 0 , 4 , 0 ) T . Thus T a , 5 = 0 T_{a,5}=0 T a , 5 = 0 1 ≤ a ≤ 4 1\leq a\leq4 1 ≤ a ≤ 4 T 1 , 5 = T 2 , 5 = 0 T_{1,5}=T_{2,5}=0 T 1 , 5 = T 2 , 5 = 0 H 0 = H 1 = H 2 H_0=H_1=H_2 H 0 = H 1 = H 2
Doubled-order identities
The root-lattice construction of the preceding section can also be carried out with 2 ℓ 2\ell 2 ℓ 2 ℓ 2\ell 2 ℓ ℓ = 2 \ell=2 ℓ = 2 2 ℓ 2\ell 2 ℓ § 4 , which involve theta series at τ \tau τ 2 τ 2\tau 2 τ A 3 A_3 A 3 M ℓ , r M_{\ell,r} M ℓ , r ξ \xi ξ R R R (64) and (65) , together with the identities and integral matrix in (66) and (67) . Throughout this section, d = ℓ − 1 d=\ell-1 d = ℓ − 1 ν = 2 ℓ \nu=2\ell ν = 2 ℓ L L L § 5 . For a matrix x x x Z / ν Z \mathbb{Z}/\nu\mathbb{Z} Z / ν Z x ≡ 0 ( m o d ℓ ) x\equiv0\pmod{\ell} x ≡ 0 ( mod ℓ ) x x x ℓ Z / ν Z \ell\mathbb{Z}/\nu\mathbb{Z} ℓ Z / ν Z
The 2ℓ-fold identity
Theorem 6.1. Let ℓ ≥ 2 \ell\geq2 ℓ ≥ 2 n ≥ 1 n\geq1 n ≥ 1 τ ∈ H n \tau\in\mathfrak{H}_n τ ∈ H n z ∈ C d , n z\in\mathbb{C}^{d,n} z ∈ C d , n λ , κ ∈ ( Z / ℓ Z ) n \lambda,\kappa\in(\mathbb{Z}/\ell\mathbb{Z})^n λ , κ ∈ ( Z / ℓ Z ) n λ ⋅ κ ≢ 0 ( m o d ℓ ) \lambda\cdot\kappa\not\equiv0\pmod{\ell} λ ⋅ κ ≡ 0 ( mod ℓ )
∑ a ∈ ( Z / ℓ Z ) n , c ∈ { 0 , 1 } n b 1 , … , b d − 1 ∈ ( Z / 2 ℓ Z ) ξ λ ⋅ a + κ ⋅ ( b 1 + ⋯ + b d − 1 + c ) Θ M ℓ , n 2 ℓ [ λ a ℓ B 2 ℓ ] ( τ , z ) = 0. (86) \sum_{\substack{a\in(\mathbb{Z}/\ell\mathbb{Z})^n,\ c\in\{0,1\}^n\\ b_1,\ldots,b_{d-1}\in(\mathbb{Z}/2\ell\mathbb{Z})}}
\xi^{\lambda\cdot a+\kappa\cdot(b_1+\cdots+b_{d-1}+c)}
\Theta^{2\ell}_{M_{\ell,n}}\left[\begin{matrix} \lambda a \\ \ell& B \\ 2\ell\end{matrix}\right](\tau,z)=0. \tag*{(86)} a ∈ ( Z / ℓ Z ) n , c ∈ { 0 , 1 } n b 1 , … , b d − 1 ∈ ( Z /2 ℓ Z ) ∑ ξ λ ⋅ a + κ ⋅ ( b 1 + ⋯ + b d − 1 + c ) Θ M ℓ , n 2 ℓ λa ℓ 2 ℓ B ( τ , z ) = 0. (86) Here B = [ b 1 , … , b d − 1 , c ] B=[b_1,\ldots,b_{d-1},c] B = [ b 1 , … , b d − 1 , c ] d = 1 d=1 d = 1 b 1 , … , b d − 1 b_1,\ldots,b_{d-1} b 1 , … , b d − 1 ξ \xi ξ ℓ \ell ℓ
As in the proof of Theorem 5.3, we first need to establish a character projection formula. We write ζ = e ( 1 / ν ) \zeta=e^{(1/\nu)} ζ = e ( 1/ ν )
X n † = { x ∈ ( Z / ν Z ) d , n : 1 T x ≡ 0 ( m o d ℓ ) } , Y n † = ( Z / ν Z ) d , n { 1 η : η ∈ ( 2 Z / ν Z ) 1 , n } . (87) X_n^\dagger=\{x\in(\mathbb{Z}/\nu\mathbb{Z})^{d,n}: \mathbf{1}^T x\equiv0\pmod{\ell}\},\qquad
Y_n^\dagger=\frac{(\mathbb{Z}/\nu\mathbb{Z})^{d,n}}{\{\mathbf{1}\eta:\eta\in(2\mathbb{Z}/\nu\mathbb{Z})^{1,n}\}}. \tag*{(87)} X n † = { x ∈ ( Z / ν Z ) d , n : 1 T x ≡ 0 ( mod ℓ )} , Y n † = { 1 η : η ∈ ( 2 Z / ν Z ) 1 , n } ( Z / ν Z ) d , n . (87) Both are finite abelian groups of order 2 n ν ( d − 1 ) n 2^n\nu^{(d-1)n} 2 n ν ( d − 1 ) n E ν = ν ( R − I d ) − 1 = 2 E E_\nu=\nu(R-I_d)^{-1}=2E E ν = ν ( R − I d ) − 1 = 2 E (68) give
E ν T Z d = ν Z d + 2 Z 1. E_\nu^T\mathbb{Z}^{d}=\nu\mathbb{Z}^{d}+2\mathbb{Z}\mathbf{1}. E ν T Z d = ν Z d + 2 Z 1 . Reduction of the first equality in (68) modulo ν \nu ν
( R − I d ) ⋅ ( Z / ν Z ) d , n = { x ∈ ( Z / ν Z ) d , n : 1 T x ≡ 0 ( m o d ℓ ) } = X n † . (88) (R-I_d)\cdot(\mathbb{Z}/\nu\mathbb{Z})^{d,n}
=\{x\in(\mathbb{Z}/\nu\mathbb{Z})^{d,n}:\mathbf{1}^T x\equiv0\pmod{\ell}\}
=X_n^\dagger. \tag*{(88)} ( R − I d ) ⋅ ( Z / ν Z ) d , n = { x ∈ ( Z / ν Z ) d , n : 1 T x ≡ 0 ( mod ℓ )} = X n † . (88) The map x ↦ R x x\mapsto Rx x ↦ R x X n † X_n^\dagger X n †
