Introduction
Let q = p b q = p^b q = p b p p p b b b C C C F q \mathbb{F}_q F q q q q Z ( C / F q , s ) Z(C / \mathbb{F}_q, s) Z ( C / F q , s ) exp ( ∑ m = 1 ∞ # C ( F q m ) s m m ) \exp \left(\sum_{m=1}^\infty \#C(\mathbb{F}_{q^m}) \frac{s^m}{m}\right) exp ( ∑ m = 1 ∞ # C ( F q m ) m s m ) [24 ] ), the zeta function is a rational function of the form
P C ( s ) ( 1 − s ) ( 1 − q s ) , P C ( s ) = ∏ i = 1 2 g ( 1 − α i s ) ∈ 1 + s Z [ s ] , \frac{P_C(s)}{(1-s)(1-qs)}, \quad P_C(s) =
\prod_{i=1}^{2g} (1-\alpha_i s) \in 1 + s\mathbb{Z}[s], ( 1 − s ) ( 1 − q s ) P C ( s ) , P C ( s ) = i = 1 ∏ 2 g ( 1 − α i s ) ∈ 1 + s Z [ s ] , where g g g C C C α i \alpha_i α i q \sqrt{q} q
The q q q α i \alpha_i α i P C ( s ) = 1 + c 1 s + ⋯ + c 2 g s 2 g P_C(s) = 1 + c_1 s + \dots + c_{2g} s^{2g} P C ( s ) = 1 + c 1 s + ⋯ + c 2 g s 2 g q q q { ( i , v q ( c i ) ) ∣ i = 0 , … , 2 g } \{(i, v_q(c_i)) \mid i = 0, \dots, 2g\} {( i , v q ( c i )) ∣ i = 0 , … , 2 g } c 0 = 1 c_0 = 1 c 0 = 1 v q v_q v q q q q Q ‾ p \overline{\mathbb{Q}}_p Q p v q ( q ) = 1 v_q(q) = 1 v q ( q ) = 1 ( 0 , 0 ) (0,0) ( 0 , 0 ) ( 2 g , g ) (2g, g) ( 2 g , g ) h h h λ \lambda λ h h h 1 − λ 1 - \lambda 1 − λ N P ( C / F q ) \mathop{\mathrm{NP}}(C / \mathbb{F}_q) NP ( C / F q ) q q q C C C C C C C C C F q \mathbb{F}_q F q q q q N P ( C ) \mathop{\mathrm{NP}}(C) NP ( C ) C C C C C C
Let f ∈ F ‾ p ( x ) f \in \overline{\mathbb{F}}_p(x) f ∈ F p ( x ) ℓ ≥ 1 \ell \ge 1 ℓ ≥ 1 ℓ \ell ℓ X f , ℓ X_{f, \ell} X f , ℓ
X f , ℓ : y p ℓ − y = f . X_{f, \ell} : y^{p^\ell} - y = f. X f , ℓ : y p ℓ − y = f . All elementary abelian p p p P 1 \mathbb{P}^1 P 1 [18 ] ), making them a subject of geometric interest.
This paper is concerned with elementary abelian p p p P 1 \mathbb{P}^1 P 1 0 0 0 ∞ \infty ∞ X f , ℓ X_{f, \ell} X f , ℓ f ∈ F ‾ p [ x ± 1 ] f \in \overline{\mathbb{F}}_p[x^{\pm1}] f ∈ F p [ x ± 1 ] f ( x ) = ∑ i = − d 1 d 2 a i x i f(x) = \sum_{i=-d_1}^{d_2} a_i x^i f ( x ) = ∑ i = − d 1 d 2 a i x i d 1 , d 2 ≥ 1 d_1, d_2 \ge 1 d 1 , d 2 ≥ 1 a − d 1 a d 2 ≠ 0 a_{-d_1}a_{d_2}\ne 0 a − d 1 a d 2 = 0 p ∤ d 1 d 2 p \nmid d_1 d_2 p ∤ d 1 d 2 d 1 d_1 d 1 0 0 0 d 2 d_2 d 2 ∞ \infty ∞ g = ( p ℓ − 1 ) ( d 1 + d 2 ) 2 g=\frac{(p^\ell-1)(d_1+d_2)}{2} g = 2 ( p ℓ − 1 ) ( d 1 + d 2 ) X F ‾ p , ℓ ( d 1 , d 2 ) \mathscr{X}_{\overline{\mathbb{F}}_p, \ell}(d_1, d_2) X F p , ℓ ( d 1 , d 2 ) ℓ \ell ℓ F ‾ p \overline{\mathbb{F}}_p F p
G N P F ‾ p , ℓ ( d 1 , d 2 ) ≔ inf X ∈ X F ‾ p , ℓ ( d 1 , d 2 ) N P ( X ) . \begin{equation}
\mathrm{GNP}_{\overline{\mathbb{F}}_p, \ell}(d_1, d_2) \coloneqq
\inf_{X \in \mathscr{X}_{\overline{\mathbb{F}}_p, \ell}(d_1, d_2)} \mathop{\mathrm{NP}}(X). \tag*{(1)}
\end{equation} GNP F p , ℓ ( d 1 , d 2 ) : = X ∈ X F p , ℓ ( d 1 , d 2 ) inf NP ( X ) . (1) Here, the infimum is taken with respect to the standard pointwise partial ordering on the set ( R ≥ 0 ) [ 0 , 2 g ] (\mathbb{R}_{\ge 0})^{[0,2g]} ( R ≥ 0 ) [ 0 , 2 g ] h : [ 0 , 2 g ] → R ≥ 0 h : [0,2g] \to \mathbb{R}_{\ge 0} h : [ 0 , 2 g ] → R ≥ 0 X X X N P ( X ) = G N P F ‾ p , ℓ ( d 1 , d 2 ) \mathop{\mathrm{NP}}(X) = \mathrm{GNP}_{\overline{\mathbb{F}}_p, \ell}(d_1, d_2) NP ( X ) = GNP F p , ℓ ( d 1 , d 2 ) X X X
The subject of determining the shape of G N P F ‾ p , ℓ ( d 1 , d 2 ) \mathop{\mathrm{GNP}}_{\overline{\mathbb{F}}_p, \ell}(d_1, d_2) GNP F p , ℓ ( d 1 , d 2 ) X F ‾ p , ℓ ( d 1 , d 2 ) \mathscr{X}_{\overline{\mathbb{F}}_p, \ell}(d_1, d_2) X F p , ℓ ( d 1 , d 2 ) [15 ] yield a universal lower bound known as the Hodge polygon. However, this lower bound is only sharp for very specific cases.