⟨ x , s ⟩ ν = ζ tr ( x T s ) (89) \langle x,s\rangle_\nu=\zeta^{\operatorname{tr}(x^T s)} \tag*{(89)} ⟨ x , s ⟩ ν = ζ tr ( x T s ) (89) for ( x , s ) ∈ X n † × Y n † (x,s)\in X_n^\dagger\times Y_n^\dagger ( x , s ) ∈ X n † × Y n †
Lemma 6.2. Formula (89) defines a well-defined bicharacter on X n † × Y n † X_n^\dagger\times Y_n^\dagger X n † × Y n † x ≠ 0 x\neq0 x = 0 s ∈ Y n † s\in Y_n^\dagger s ∈ Y n † ⟨ x , s ⟩ ν ≠ 1 \langle x,s\rangle_\nu\neq1 ⟨ x , s ⟩ ν = 1
Proof. If s s s s + 1 η s+1\eta s + 1 η η ∈ ( 2 Z / ν Z ) 1 , n \eta\in(2\mathbb{Z}/\nu\mathbb{Z})^{1,n} η ∈ ( 2 Z / ν Z ) 1 , n
tr ( x T 1 η ) = tr ( ( 1 T x ) T η ) ≡ 0 ( m o d ν ) , \operatorname{tr}(x^{T}1\eta)=\operatorname{tr}((1^{T}x)^{T}\eta)\equiv0\pmod{\nu}, tr ( x T 1 η ) = tr (( 1 T x ) T η ) ≡ 0 ( mod ν ) , because 1 T x ≡ 0 ( m o d ℓ ) 1^{T}x\equiv0\pmod{\ell} 1 T x ≡ 0 ( mod ℓ ) η ≡ 0 ( m o d 2 ) \eta\equiv0\pmod{2} η ≡ 0 ( mod 2 ) x ≠ 0 x\ne0 x = 0 y ∈ ( Z / ν Z ) d , n y\in(\mathbb{Z}/\nu\mathbb{Z})^{d,n} y ∈ ( Z / ν Z ) d , n tr ( x T y ) ≢ 0 ( m o d ν ) \operatorname{tr}(x^{T}y)\not\equiv0\pmod{\nu} tr ( x T y ) ≡ 0 ( mod ν ) y y y Y n † \mathcal{Y}^{\dagger}_{n} Y n † x x x
Lemma 6.3. Suppose that F x , s F_{x,s} F x , s x ∈ X n † x\in\mathcal{X}^{\dagger}_{n} x ∈ X n † s ∈ Y n † s\in\mathcal{Y}^{\dagger}_{n} s ∈ Y n †
2 n ν ( d − 1 ) n F R − 1 x , t = ∑ y ∈ X n † , s ∈ Y n † ⟨ y − x , s − t ⟩ ν F y , s , (90) 2^{n}\nu^{(d-1)n}F_{R^{-1}x,t}=\sum_{\substack{y\in\mathcal{X}^{\dagger}_{n},\,s\in\mathcal{Y}^{\dagger}_{n}}}\langle y-x,s-t\rangle_{\nu}F_{y,s}, \tag*{(90)} 2 n ν ( d − 1 ) n F R − 1 x , t = y ∈ X n † , s ∈ Y n † ∑ ⟨ y − x , s − t ⟩ ν F y , s , (90) for every x ∈ X n † x\in\mathcal{X}^{\dagger}_{n} x ∈ X n † t ∈ Y n † t\in\mathcal{Y}^{\dagger}_{n} t ∈ Y n † λ , κ ∈ ( Z / ℓ Z ) 1 , n \lambda,\kappa\in(\mathbb{Z}/\ell\mathbb{Z})^{1,n} λ , κ ∈ ( Z / ℓ Z ) 1 , n λ κ T ≢ 0 ( m o d ℓ ) \lambda\kappa^{T}\not\equiv0\pmod{\ell} λ κ T ≡ 0 ( mod ℓ )
∑ a ∈ ( Z / ℓ Z ) 1 , n t ∈ Y n † ζ λ a T ⟨ 2 r κ , t ⟩ ν F 2 r a , t = 0. (91) \sum_{\substack{a\in(\mathbb{Z}/\ell\mathbb{Z})^{1,n}\\t\in\mathcal{Y}^{\dagger}_{n}}}\zeta^{\lambda a^{T}}\langle2r\kappa,t\rangle_{\nu}F_{2ra,t}=0. \tag*{(91)} a ∈ ( Z / ℓ Z ) 1 , n t ∈ Y n † ∑ ζ λ a T ⟨ 2 r κ , t ⟩ ν F 2 r a , t = 0. (91) Proof. Since R − 1 r = r − ℓ e d R^{-1}r=r-\ell e_{d} R − 1 r = r − ℓ e d F 2 R − 1 r a , t = F 2 r a , t F_{2R^{-1}ra,t}=F_{2ra,t} F 2 R − 1 r a , t = F 2 r a , t 2 r a 2ra 2 r a 2 r κ 2r\kappa 2 r κ X n † \mathcal{X}^{\dagger}_{n} X n † ( Z / ℓ Z ) d , n → ( 2 Z / ν Z ) d , n (\mathbb{Z}/\ell\mathbb{Z})^{d,n}\to(2\mathbb{Z}/\nu\mathbb{Z})^{d,n} ( Z / ℓ Z ) d , n → ( 2 Z / ν Z ) d , n v ↦ 2 v v\mapsto2v v ↦ 2 v (90) , with x = 2 r a x=2ra x = 2 r a ζ λ a T ⟨ 2 r κ , t ⟩ ν \zeta^{\lambda a^{T}}\langle2r\kappa,t\rangle_{\nu} ζ λ a T ⟨ 2 r κ , t ⟩ ν a a a t t t Lemma 6.2 and character orthogonality on Y n † \mathcal{Y}^{\dagger}_{n} Y n †
∑ t ∈ Y n † ⟨ 2 r κ , t ⟩ ν ⟨ y − 2 r a , − t ⟩ ν = ∑ t ∈ Y n † ⟨ 2 r κ − ( y − 2 r a ) , t ⟩ ν = { 2 n ν ( d − 1 ) n , y − 2 r a = 2 r κ , 0 , otherwise . \sum_{t\in\mathcal{Y}^{\dagger}_{n}}\langle2r\kappa,t\rangle_{\nu}\langle y-2ra,-t\rangle_{\nu}
=\sum_{t\in\mathcal{Y}^{\dagger}_{n}}\langle2r\kappa-(y-2ra),t\rangle_{\nu}
=
\begin{cases}
2^{n}\nu^{(d-1)n}, & y-2ra=2r\kappa,\\
0, & \text{otherwise}.
\end{cases} t ∈ Y n † ∑ ⟨ 2 r κ , t ⟩ ν ⟨ y − 2 r a , − t ⟩ ν = t ∈ Y n † ∑ ⟨ 2 r κ − ( y − 2 r a ) , t ⟩ ν = { 2 n ν ( d − 1 ) n , 0 , y − 2 r a = 2 r κ , otherwise . The value y = 2 r ( a + κ ) y=2r(a+\kappa) y = 2 r ( a + κ ) X n † \mathcal{X}^{\dagger}_{n} X n † S S S (91) , the left-hand side of the summed Fourier relation is 2 n ν ( d − 1 ) n S 2^{n}\nu^{(d-1)n}S 2 n ν ( d − 1 ) n S
2 n ν ( d − 1 ) n ∑ a , s ζ λ a T ⟨ 2 r κ , s ⟩ ν F 2 r ( a + κ ) , s = 2 n ν ( d − 1 ) n ζ − λ κ T S . 2^{n}\nu^{(d-1)n}\sum_{a,s}\zeta^{\lambda a^{T}}\langle2r\kappa,s\rangle_{\nu}F_{2r(a+\kappa),s}
=2^{n}\nu^{(d-1)n}\zeta^{-\lambda\kappa^{T}}S. 2 n ν ( d − 1 ) n a , s ∑ ζ λ a T ⟨ 2 r κ , s ⟩ ν F 2 r ( a + κ ) , s = 2 n ν ( d − 1 ) n ζ − λ κ T S . Since ζ − λ κ T ≠ 1 \zeta^{-\lambda\kappa^{T}}\ne1 ζ − λ κ T = 1 S = 0 S=0 S = 0
Proof of Theorem 6.1 . For x ∈ X n † x\in\mathcal{X}^{\dagger}_{n} x ∈ X n † s ∈ Y n † s\in\mathcal{Y}^{\dagger}_{n} s ∈ Y n †
γ s = − L s ν , F x , s = Θ M ℓ , n ν [ x ν γ s ] ( τ , z ) . (92) \gamma_{s}=-\frac{Ls}{\nu},\qquad F_{x,s}=\Theta^{\nu}_{M_{\ell,n}}\left[\begin{matrix}\frac{x}{\nu}\\ \gamma_{s}\end{matrix}\right](\tau,z). \tag*{(92)} γ s = − ν L s , F x , s = Θ M ℓ , n ν [ ν x γ s ] ( τ , z ) . (92) This is well defined. Indeed, replacing s s s s + 1 η s+1\eta s + 1 η η ∈ ( 2 Z ) 1 , n \eta\in(2\mathbb{Z})^{1,n} η ∈ ( 2 Z ) 1 , n γ s \gamma_{s} γ s − r η / ν = M ℓ − 1 ( − e d η / 2 ) -r\eta/\nu=M_{\ell}^{-1}(-e_{d}\eta/2) − r η / ν = M ℓ − 1 ( − e d η /2 ) Lemma 2.6 , the corresponding factor in a single theta series is ε ( − 1 2 ν tr ( x T e d η ) ) \varepsilon\left(-\frac{1}{2\nu}\operatorname{tr}(x^{T}e_{d}\eta)\right) ε ( − 2 ν 1 tr ( x T e d η ) ) ν \nu ν x x x s s s ν k \nu k ν k k ∈ Z d , n k\in\mathbb{Z}^{d,n} k ∈ Z d , n γ s \gamma_{s} γ s − L k -Lk − L k Lemma 2.6 , the latter change multiplies a single theta series by
ζ − tr ( x T M ℓ L k ) , \zeta^{-\operatorname{tr}(x^{T}M_{\ell}Lk)}, ζ − tr ( x T M ℓ L k ) , whose ν \nu ν F x , s F_{x,s} F x , s
We apply Theorem 2.7 with
m = d , μ = ν , M 1 = M 2 = M ℓ , A = I d ν + 1 ν J ν ⊗ ( R − I d ) , m=d,\qquad\mu=\nu,\qquad M_{1}=M_{2}=M_{\ell},\qquad A=I_{d\nu}+\frac{1}{\nu}J_{\nu}\otimes(R-I_{d}), m = d , μ = ν , M 1 = M 2 = M ℓ , A = I d ν + ν 1 J ν ⊗ ( R − I d ) , where J ν J_{\nu} J ν ν × ν \nu\times\nu ν × ν A T M ℓ ( ν ) A = M ℓ ( ν ) A^{T}M_{\ell}^{(\nu)}A=M_{\ell}^{(\nu)} A T M ℓ ( ν ) A = M ℓ ( ν ) Theorem 5.3 . The integral matrix A ~ \widetilde{A} A
A ~ ⋅ ( x 1 , … , x ν ) T = ( x 1 − x ν , … , x ν − 1 − x ν , x ν E ν T ) T , \widetilde{A}\cdot(x_{1},\ldots,x_{\nu})^{T}=(x_{1}-x_{\nu},\ldots,x_{\nu-1}-x_{\nu},x_{\nu}E_{\nu}^{T})^{T}, A ⋅ ( x 1 , … , x ν ) T = ( x 1 − x ν , … , x ν − 1 − x ν , x ν E ν T ) T , for all x 1 , … , x ν ∈ Z 1 , d . x_{1},\ldots,x_{\nu}\in\mathbb{Z}^{1,d}. x 1 , … , x ν ∈ Z 1 , d . h = [ h 1 , … , h ν ] ∈ Z d ν , n h=[h_1,\ldots,h_\nu]\in\mathbb{Z}^{d\nu,n} h = [ h 1 , … , h ν ] ∈ Z d ν , n
A h − h = [ ( R − I d ) u ν , … , ( R − I d ) u ν ] , u = h 1 + ⋯ + h ν , Ah-h=\left[\frac{(R-I_d)u}{\nu},\ldots,\frac{(R-I_d)u}{\nu}\right],\qquad u=h_1+\cdots+h_\nu, A h − h = [ ν ( R − I d ) u , … , ν ( R − I d ) u ] , u = h 1 + ⋯ + h ν , and every u ∈ Z d , n u\in\mathbb{Z}^{d,n} u ∈ Z d , n A ~ v \widetilde{A}v A v v v v Z d , n \mathbb{Z}^{d,n} Z d , n E ν − 1 Z d , n = ν − 1 ( R − I d ) Z d , n E_\nu^{-1}\mathbb{Z}^{d,n}=\nu^{-1}(R-I_d)\mathbb{Z}^{d,n} E ν − 1 Z d , n = ν − 1 ( R − I d ) Z d , n
Z d ν , n + A Z d ν , n = A ~ − 1 Z d ν , n = Z d ν , n + { [ ( R − I d ) u ν , … , ( R − I d ) u ν ] : u ∈ Z d , n } . \mathbb{Z}^{d\nu,n}+A\mathbb{Z}^{d\nu,n}=\widetilde{A}^{-1}\mathbb{Z}^{d\nu,n}=\mathbb{Z}^{d\nu,n}+\left\{\left[\frac{(R-I_d)u}{\nu},\ldots,\frac{(R-I_d)u}{\nu}\right]:u\in\mathbb{Z}^{d,n}\right\}. Z d ν , n + A Z d ν , n = A − 1 Z d ν , n = Z d ν , n + { [ ν ( R − I d ) u , … , ν ( R − I d ) u ] : u ∈ Z d , n } . Thus all the hypotheses of Theorem 2.7 are satisfied. Moreover, ∣ det A ~ ∣ = ∣ det E ν ∣ = 2 ℓ d ℓ − 1 = 2 ν d ν − 1 |\det\widetilde{A}|=|\det E_\nu|=2^{\ell}d^{\ell-1}=2^\nu d^{\nu-1} ∣ det A ∣ = ∣ det E ν ∣ = 2 ℓ d ℓ − 1 = 2 ν d ν − 1