As d 1 d_1 d 1 d 2 d_2 d 2 p p p G N P F ‾ p , ℓ ( d 1 , d 2 ) \mathrm{GNP}_{\overline{\mathbb{F}}_p, \ell}(d_1, d_2) GNP F p , ℓ ( d 1 , d 2 ) d 1 d_1 d 1 d 2 d_2 d 2 G N P F ‾ p , ℓ ( d 1 , d 2 ) \mathrm{GNP}_{\overline{\mathbb{F}}_p, \ell}(d_1, d_2) GNP F p , ℓ ( d 1 , d 2 ) p → ∞ p \to \infty p → ∞ 1 1 1 ℓ = 1 \ell = 1 ℓ = 1 [25 ] for all p p p [16 ] .
Most recently, Zhu discovered the locus of generic elementary abelian p p p [26 ] . Consider a polynomial f ∈ Q ‾ [ x ] f \in \overline{\mathbb{Q}}[x] f ∈ Q [ x ] p \mathfrak{p} p K K K f f f f f f p \mathfrak{p} p f ‾ \overline{f} f f f f k p = O K / p k_{\mathfrak{p}} = \mathcal{O}_K / \mathfrak{p} k p = O K / p X f ‾ , ℓ X_{\overline{f}, \ell} X f , ℓ [26 Theorem 1.2] that an open condition on polynomials in Q ‾ [ x ] \overline{\mathbb{Q}}[x] Q [ x ] p \mathfrak{p} p p p p
Our first main theorem extends Zhu’s global result on elementary abelian p p p
Theorem. Let d 1 , d 2 ≥ 1 d_1, d_2 \ge 1 d 1 , d 2 ≥ 1 I = { − d 1 , … , d 2 } I = \{-d_1, \dots, d_2\} I = { − d 1 , … , d 2 } A Q I \mathbb{A}^I_{\mathbb{Q}} A Q I Q \mathbb{Q} Q I I I a ∈ A Q I ( Q ‾ ) a \in \mathbb{A}^I_{\mathbb{Q}}(\overline{\mathbb{Q}}) a ∈ A Q I ( Q ) f a = ∑ i ∈ I a i x i f_a = \sum_{i \in I} a_i x^i f a = ∑ i ∈ I a i x i U ⊆ A Q I \mathcal{U} \subseteq \mathbb{A}_{\mathbb{Q}}^I U ⊆ A Q I Q \mathbb{Q} Q a ∈ A Q I ( Q ‾ ) a \in \mathbb{A}_\mathbb{Q}^I(\overline{\mathbb{Q}}) a ∈ A Q I ( Q ) a − d 1 a d 2 ≠ 0 a_{-d_1}a_{d_2} \ne 0 a − d 1 a d 2 = 0
a ∈ U ( Q ‾ ) a \in \mathcal{U}(\overline{\mathbb{Q}}) a ∈ U ( Q )
There is a constant N a N_a N a p > N a p > N_a p > N a K / Q ( a i ∣ i ∈ I ) K / \mathbb{Q}(a_i \mid i \in I) K / Q ( a i ∣ i ∈ I ) p \mathfrak{p} p K K K p p p f ‾ a = f a ( m o d p ) \overline{f}_a = f_a \pmod{\mathfrak{p}} f a = f a ( mod p ) X f ‾ a , ℓ X_{\overline{f}_a, \ell} X f a , ℓ ℓ \ell ℓ [ k p : F p ] [k_\mathfrak{p}: \mathbb{F}_p] [ k p : F p ]
This extension cannot be obtained merely by treating positive and negative terms of f f f U \mathcal{U} U Remark 4.6 .