The two quotient groups in Theorem 2.7 are parametrized by X n † \mathcal{X}_n^\dagger X n † Y n † \mathcal{Y}_n^\dagger Y n †
X n † ⟶ ( A ~ A ) − 1 Z d ν , n Z d ν , n , v ⟼ [ v ν , … , v ν ] + Z d ν , n , \mathcal{X}_n^\dagger\longrightarrow\frac{(\widetilde{A}A)^{-1}\mathbb{Z}^{d\nu,n}}{\mathbb{Z}^{d\nu,n}},\qquad v\longmapsto\left[\frac{v}{\nu},\ldots,\frac{v}{\nu}\right]+\mathbb{Z}^{d\nu,n}, X n † ⟶ Z d ν , n ( A A ) − 1 Z d ν , n , v ⟼ [ ν v , … , ν v ] + Z d ν , n , and the second parametrization is
Y n † ⟶ A − 1 M ℓ ( ν ) − 1 Z d ν , n A − 1 M ℓ ( ν ) − 1 A ~ T Z d ν , n , u ⟼ β ( u ) + A − 1 M ℓ ( ν ) − 1 A ~ T Z d ν , n , \mathcal{Y}_n^\dagger\longrightarrow\frac{A^{-1}M_\ell^{(\nu)-1}\mathbb{Z}^{d\nu,n}}{A^{-1}M_\ell^{(\nu)-1}\widetilde{A}^{T}\mathbb{Z}^{d\nu,n}},\qquad u\longmapsto\beta(u)+A^{-1}M_\ell^{(\nu)-1}\widetilde{A}^{T}\mathbb{Z}^{d\nu,n}, Y n † ⟶ A − 1 M ℓ ( ν ) − 1 A T Z d ν , n A − 1 M ℓ ( ν ) − 1 Z d ν , n , u ⟼ β ( u ) + A − 1 M ℓ ( ν ) − 1 A T Z d ν , n , where
β ( u ) = [ γ u , … , γ u , γ u + M ℓ − 1 u ] . \beta(u)=[\gamma_u,\ldots,\gamma_u,\gamma_u+M_\ell^{-1}u]. β ( u ) = [ γ u , … , γ u , γ u + M ℓ − 1 u ] . The proof that these maps are group isomorphisms is analogous to the proof for the maps from X n \mathcal{X}_n X n Y n \mathcal{Y}_n Y n Theorem 5.3 .
For fixed x ∈ X n † x\in\mathcal{X}_n^\dagger x ∈ X n † t ∈ Y n † t\in\mathcal{Y}_n^\dagger t ∈ Y n † Theorem 2.7 to
a i = x ν , b i = R γ t , z i = R z ( 1 ≤ i ≤ ν ) . a_i=\frac{x}{\nu},\qquad b_i=R\gamma_t,\qquad z_i=Rz\qquad(1\leq i\leq\nu). a i = ν x , b i = R γ t , z i = R z ( 1 ≤ i ≤ ν ) . Since A A A R R R
a i ′ = R − 1 x ν , b i ′ = γ t , w i = z ( 1 ≤ i ≤ ν ) . a_i'=\frac{R^{-1}x}{\nu},\qquad b_i'=\gamma_t,\qquad w_i=z\qquad(1\leq i\leq\nu). a i ′ = ν R − 1 x , b i ′ = γ t , w i = z ( 1 ≤ i ≤ ν ) . Because R − 1 x ∈ X n † R^{-1}x\in\mathcal{X}_n^\dagger R − 1 x ∈ X n † v = y − R − 1 x v=y-R^{-1}x v = y − R − 1 x
[ y − R − 1 x ν , … , y − R − 1 x ν ] , y ∈ X n † . \left[\frac{y-R^{-1}x}{\nu},\ldots,\frac{y-R^{-1}x}{\nu}\right],\qquad y\in\mathcal{X}_n^\dagger. [ ν y − R − 1 x , … , ν y − R − 1 x ] , y ∈ X n † . For the representative indexed by u u u s = t + u s=t+u s = t + u u = s − t u=s-t u = s − t Y n † \mathcal{Y}_n^\dagger Y n † I d − M ℓ L = R T I_d-M_\ell L=R^T I d − M ℓ L = R T
M ℓ ( ν γ u + M ℓ − 1 u ) = R T u . M_\ell(\nu\gamma_u+M_\ell^{-1}u)=R^T u. M ℓ ( ν γ u + M ℓ − 1 u ) = R T u . The exterior factor in (14) is therefore ζ − tr ( x T u ) \zeta^{-\operatorname{tr}(x^Tu)} ζ − tr ( x T u ) M ℓ − 1 u M_\ell^{-1}u M ℓ − 1 u ζ tr ( y T u ) \zeta^{\operatorname{tr}(y^Tu)} ζ tr ( y T u ) ⟨ y − x , s − t ⟩ ν \langle y-x,s-t\rangle_\nu ⟨ y − x , s − t ⟩ ν 2 n ν ( d − 1 ) n F R − 1 x , t 2^{n\nu}(d-1)^nF_{R^{-1}x,t} 2 n ν ( d − 1 ) n F R − 1 x , t
Θ M ℓ , n [ v , R w ] ( τ , R z ) = Θ M ℓ , n [ R − 1 v , w ] ( τ , z ) . \Theta_{M_\ell,n}[v,Rw](\tau,Rz)=\Theta_{M_\ell,n}[R^{-1}v,w](\tau,z). Θ M ℓ , n [ v , R w ] ( τ , R z ) = Θ M ℓ , n [ R − 1 v , w ] ( τ , z ) . Thus Theorem 2.7 yields
2 n ν ( d − 1 ) n F R − 1 x , t = ∑ y ∈ X n † , s ∈ Y n † ⟨ y − x , s − t ⟩ ν F y , s . (93) 2^{n\nu}(d-1)^nF_{R^{-1}x,t}=\sum_{\substack{y\in\mathcal{X}_n^\dagger,\,s\in\mathcal{Y}_n^\dagger}}\langle y-x,s-t\rangle_\nu F_{y,s}. \tag*{(93)} 2 n ν ( d − 1 ) n F R − 1 x , t = y ∈ X n † , s ∈ Y n † ∑ ⟨ y − x , s − t ⟩ ν F y , s . (93) which is precisely the relation in (90) . Lemma 6.3 therefore gives (91) for the F x , s F_{x,s} F x , s (92) .