We construct a combinatorial cost polygon G L ( d 1 , d 2 , p ) \mathop{\mathrm{GL}}(d_1,d_2,p) GL ( d 1 , d 2 , p ) 2.5 , so that for p p p
G N P F ˉ p , ℓ ( d 1 , d 2 ) ( x ) = ( p ℓ − 1 ) G L ( d 1 , d 2 , p ) ( x p ℓ − 1 ) . \mathop{\mathrm{GNP}}_{\bar{F}_p,\ell}(d_1,d_2)(x)=(p^\ell-1)\mathop{\mathrm{GL}}(d_1,d_2,p)
\left(\frac{x}{p^\ell-1}\right). GNP F ˉ p , ℓ ( d 1 , d 2 ) ( x ) = ( p ℓ − 1 ) GL ( d 1 , d 2 , p ) ( p ℓ − 1 x ) . The same global open condition also controls T T T U \mathcal U U [19 ] extended in Proposition 6.4 to the range p ≥ 2 D p\ge2D p ≥ 2 D Theorem 6.5 . We use the C C C L L L § 3.3 and write π m = ζ p m − 1 \pi_m=\zeta_{p^m}-1 π m = ζ p m − 1 ζ p m \zeta_{p^m} ζ p m p m p^m p m
Corollary 1.2 (Simultaneous T T T I I I f a f_a f a U \mathcal{U} U (?) , and let a ∈ A Q I ( Q ‾ ) a\in\mathbb{A}^I_{\mathbb{Q}}(\overline{\mathbb{Q}}) a ∈ A Q I ( Q ) a − d 1 a d 2 ≠ 0 a_{-d_1}a_{d_2}\ne 0 a − d 1 a d 2 = 0 a ∈ U ( Q ‾ ) a\in \mathcal{U}(\overline{\mathbb{Q}}) a ∈ U ( Q ) N a N_a N a p > N a p>N_a p > N a K K K f a f_a f a p \mathfrak{p} p K K K p p p f ˉ a \bar f_a f ˉ a C ∗ ( f ˉ a ; T , s ) , C ∗ ( f ˉ a ; T , s ) ∣ T = π m , L ∗ ( f ˉ a ; T , s ) ∣ T = π m C^*(\bar f_a;T,s),\qquad
\left.C^*(\bar f_a;T,s)\right|_{T=\pi_m},\qquad
\left.L^*(\bar f_a;T,s)\right|_{T=\pi_m} C ∗ ( f ˉ a ; T , s ) , C ∗ ( f ˉ a ; T , s ) T = π m , L ∗ ( f ˉ a ; T , s ) T = π m m ≥ 1 m\ge1 m ≥ 1 T T T π m \pi_m π m d 1 d_1 d 1 d 2 d_2 d 2 L L L [ 0 , p m − 1 ( d 1 + d 2 ) ] [0,p^{m-1}(d_1+d_2)] [ 0 , p m − 1 ( d 1 + d 2 )]
In particular, the bound N a N_a N a m m m K K K L L L p m p^m p m f ˉ a \bar f_a f ˉ a [21 ] study Newton slopes and spectral halos for higher-rank Artin–Schreier–Witt towers. Tang [23 ] proves slope uniformity for strongly genus-stable higher-rank towers and slope stability under an additional coefficient-growth condition. These works study character variation in fixed characteristic for towers ramified only at infinity. Our T T T Q \mathbb{Q} Q
Our proof is based on valuation estimates for Dwork Fredholm determinants. § 2 establishes the combinatorial results that identify the minimal terms and prevent their cancellation. § 3 reviews the relevant Dwork theory and T T T § 4 constructs the polynomials controlling genericity and extracts factors independent of p p p § 5 proves the compatibility of the genericity criterion with finite extensions of the base field. § 6 combines these results to prove (?) . After establishing in Proposition 6.4 simultaneous genericity for the T T T C C C C C C L L L [19 ] , we prove Corollary 1.2 in Theorem 6.5 .
Combinatorics of minimal terms in Fredholm determinants
In this section, we prove the combinatorial facts needed for analyzing the Fredholm determinants in the sections that follow. We first describe integer representations and minimal representations in § 2.1 . All of the key results are in § 2.2 and § 2.3 . § 2.2 will determine which index sets and permutations contribute to the valuations in the Fredholm determinant. It will also determine which points are vertices on the generic Newton polygon. Finally, we will choose a monomial order in § 2.3 that singles out a noncancelling term in the relevant part of the Fredholm determinant’s coefficients.
For any integer N N N d d d [ N ] d [N]_d [ N ] d N N N d d d [ N ] d [N]_d [ N ] d { 0 , … , d − 1 } \{0, \dots, d-1\} { 0 , … , d − 1 } N N N d d d d 1 d_1 d 1 d 2 d_2 d 2
D = \mathop{\mathrm{lcm}}(d_1,d_2),\quad
I = \{-d_1, \dots, d_2\}, \quad \mbox{and} \quad
I_0 = I \setminus \{0\}.
Minimal integer representations
The Dwork operator used in S:dwork-work,S:ck_formula is built out of a p p p F ( s ) = ∑ N ∈ Z G N s N F(s) = \sum_{N \in \mathbb{Z}} G_N s^N F ( s ) = ∑ N ∈ Z G N s N G N G_N G N N N N
In computational mathematics, a change-making problem concerns the minimization of a summation ∑ i t i \sum_i t_i ∑ i t i t i t_i t i ∑ i t i c i = N \sum_i t_i c_i = N ∑ i t i c i = N { c i } \{c_i\} { c i } N N N c i c_i c i N N N
Definition 2.1. For N ∈ Z N \in \mathbb{Z} N ∈ Z
T ( N ) = { t = ( t i ) ∈ Z ≥ 0 I 0 ∣ ∑ i ∈ I 0 i t i = N } , ∣ t ∣ = ∑ i ∈ I 0 t i , M ( N ) = min t ∈ T ( N ) ∣ t ∣ , \mathcal{T}(N) = \left\{\mathbf{t}= (t_i) \in \mathbb{Z}_{\ge 0}^{I_0} \mid
\sum_{i \in I_0} i t_i = N\right\}, \quad
|\mathbf{t}| = \sum_{i \in I_0} t_i, \quad
M(N) = \min_{\mathbf{t}\in \mathcal{T}(N)} |\mathbf{t}|, T ( N ) = { t = ( t i ) ∈ Z ≥ 0 I 0 ∣ i ∈ I 0 ∑ i t i = N } , ∣ t ∣ = i ∈ I 0 ∑ t i , M ( N ) = t ∈ T ( N ) min ∣ t ∣ , and S o l ( N ) = { t ∈ T ( N ) ∣ ∣ t ∣ = M ( N ) } \mathop{\mathrm{Sol}}(N) = \left\{\mathbf{t}\in \mathcal{T}(N) \mid |\mathbf{t}| = M(N)\right\} Sol ( N ) = { t ∈ T ( N ) ∣ ∣ t ∣ = M ( N ) } t ∈ T ( N ) \mathbf{t}\in \mathcal{T}(N) t ∈ T ( N ) N N N t ∈ S o l ( N ) \mathbf{t}\in \mathop{\mathrm{Sol}}(N) t ∈ Sol ( N ) t \mathbf{t} t N N N N N N
We first prove two lemmas about M ( N ) M(N) M ( N ) S o l ( N ) \mathop{\mathrm{Sol}}(N) Sol ( N )
Lemma 2.2. For any N ∈ Z N \in \mathbb{Z} N ∈ Z d = d 1 d = d_1 d = d 1 N < 0 N<0 N < 0 d = d 2 d = d_2 d = d 2 N ≥ 0 N \ge 0 N ≥ 0 M ( N ) = ⌈ ∣ N ∣ / d ⌉ M(N) = \left\lceil |N|/d \right\rceil M ( N ) = ⌈ ∣ N ∣/ d ⌉ N N N t ∈ S o l ( N ) \mathbf{t}\in \mathop{\mathrm{Sol}}(N) t ∈ Sol ( N ) t i = 0 t_i = 0 t i = 0 i i i s g n ( i ) ≠ s g n ( N ) \mathop{\mathrm{sgn}}(i) \ne \mathop{\mathrm{sgn}}(N) sgn ( i ) = sgn ( N )
There is a distinguished solution t + = ( t i + ) i ∈ I 0 \mathbf{t}^+ = (t^+_i)_{i \in I_0} t + = ( t i + ) i ∈ I 0 N N N
If N ≥ 0 N \ge 0 N ≥ 0 t d 2 + = ⌊ N / d 2 ⌋ t^+_{d_2} = \lfloor N/d_2\rfloor t d 2 + = ⌊ N / d 2 ⌋ t [ N ] d 2 + = 1 t^+_{[N]_{d_2}} = 1 t [ N ] d 2 + = 1 [ N ] d 2 ≠ 0 [N]_{d_2} \ne 0 [ N ] d 2 = 0 t j + = 0 t^+_j = 0 t j + = 0 j j j
If N < 0 N < 0 N < 0 t − d 1 + = ⌊ − N / d 1 ⌋ t^+_{-d_1} = \lfloor -N / d_1\rfloor t − d 1 + = ⌊ − N / d 1 ⌋ t − [ − N ] d 1 + = 1 t^+_{-[-N]_{d_1}}=1 t − [ − N ] d 1 + = 1 [ − N ] d 1 ≠ 0 [-N]_{d_1} \ne 0 [ − N ] d 1 = 0 t j + = 0 t^+_j = 0 t j + = 0 j j j
Proof. Suppose N ≥ 0 N\ge0 N ≥ 0 t ∈ S o l ( N ) \mathbf{t}\in \mathop{\mathrm{Sol}}(N) t ∈ Sol ( N ) T + = ∑ i > 0 t i T_+ = \sum_{i > 0} t_i T + = ∑ i > 0 t i T − = ∑ i < 0 t i T_- = \sum_{i < 0} t_i T − = ∑ i < 0 t i N ≤ d 2 T + N \le d_2 T_+ N ≤ d 2 T +