It remains to choose coordinates on Y n † \mathcal{Y}_{n}^{\dagger} Y n † L L L t t t B = − L t B=-Lt B = − L t ν \nu ν t t t t + 1 η t+\mathbf{1}\eta t + 1 η B B B B − r η B-r\eta B − r η d = ℓ − 1 d=\ell-1 d = ℓ − 1 ℓ \ell ℓ B B B { 0 , 1 } n \{0,1\}^{n} { 0 , 1 } n d − 1 d-1 d − 1 ( Z / ν Z ) n (\mathbb{Z}/\nu\mathbb{Z})^{n} ( Z / ν Z ) n
B = [ b 1 , … , b d − 1 , c ] B=[b_{1},\ldots,b_{d-1},c] B = [ b 1 , … , b d − 1 , c ] used in the statement. Finally, − r T L − 1 = ( 1 , … , 1 , − d ) ≡ 1 T ( m o d ℓ ) -r^{T}L^{-1}=(1,\ldots,1,-d)\equiv\mathbf{1}^{T}\pmod{\ell} − r T L − 1 = ( 1 , … , 1 , − d ) ≡ 1 T ( mod ℓ ) ⟨ 2 r κ , t ⟩ ν = ξ κ ⋅ ( b 1 + ⋯ + b d − 1 + c ) \langle2r\kappa,t\rangle_{\nu}=\xi^{\kappa\cdot(b_{1}+\cdots+b_{d-1}+c)} ⟨ 2 r κ , t ⟩ ν = ξ κ ⋅ ( b 1 + ⋯ + b d − 1 + c ) (91) and (92) yields (86) . □
Quartic identities
We now specialize to ℓ = 2 \ell=2 ℓ = 2 d = 1 d=1 d = 1 M 2 = ( 2 ) M_{2}=(2) M 2 = ( 2 ) F 2 = Z / 2 Z \mathbb{F}_{2}=\mathbb{Z}/2\mathbb{Z} F 2 = Z /2 Z p , q ∈ F 2 n p,q\in\mathbb{F}_{2}^{n} p , q ∈ F 2 n
Φ p , q ( n ) ( τ , z ) : = Θ ( 2 ) , n [ p / 2 q / 4 ] ( τ , z ) , φ p , q ( n ) ( τ ) : = Φ p , q ( n ) ( τ , 0 ) . (94) \Phi_{p,q}^{(n)}(\tau,z):=\Theta_{(2),n}\left[\begin{matrix}p/2\\q/4\end{matrix}\right](\tau,z),\qquad\varphi_{p,q}^{(n)}(\tau):=\Phi_{p,q}^{(n)}(\tau,0). \tag*{(94)} Φ p , q ( n ) ( τ , z ) := Θ ( 2 ) , n [ p /2 q /4 ] ( τ , z ) , φ p , q ( n ) ( τ ) := Φ p , q ( n ) ( τ , 0 ) . (94) Changing the lift of p p p q q q q + 2 k q+2k q + 2 k k ∈ Z n k\in\mathbb{Z}^{n} k ∈ Z n ( − 1 ) p ⋅ k (-1)^{p\cdot k} ( − 1 ) p ⋅ k
Corollary 6.4. Let n ≥ 1 n\geq1 n ≥ 1 τ ∈ H n \tau\in\mathfrak{H}_{n} τ ∈ H n z ∈ C 1 , n z\in\mathbb{C}^{1,n} z ∈ C 1 , n λ , κ ∈ F 2 n \lambda,\kappa\in\mathbb{F}_{2}^{n} λ , κ ∈ F 2 n λ ⋅ κ = 1 \lambda\cdot\kappa=1 λ ⋅ κ = 1
∑ p , q ∈ F 2 n ( − 1 ) λ ⋅ p + κ ⋅ q Φ p , q ( n ) ( τ , z ) 4 = 0. (95) \sum_{p,q\in\mathbb{F}_{2}^{n}}(-1)^{\lambda\cdot p+\kappa\cdot q}\Phi_{p,q}^{(n)}(\tau,z)^{4}=0. \tag*{(95)} p , q ∈ F 2 n ∑ ( − 1 ) λ ⋅ p + κ ⋅ q Φ p , q ( n ) ( τ , z ) 4 = 0. (95) Moreover,
p ⋅ q = 1 ⟹ φ p , q ( n ) ( τ ) = 0. (96) p\cdot q=1\quad\Longrightarrow\quad\varphi_{p,q}^{(n)}(\tau)=0. \tag*{(96)} p ⋅ q = 1 ⟹ φ p , q ( n ) ( τ ) = 0. (96) Proof. Equation (95) follows from Theorem 6.1 upon specializing to ℓ = 2 \ell=2 ℓ = 2 a = p a=p a = p c = q c=q c = q (96) , the involution g ↦ − g − p g\mapsto-g-p g ↦ − g − p p ⋅ q = 1 p\cdot q=1 p ⋅ q = 1 p i = 1 p_i=1 p i = 1 i i i 2 g i = − 1 2g_i=-1 2 g i = − 1 ( − 1 ) p ⋅ q = − 1 (-1)^{p\cdot q}=-1 ( − 1 ) p ⋅ q = − 1
In degree one, (95) gives an elliptic-variable form of Jacobi’s quartic identity.
Corollary 6.5. For τ ∈ H 1 \tau\in\mathfrak{H}_{1} τ ∈ H 1 z ∈ C z\in\mathbb{C} z ∈ C
Φ 0 , 0 ( 1 ) ( τ , z ) 4 + Φ 1 , 1 ( 1 ) ( τ , z ) 4 = Φ 1 , 0 ( 1 ) ( τ , z ) 4 + Φ 0 , 1 ( 1 ) ( τ , z ) 4 . (97) \Phi_{0,0}^{(1)}(\tau,z)^{4}+\Phi_{1,1}^{(1)}(\tau,z)^{4}=\Phi_{1,0}^{(1)}(\tau,z)^{4}+\Phi_{0,1}^{(1)}(\tau,z)^{4}. \tag*{(97)} Φ 0 , 0 ( 1 ) ( τ , z ) 4 + Φ 1 , 1 ( 1 ) ( τ , z ) 4 = Φ 1 , 0 ( 1 ) ( τ , z ) 4 + Φ 0 , 1 ( 1 ) ( τ , z ) 4 . (97) At z = 0 z=0 z = 0
φ 0 , 0 ( 1 ) ( τ ) 4 = φ 1 , 0 ( 1 ) ( τ ) 4 + φ 0 , 1 ( 1 ) ( τ ) 4 . (98) \varphi_{0,0}^{(1)}(\tau)^{4}=\varphi_{1,0}^{(1)}(\tau)^{4}+\varphi_{0,1}^{(1)}(\tau)^{4}. \tag*{(98)} φ 0 , 0 ( 1 ) ( τ ) 4 = φ 1 , 0 ( 1 ) ( τ ) 4 + φ 0 , 1 ( 1 ) ( τ ) 4 . (98) Proof. For n = 1 n=1 n = 1 λ = κ = 1 \lambda=\kappa=1 λ = κ = 1 (95) are
Φ 0 , 0 ( 1 ) ( τ , z ) 4 − Φ 0 , 1 ( 1 ) ( τ , z ) 4 − Φ 1 , 0 ( 1 ) ( τ , z ) 4 + Φ 1 , 1 ( 1 ) ( τ , z ) 4 = 0. \Phi_{0,0}^{(1)}(\tau,z)^{4}-\Phi_{0,1}^{(1)}(\tau,z)^{4}-\Phi_{1,0}^{(1)}(\tau,z)^{4}+\Phi_{1,1}^{(1)}(\tau,z)^{4}=0. Φ 0 , 0 ( 1 ) ( τ , z ) 4 − Φ 0 , 1 ( 1 ) ( τ , z ) 4 − Φ 1 , 0 ( 1 ) ( τ , z ) 4 + Φ 1 , 1 ( 1 ) ( τ , z ) 4 = 0. Rearrangement gives (97) . At z = 0 z=0 z = 0 (96) gives φ 1 , 1 ( 1 ) = 0 \varphi_{1,1}^{(1)}=0 φ 1 , 1 ( 1 ) = 0 (98) . □
For an explicit q q q q = e ( τ ) q=e(\tau) q = e ( τ ) u = e ( 2 z ) u=e(2z) u = e ( 2 z ) u α = e ( 2 α z ) u^{\alpha}=e(2\alpha z) u α = e ( 2 α z ) (97) is
( ∑ k ∈ Z q k 2 u k ) 4 + ( ∑ k ∈ Z ( − 1 ) k q ( k + 1 / 2 ) 2 u k + 1 / 2 ) 4 = ( ∑ k ∈ Z q ( k + 1 / 2 ) 2 u k + 1 / 2 ) 4 + ( ∑ k ∈ Z ( − 1 ) k q k 2 u k ) 4 . (99) \left(\sum_{k\in\mathbb{Z}}q^{k^{2}}u^{k}\right)^{4}
+\left(\sum_{k\in\mathbb{Z}}(-1)^{k}q^{(k+1/2)^{2}}u^{k+1/2}\right)^{4}
=
\left(\sum_{k\in\mathbb{Z}}q^{(k+1/2)^{2}}u^{k+1/2}\right)^{4}
+\left(\sum_{k\in\mathbb{Z}}(-1)^{k}q^{k^{2}}u^{k}\right)^{4}. \tag*{(99)} ( k ∈ Z ∑ q k 2 u k ) 4 + ( k ∈ Z ∑ ( − 1 ) k q ( k + 1/2 ) 2 u k + 1/2 ) 4 = ( k ∈ Z ∑ q ( k + 1/2 ) 2 u k + 1/2 ) 4 + ( k ∈ Z ∑ ( − 1 ) k q k 2 u k ) 4 . (99) Here Φ 1 , 1 ( 1 ) \Phi^{(1)}_{1,1} Φ 1 , 1 ( 1 ) i 4 = 1 i^4=1 i 4 = 1 [4 Theorem 1.2] proved an identity whose specialization x = y = u x=y=u x = y = u (99) . Jacobi [12 p. 90, Eq. (14)] gives the classical theta-constant identity (98) . In q q q
( ∑ k ∈ Z q k 2 ) 4 = ( ∑ k ∈ Z q ( k + 1 / 2 ) 2 ) 4 + ( ∑ k ∈ Z ( − 1 ) k q k 2 ) 4 . \left(\sum_{k\in\mathbb{Z}}q^{k^2}\right)^4=\left(\sum_{k\in\mathbb{Z}}q^{(k+1/2)^2}\right)^4+\left(\sum_{k\in\mathbb{Z}}(-1)^kq^{k^2}\right)^4. ( k ∈ Z ∑ q k 2 ) 4 = ( k ∈ Z ∑ q ( k + 1/2 ) 2 ) 4 + ( k ∈ Z ∑ ( − 1 ) k q k 2 ) 4 . We finally specialize to degree two. We write
τ = ( τ 1 τ 0 τ 0 τ 2 ) ∈ H 2 , Q = e ( τ 1 ) , S = e ( τ 0 ) , T = e ( τ 2 ) . \tau=\begin{pmatrix}\tau_1&\tau_0\\\tau_0&\tau_2\end{pmatrix}\in\mathcal{H}_2,\qquad Q=e(\tau_1),\qquad S=e(\tau_0),\qquad T=e(\tau_2). τ = ( τ 1 τ 0 τ 0 τ 2 ) ∈ H 2 , Q = e ( τ 1 ) , S = e ( τ 0 ) , T = e ( τ 2 ) . For binary representatives p = ( p 1 , p 2 ) \mathbf{p}=(p_1,p_2) p = ( p 1 , p 2 ) q = ( q 1 , q 2 ) \mathbf{q}=(q_1,q_2) q = ( q 1 , q 2 ) (94) gives