∣ t ∣ = T + + T − ≥ T + ≥ ⌈ N / d 2 ⌉ . |\mathbf{t}| = T_+ + T_- \ge T_+ \ge \left\lceil N/d_2\right\rceil. ∣ t ∣ = T + + T − ≥ T + ≥ ⌈ N / d 2 ⌉ . The division algorithm gives the representation t + \mathbf{t}^+ t + N N N ∣ t ∣ = ⌈ N / d 2 ⌉ |\mathbf{t}| = \left\lceil N / d_2 \right\rceil ∣ t ∣ = ⌈ N / d 2 ⌉ T − = 0 T_- = 0 T − = 0 ∣ t ∣ = T + + T − > T + ≥ ⌈ N / d 2 ⌉ |\mathbf{t}| = T_+ + T_- > T_+ \ge \left\lceil N / d_2\right\rceil ∣ t ∣ = T + + T − > T + ≥ ⌈ N / d 2 ⌉ N < 0 N<0 N < 0
Lemma 2.3. For nonnegative integers d d d n n n T d ( n ) \mathcal{T}_d(n) T d ( n ) ( t 1 , … , t d ) ∈ Z ≥ 0 d (t_1, \dots, t_d) \in \mathbb{Z}_{\ge 0}^d ( t 1 , … , t d ) ∈ Z ≥ 0 d ∑ i = 1 d t i i = n \sum_{i=1}^d t_i i = n ∑ i = 1 d t i i = n T 0 ( 0 ) = { ( ) } \mathcal{T}_0(0) = \{()\} T 0 ( 0 ) = {( )} T 0 ( n ) = ∅ \mathcal{T}_0(n) = \varnothing T 0 ( n ) = ∅ n n n
Let N ∈ Z ∖ { 0 } N \in \mathbb{Z}\setminus\{0\} N ∈ Z ∖ { 0 } d = d 1 d = d_1 d = d 1 d = d 2 d = d_2 d = d 2 N N N Δ = d M ( N ) − ∣ N ∣ \Delta = d M(N) - |N| Δ = d M ( N ) − ∣ N ∣ M ( N ) ≥ d − 1 M(N) \ge d-1 M ( N ) ≥ d − 1 S o l ( N ) \mathop{\mathrm{Sol}}(N) Sol ( N ) T d − 1 ( Δ ) \mathcal{T}_{d-1}(\Delta) T d − 1 ( Δ ) S o l ( N ) → T d − 1 ( Δ ) \mathop{\mathrm{Sol}}(N) \to \mathcal{T}_{d-1}(\Delta) Sol ( N ) → T d − 1 ( Δ ) ( t i ) i ∈ I 0 ↦ ( t s g n ( N ) ( d − i ) ) i = 1 d − 1 (t_i)_{i \in I_0} \mapsto (t_{\mathop{\mathrm{sgn}}(N)(d-i)})_{i=1}^{d-1} ( t i ) i ∈ I 0 ↦ ( t sgn ( N ) ( d − i ) ) i = 1 d − 1 u = ( u j ) ∈ T d − 1 ( Δ ) \mathbf{u} = (u_j) \in \mathcal{T}_{d-1}(\Delta) u = ( u j ) ∈ T d − 1 ( Δ ) t s g n ( N ) j = u d − j t_{\mathop{\mathrm{sgn}}(N)j} = u_{d-j} t sgn ( N ) j = u d − j j = 1 , … , d − 1 j=1,\dots,d-1 j = 1 , … , d − 1 t s g n ( N ) d = M ( N ) − ∣ u ∣ t_{\mathop{\mathrm{sgn}}(N)d} = M(N) - |\mathbf{u}| t sgn ( N ) d = M ( N ) − ∣ u ∣
Proof. Every minimal representation of N N N M ( N ) M(N) M ( N ) s g n ( N ) \mathop{\mathrm{sgn}}(N) sgn ( N ) Lemma 2.2 . Therefore
Δ = d M ( N ) − ∣ N ∣ = ∑ i = 1 d − 1 ( t s g n ( N ) ( d − i ) ) i . \Delta = d M(N) - |N| = \sum_{i=1}^{d-1} (t_{\mathop{\mathrm{sgn}}(N)(d-i)})i. Δ = d M ( N ) − ∣ N ∣ = i = 1 ∑ d − 1 ( t sgn ( N ) ( d − i ) ) i . The stated formulas give the inverse map. ◻
Permutation costs and minimizing index sets
Evaluating the valuation of the Fredholm determinant’s coefficients concerns the determinants of finite principal minors of the infinite matrix ( G p i − j ) i , j ∈ Z (G_{pi-j})_{i,j \in \mathbb{Z}} ( G p i − j ) i , j ∈ Z p p p G N G_N G N M ( N ) M(N) M ( N ) p p p ∑ ℓ ∈ I M ( p ℓ − σ ( ℓ ) ) \sum_{\ell \in \mathcal{I}} M(p\ell-\sigma(\ell)) ∑ ℓ ∈ I M ( p ℓ − σ ( ℓ )) I \mathcal{I} I Z \mathbb{Z} Z σ \sigma σ I \mathcal{I} I Proposition 2.6 provides a concrete solution when I \mathcal{I} I Proposition 2.11 demonstrates that for sufficiently large p p p I \mathcal{I} I Corollary 2.13 establishes the convexity of the generic construction, while Proposition 2.14 will ultimately validate a reduction we employ in § 5 .
We inherit the notation of Definition 2.1 from the previous subsection.
Definition 2.4. Let p p p I \mathcal{I} I Z \mathbb{Z} Z σ \sigma σ I \mathcal{I} I
S p ( I ) ≔ min σ ∈ S y m ( I ) ∑ ℓ ∈ I M ( p ℓ − σ ( ℓ ) ) . S^p(\mathcal{I}) \coloneqq \min_{\sigma \in \mathrm{Sym}(\mathcal{I})}
\sum_{\ell \in \mathcal{I}} M(p\ell - \sigma(\ell)). S p ( I ) : = σ ∈ Sym ( I ) min ℓ ∈ I ∑ M ( p ℓ − σ ( ℓ )) . For a nonnegative integer k k k
S p ( k ) ≔ min I ∈ ( Z k ) S p ( I ) , S^p(k) \coloneqq \min_{\mathcal{I}\in \binom{\mathbb{Z}}{k}} S^p(\mathcal{I}), S p ( k ) : = I ∈ ( k Z ) min S p ( I ) , and
Σ p ( I ) = { σ ∈ S y m ( I ) ∣ ∑ ℓ ∈ I M ( p ℓ − σ ( ℓ ) ) = S p ( I ) } , \mathbf{\Sigma}^p(\mathcal{I}) = \left\{\sigma \in \mathrm{Sym}(\mathcal{I}) \mid
\sum_{\ell \in \mathcal{I}}M(p\ell - \sigma(\ell)) = S^p(\mathcal{I})\right\}, Σ p ( I ) = { σ ∈ Sym ( I ) ∣ ℓ ∈ I ∑ M ( p ℓ − σ ( ℓ )) = S p ( I ) } , where ( Z k ) \binom{\mathbb{Z}}{k} ( k Z ) Z \mathbb{Z} Z k k k σ ℓ \sigma\ell σ ℓ σ ( ℓ ) \sigma(\ell) σ ( ℓ )
The function S p ( k ) S^p(k) S p ( k ) y y y
Definition 2.5 (Combinatorial cost polygon). For a prime number p p p d 1 , d 2 ≥ 1 d_1, d_2 \ge 1 d 1 , d 2 ≥ 1 G L ( d 1 , d 2 , p ) \mathop{\mathrm{GL}}(d_1,d_2,p) GL ( d 1 , d 2 , p )
{ ( k , S p ( k ) p − 1 ) ∣ k ∈ { 1 , … , d 1 + d 2 } } ∪ { ( 0 , 0 ) } . \left\{ \left(k, \frac{S^p(k)}{p-1}\right) \mid
k \in \{1, \dots, d_1+d_2\}\right\} \cup \{(0,0)\}. { ( k , p − 1 S p ( k ) ) ∣ k ∈ { 1 , … , d 1 + d 2 } } ∪ {( 0 , 0 )} . We call this the combinatorial cost polygon of ( d 1 , d 2 ) (d_1, d_2) ( d 1 , d 2 ) p p p
When d 1 d_1 d 1 d 2 d_2 d 2 V p ⊆ { 1 , … , d 1 + d 2 − 1 } V_p \subseteq \{1, \dots, d_1+d_2 - 1\} V p ⊆ { 1 , … , d 1 + d 2 − 1 } G L ( d 1 , d 2 , p ) \mathop{\mathrm{GL}}(d_1,d_2,p) GL ( d 1 , d 2 , p )
One of the goals of this subsection is to prove the convexity of the function S p ( k ) S^p(k) S p ( k ) k k k V p V_p V p Corollary 2.13 .
We begin our exploration by deriving a more computable expression for S p ( I ) S^p(\mathcal{I}) S p ( I ) I \mathcal{I} I
Proposition 2.6. Suppose p p p d 1 d_1 d 1 d 2 d_2 d 2 k 1 k_1 k 1 k 2 k_2 k 2 − d 1 ≤ − k 1 ≤ 0 ≤ k 2 ≤ d 2 -d_1 \le -k_1 \le 0 \le k_2 \le d_2 − d 1 ≤ − k 1 ≤ 0 ≤ k 2 ≤ d 2 I = [ − k 1 , k 2 ] ∩ Z \mathcal{I}= [-k_1, k_2] \cap \mathbb{Z} I = [ − k 1 , k 2 ] ∩ Z S p ( I ) = ∑ ℓ = 1 k 1 ⌈ p ℓ − k 1 d 1 ⌉ + ∑ ℓ = 1 k 2 ⌈ p ℓ − k 2 d 2 ⌉ . S^p(\mathcal{I}) = \sum_{\ell=1}^{k_1}
\left\lceil \frac{p\ell - k_1}{d_1} \right\rceil +