ϕ p , q ( 2 ) ( τ ) = i p 1 q 1 + p 2 q 2 ∑ j , k ∈ Z ( − 1 ) q 1 j + q 2 k Q ( j + p 1 / 2 ) 2 S 2 ( j + p 1 / 2 ) ( k + p 2 / 2 ) T ( k + p 2 / 2 ) 2 . (100) \phi^{(2)}_{\mathbf{p},\mathbf{q}}(\tau)=i^{p_1q_1+p_2q_2}\sum_{j,k\in\mathbb{Z}}(-1)^{q_1j+q_2k}Q^{(j+p_1/2)^2}S^{2(j+p_1/2)(k+p_2/2)}T^{(k+p_2/2)^2}. \tag*{(100)} ϕ p , q ( 2 ) ( τ ) = i p 1 q 1 + p 2 q 2 j , k ∈ Z ∑ ( − 1 ) q 1 j + q 2 k Q ( j + p 1 /2 ) 2 S 2 ( j + p 1 /2 ) ( k + p 2 /2 ) T ( k + p 2 /2 ) 2 . (100) We use the abbreviations
0 = ( 0 , 0 ) , e 1 = ( 1 , 0 ) , e 2 = ( 0 , 1 ) , e = ( 1 , 1 ) in F 2 2 . \mathbf{0}=(0,0),\qquad\mathbf{e}_1=(1,0),\qquad\mathbf{e}_2=(0,1),\qquad\mathbf{e}=(1,1)\quad\text{in }\mathbb{F}_2^2. 0 = ( 0 , 0 ) , e 1 = ( 1 , 0 ) , e 2 = ( 0 , 1 ) , e = ( 1 , 1 ) in F 2 2 . The ten pairs with p ⋅ q = 0 \mathbf{p}\cdot\mathbf{q}=0 p ⋅ q = 0
( p , q ) = ( 0 , 0 ) , ( 0 , e 1 ) , ( 0 , e 2 ) , ( 0 , e ) , ( e 1 , 0 ) , ( e 1 , e 2 ) , ( e 2 , 0 ) , ( e 2 , e 1 ) , ( e , 0 ) , ( e , e ) . (101) (\mathbf{p},\mathbf{q})=(\mathbf{0},\mathbf{0}),(\mathbf{0},\mathbf{e}_1),(\mathbf{0},\mathbf{e}_2),(\mathbf{0},\mathbf{e}),(\mathbf{e}_1,\mathbf{0}),(\mathbf{e}_1,\mathbf{e}_2),(\mathbf{e}_2,\mathbf{0}),(\mathbf{e}_2,\mathbf{e}_1),(\mathbf{e},\mathbf{0}),(\mathbf{e},\mathbf{e}). \tag*{(101)} ( p , q ) = ( 0 , 0 ) , ( 0 , e 1 ) , ( 0 , e 2 ) , ( 0 , e ) , ( e 1 , 0 ) , ( e 1 , e 2 ) , ( e 2 , 0 ) , ( e 2 , e 1 ) , ( e , 0 ) , ( e , e ) . (101) The relations in the following theorem are the classical genus-two quartic relations; see Cléry, van der Geer, and Grushevsky [5 ] .
Theorem 6.6. The fourth powers of the theta series defined by (100) and indexed by (101) satisfy the following five independent relations:
ϕ 0 , 0 4 = ϕ 0 , e 2 4 + ϕ e 2 , 0 4 + ϕ e , e 4 , \phi^4_{\mathbf{0},\mathbf{0}}=\phi^4_{\mathbf{0},\mathbf{e}_2}+\phi^4_{\mathbf{e}_2,\mathbf{0}}+\phi^4_{\mathbf{e},\mathbf{e}}, ϕ 0 , 0 4 = ϕ 0 , e 2 4 + ϕ e 2 , 0 4 + ϕ e , e 4 , ϕ 0 , 0 4 = ϕ 0 , e 2 4 + ϕ e 2 , e 1 4 + ϕ e , 0 4 , \phi^4_{\mathbf{0},\mathbf{0}}=\phi^4_{\mathbf{0},\mathbf{e}_2}+\phi^4_{\mathbf{e}_2,\mathbf{e}_1}+\phi^4_{\mathbf{e},\mathbf{0}}, ϕ 0 , 0 4 = ϕ 0 , e 2 4 + ϕ e 2 , e 1 4 + ϕ e , 0 4 , ϕ 0 , 0 4 = ϕ 0 , e 1 4 + ϕ e 1 , 0 4 + ϕ e , e 4 , \phi^4_{\mathbf{0},\mathbf{0}}=\phi^4_{\mathbf{0},\mathbf{e}_1}+\phi^4_{\mathbf{e}_1,\mathbf{0}}+\phi^4_{\mathbf{e},\mathbf{e}}, ϕ 0 , 0 4 = ϕ 0 , e 1 4 + ϕ e 1 , 0 4 + ϕ e , e 4 , ϕ 0 , 0 4 = ϕ 0 , e 1 4 + ϕ e 1 , e 2 4 + ϕ e 2 , 0 4 , \phi^4_{\mathbf{0},\mathbf{0}}=\phi^4_{\mathbf{0},\mathbf{e}_1}+\phi^4_{\mathbf{e}_1,\mathbf{e}_2}+\phi^4_{\mathbf{e}_2,\mathbf{0}}, ϕ 0 , 0 4 = ϕ 0 , e 1 4 + ϕ e 1 , e 2 4 + ϕ e 2 , 0 4 , ϕ 0 , 0 4 = ϕ 0 , e 4 + ϕ e 2 , e 1 4 + ϕ e 1 , 0 4 . \phi^4_{\mathbf{0},\mathbf{0}}=\phi^4_{\mathbf{0},\mathbf{e}}+\phi^4_{\mathbf{e}_2,\mathbf{e}_1}+\phi^4_{\mathbf{e}_1,\mathbf{0}}. ϕ 0 , 0 4 = ϕ 0 , e 4 + ϕ e 2 , e 1 4 + ϕ e 1 , 0 4 . (We suppress the superscript ( 2 ) (2) ( 2 ) τ \tau τ
Proof. We order the ten fourth powers according to the lexicographic list of even characteristics used by Cléry, van der Geer, and Grushevsky [5 Section 3] :
( x 1 , … , x 10 ) = ( ϕ 0 , 0 4 , ϕ 0 , e 2 4 , ϕ 0 , e 1 4 , ϕ 0 , e 4 , ϕ e 2 , 0 4 , ϕ e 2 , e 1 4 , ϕ e 1 , 0 4 , ϕ e 1 , e 2 4 , ϕ e , 0 4 , ϕ e , e 4 ) . (x_1,\ldots,x_{10})=(\phi^4_{\mathbf{0},\mathbf{0}},\phi^4_{\mathbf{0},\mathbf{e}_2},\phi^4_{\mathbf{0},\mathbf{e}_1},\phi^4_{\mathbf{0},\mathbf{e}},\phi^4_{\mathbf{e}_2,\mathbf{0}},\phi^4_{\mathbf{e}_2,\mathbf{e}_1},\phi^4_{\mathbf{e}_1,\mathbf{0}},\phi^4_{\mathbf{e}_1,\mathbf{e}_2},\phi^4_{\mathbf{e},\mathbf{0}},\phi^4_{\mathbf{e},\mathbf{e}}). ( x 1 , … , x 10 ) = ( ϕ 0 , 0 4 , ϕ 0 , e 2 4 , ϕ 0 , e 1 4 , ϕ 0 , e 4 , ϕ e 2 , 0 4 , ϕ e 2 , e 1 4 , ϕ e 1 , 0 4 , ϕ e 1 , e 2 4 , ϕ e , 0 4 , ϕ e , e 4 ) . The six admissible pairs ( λ , κ ) (\lambda,\kappa) ( λ , κ )