\sum_{\ell = 1}^{k_2} \left\lceil \frac{p\ell - k_2}{d_2} \right\rceil. S p ( I ) = ∑ ℓ = 1 k 1 ⌈ d 1 p ℓ − k 1 ⌉ + ∑ ℓ = 1 k 2 ⌈ d 2 p ℓ − k 2 ⌉ . σ ∈ Σ p ( I ) \sigma \in \mathbf{\Sigma}^p(\mathcal{I}) σ ∈ Σ p ( I ) σ ( 0 ) = 0 \sigma(0) = 0 σ ( 0 ) = 0 ⌈ p ℓ − σ ℓ d 2 ⌉ = ⌈ p ℓ − k 2 d 2 ⌉ \left\lceil \frac{p\ell - \sigma\ell}{d_2}\right\rceil =
\left\lceil \frac{p\ell - k_2}{d_2}\right\rceil ⌈ d 2 p ℓ − σ ℓ ⌉ = ⌈ d 2 p ℓ − k 2 ⌉ ℓ \ell ℓ ⌈ p ( − ℓ ) + σ ℓ d 1 ⌉ = ⌈ p ( − ℓ ) − k 1 d 1 ⌉ \left\lceil \frac{p(-\ell) + \sigma\ell}{d_1}\right\rceil
= \left\lceil \frac{p(-\ell) - k_1}{d_1}\right\rceil ⌈ d 1 p ( − ℓ ) + σ ℓ ⌉ = ⌈ d 1 p ( − ℓ ) − k 1 ⌉ ℓ \ell ℓ
Proof. Note that the righthand side is certainly a lower bound for S p ( I ) S^p(\mathcal{I}) S p ( I ) ℓ \ell ℓ p ℓ − σ ℓ ≥ p ℓ − k 2 p\ell - \sigma\ell \ge p\ell - k_2 p ℓ − σ ℓ ≥ p ℓ − k 2 M ( p ℓ − σ ℓ ) ≥ ⌈ p ℓ − k 2 d 2 ⌉ M(p\ell - \sigma\ell) \ge \left\lceil
\frac{p\ell-k_2}{d_2}\right\rceil M ( p ℓ − σ ℓ ) ≥ ⌈ d 2 p ℓ − k 2 ⌉ ℓ \ell ℓ M ( p ℓ − σ ℓ ) ≥ ⌈ p ( − ℓ ) − k 1 d 1 ⌉ M(p\ell - \sigma\ell) \ge \left\lceil
\frac{p(-\ell)-k_1}{d_1}\right\rceil M ( p ℓ − σ ℓ ) ≥ ⌈ d 1 p ( − ℓ ) − k 1 ⌉ ℓ = 0 \ell = 0 ℓ = 0 M ( p ℓ − σ ℓ ) ≥ 0 M(p\ell - \sigma\ell) \ge 0 M ( p ℓ − σ ℓ ) ≥ 0 σ \sigma σ
We construct the permutation directly and refer to it as τ \tau τ ℓ = k 2 \ell = k_2 ℓ = k 2 ℓ = 1 \ell=1 ℓ = 1 I ℓ = { 1 , … , k 2 } ∖ { τ − 1 ( j ) ∣ j ≥ ℓ + 1 } I_\ell = \{1, \dots, k_2\} \setminus \{\tau^{-1}(j) \mid j \ge
\ell+1\} I ℓ = { 1 , … , k 2 } ∖ { τ − 1 ( j ) ∣ j ≥ ℓ + 1 } τ − 1 ( ℓ ) \tau^{-1}(\ell) τ − 1 ( ℓ ) i ∈ I ℓ i \in I_\ell i ∈ I ℓ [ p i − ℓ ] d 2 [pi-\ell]_{d_2} [ p i − ℓ ] d 2 p p p d 2 d_2 d 2 { p i − ℓ ∣ 1 ≤ i ≤ d 2 } \{pi-\ell \mid 1 \le i \le d_2\} { p i − ℓ ∣ 1 ≤ i ≤ d 2 } d 2 d_2 d 2 ∣ I ℓ ∣ ≤ k 2 ≤ d 2 |I_\ell| \le k_2 \le d_2 ∣ I ℓ ∣ ≤ k 2 ≤ d 2 [ p i − ℓ ] d 2 [pi-\ell]_{d_2} [ p i − ℓ ] d 2 i i i τ − 1 \tau^{-1} τ − 1
Suppose that for some i i i ⌈ p i − τ ( i ) d 2 ⌉ > ⌈ p i − k 2 d 2 ⌉ \left\lceil \frac{pi - \tau(i)}{d_2}\right\rceil >
\left\lceil \frac{pi - k_2}{d_2}\right\rceil ⌈ d 2 p i − τ ( i ) ⌉ > ⌈ d 2 p i − k 2 ⌉ τ ( i ) + 1 ≤ ℓ ≤ k 2 \tau (i) + 1 \le \ell \le k_2 τ ( i ) + 1 ≤ ℓ ≤ k 2 p i − ℓ ≡ 0 ( m o d d 2 ) pi-\ell \equiv 0 \pmod{d_2} p i − ℓ ≡ 0 ( mod d 2 ) [ p i − ℓ ] d 2 = 0 [pi-\ell]_{d_2} = 0 [ p i − ℓ ] d 2 = 0 ℓ ≠ τ ( i ) \ell \ne \tau(i) ℓ = τ ( i ) i ∉ I ℓ i \not\in I_\ell i ∈ I ℓ i ∈ { τ − 1 ( j ) ∣ j ≥ ℓ + 1 } i \in \{\tau^{-1}(j) \mid j \ge \ell+1\} i ∈ { τ − 1 ( j ) ∣ j ≥ ℓ + 1 } i = τ − 1 ( j ) i = \tau^{-1}(j) i = τ − 1 ( j ) j ≥ ℓ + 1 j \ge \ell+1 j ≥ ℓ + 1 τ ( i ) ≥ ℓ + 1 \tau(i) \ge \ell+1 τ ( i ) ≥ ℓ + 1 ℓ + 1 ≤ τ ( i ) < τ ( i ) + 1 ≤ ℓ \ell+1 \le \tau(i) < \tau(i)+1 \le \ell ℓ + 1 ≤ τ ( i ) < τ ( i ) + 1 ≤ ℓ