( e 2 , e 2 ) , ( e 2 , e ) , ( e 1 , e 1 ) , ( e 1 , e ) , ( e , e 2 ) , ( e , e 1 ) . (\mathbf{e}_2,\mathbf{e}_2),\quad(\mathbf{e}_2,\mathbf{e}),\quad(\mathbf{e}_1,\mathbf{e}_1),\quad(\mathbf{e}_1,\mathbf{e}),\quad(\mathbf{e},\mathbf{e}_2),\quad(\mathbf{e},\mathbf{e}_1). ( e 2 , e 2 ) , ( e 2 , e ) , ( e 1 , e 1 ) , ( e 1 , e ) , ( e , e 2 ) , ( e , e 1 ) . After the six odd theta constants are removed by (96) , the corresponding six instances of (95) have coefficient matrix
( 1 − 1 1 − 1 − 1 − 1 1 − 1 − 1 1 1 − 1 − 1 1 − 1 1 1 − 1 − 1 − 1 1 1 − 1 − 1 1 − 1 − 1 − 1 − 1 1 1 − 1 − 1 1 1 − 1 − 1 1 − 1 − 1 1 − 1 1 − 1 − 1 − 1 − 1 1 1 − 1 1 1 − 1 − 1 − 1 1 − 1 − 1 1 − 1 ) . \begin{pmatrix}
1&-1&1&-1&-1&-1&1&-1&-1&1\\
1&-1&-1&1&-1&1&1&-1&-1&-1\\
1&1&-1&-1&1&-1&-1&-1&-1&1\\
1&-1&-1&1&1&-1&-1&1&-1&-1\\
1&-1&1&-1&-1&-1&-1&1&1&-1\\
1&1&-1&-1&-1&1&-1&-1&1&-1
\end{pmatrix}. 1 1 1 1 1 1 − 1 − 1 1 − 1 − 1 1 1 − 1 − 1 − 1 1 − 1 − 1 1 − 1 1 − 1 − 1 − 1 − 1 1 1 − 1 − 1 − 1 1 − 1 − 1 − 1 1 1 1 − 1 − 1 − 1 − 1 − 1 − 1 − 1 1 1 − 1 − 1 − 1 − 1 − 1 1 1 1 − 1 1 − 1 − 1 − 1 . If its rows are denoted by ρ 1 , … , ρ 6 \rho_1,\ldots,\rho_6 ρ 1 , … , ρ 6
ρ 2 + ρ 5 2 , ρ 1 + ρ 4 2 , ρ 4 + ρ 6 2 , ρ 2 + ρ 3 2 , ρ 3 + ρ 5 2 . \frac{\rho_2+\rho_5}{2},\qquad\frac{\rho_1+\rho_4}{2},\qquad\frac{\rho_4+\rho_6}{2},\qquad\frac{\rho_2+\rho_3}{2},\qquad\frac{\rho_3+\rho_5}{2}. 2 ρ 2 + ρ 5 , 2 ρ 1 + ρ 4 , 2 ρ 4 + ρ 6 , 2 ρ 2 + ρ 3 , 2 ρ 3 + ρ 5 . This proves all five relations. Their coefficient vectors are independent: the coefficients of x 5 , x 8 , x 4 , x 3 , x 2 x_5,x_8,x_4,x_3,x_2 x 5 , x 8 , x 4 , x 3 , x 2
For comparison with the standard theta constants, we use the normalization
ϑ [ p q ] ( Ω , 0 ) : = ∑ g ∈ Z 2 exp ( π i ( ( g + p / 2 ) T Ω ( g + p / 2 ) + ( g + p / 2 ) T q ) ) . \vartheta\begin{bmatrix}p\\q\end{bmatrix}(\Omega,0):=\sum_{g\in\mathbb{Z}^2}\exp\left(\pi i\left((g+p/2)^{T}\Omega(g+p/2)+(g+p/2)^{T}q\right)\right). ϑ [ p q ] ( Ω , 0 ) := g ∈ Z 2 ∑ exp ( π i ( ( g + p /2 ) T Ω ( g + p /2 ) + ( g + p /2 ) T q ) ) . Definition (94) then gives the exact equality
ϕ p , q ( 2 ) ( τ ) = ϑ [ p q ] ( 2 τ , 0 ) . \phi_{p,q}^{(2)}(\tau)=\vartheta\begin{bmatrix}p\\q\end{bmatrix}(2\tau,0). ϕ p , q ( 2 ) ( τ ) = ϑ [ p q ] ( 2 τ , 0 ) . Under the ordering above, the fourth powers of these quantities are precisely the forms x i = ϑ ν i 4 x_i=\vartheta_{\nu_i}^{4} x i = ϑ ν i 4 [5 Notation 6.1] , evaluated at 2 τ 2\tau 2 τ [5 Section 6, Eq. (6.1)] show that these ten forms span a five-dimensional space. Hence their space of linear relations has dimension five, and the five independent relations displayed in the theorem form a basis. □
References
Department of Mathematics, Shanghai Normal University
Email address: rchen@shnu.edu.cn
School of Mathematical Sciences, Tongji University
Email address: haigangz@tongji.edu.cn
School of Mathematical Sciences, Key Laboratory of MEA (Ministry of Education) & Shanghai Key Laboratory of PMMP, East China Normal University
Email address, Corresponding author: zhuxiaojiemath@outlook.com