After performing a similar inductive construction for τ \tau τ ℓ \ell ℓ τ ( 0 ) = 0 \tau(0) = 0 τ ( 0 ) = 0
Next we introduce a weight function, which will aid in determining the minimizing index sets I \mathcal{I} I
Definition 2.8. We define a weight function w : Z → Q ≥ 0 w : \mathbb{Z}\to \mathbb{Q}_{\ge 0} w : Z → Q ≥ 0
w(i) =
\begin{cases*}
\frac{-i}{d_1} & if $i \le 0$ \\
\frac{i}{d_2} & if $i > 0$
\end{cases*}
and extend it to finite subsets I ⊆ Z \mathcal{I}\subseteq \mathbb{Z} I ⊆ Z w ( I ) ≔ ∑ ℓ ∈ I w ( ℓ ) w(\mathcal{I}) \coloneqq \sum_{\ell \in \mathcal{I}} w(\ell) w ( I ) : = ∑ ℓ ∈ I w ( ℓ ) k ≥ 1 k \ge 1 k ≥ 1 W k ≔ min I ∈ ( Z k ) w ( I ) W_k \coloneqq \min_{\mathcal{I}\in \binom{\mathbb{Z}}{k}} w(\mathcal{I}) W k : = min I ∈ ( k Z ) w ( I ) I ∈ ( Z k ) \mathcal{I}\in \binom{\mathbb{Z}}{k} I ∈ ( k Z ) w ( I ) = W k w(\mathcal{I}) = W_k w ( I ) = W k I \mathcal{I} I
We provide some basic properties of our weight function.
Lemma 2.9. Let k ≥ 1 k \ge 1 k ≥ 1
For any N ∈ Z N \in \mathbb{Z} N ∈ Z M ( N ) = ⌈ w ( N ) ⌉ M(N) = \lceil w(N)\rceil M ( N ) = ⌈ w ( N )⌉
If I ∈ ( Z k ) \mathcal{I}\in \binom{\mathbb{Z}}{k} I ∈ ( k Z ) I \mathcal{I} I I = { − k 1 , … , k 2 } \mathcal{I}= \{-k_1, \dots, k_2\} I = { − k 1 , … , k 2 } k 1 k_1 k 1 k 2 k_2 k 2 k − 1 k-1 k − 1 { w ( ℓ ) ∣ ℓ ∈ I ∖ { 0 } } \{w(\ell) \mid \ell \in \mathcal{I}\setminus\{0\}\} { w ( ℓ ) ∣ ℓ ∈ I ∖ { 0 }} k − 1 k-1 k − 1 { 1 / d 1 , 2 / d 1 , … } ⊔ { 1 / d 2 , 2 / d 2 , … } \{1/d_1, 2/d_1, \dots\} \sqcup \{1/d_2, 2/d_2, \dots\} { 1/ d 1 , 2/ d 1 , … } ⊔ { 1/ d 2 , 2/ d 2 , … }
There are either one or two distinct weight-minimizing sets in ( Z k ) \binom{\mathbb{Z}}{k} ( k Z )
There are two distinct weight-minimizing sets I , J ∈ ( Z k ) \mathcal{I}, \mathcal{J} \in \binom{\mathbb{Z}}{k} I , J ∈ ( k Z ) k 1 k_1 k 1 k 2 k_2 k 2 k 1 / d 1 = ( k 2 + 1 ) / d 2 k_1 / d_1 = (k_2+1) / d_2 k 1 / d 1 = ( k 2 + 1 ) / d 2 k 1 + k 2 = k − 1 k_1 + k_2 = k-1 k 1 + k 2 = k − 1 I = { − k 1 , … , k 2 } \mathcal{I}= \{-k_1, \dots, k_2\} I = { − k 1 , … , k 2 } J = { − k 1 + 1 , … , k 2 + 1 } \mathcal{J} = \{-k_1+1, \dots, k_2+1\} J = { − k 1 + 1 , … , k 2 + 1 }
( p − 1 ) w ( I ) ≤ S p ( I ) (p-1)w(\mathcal{I}) \le S^p(\mathcal{I}) ( p − 1 ) w ( I ) ≤ S p ( I ) ( p − 1 ) W k ≤ S p ( k ) (p-1)W_k \le S^p(k) ( p − 1 ) W k ≤ S p ( k )
S p ( k ) < ( p − 1 ) W k + k S^p(k) < (p-1)W_k + k S p ( k ) < ( p − 1 ) W k + k
Proof. (1) - (3) are obvious.
(4) Both of the “if-and-only-if” conditions are equivalent to { 1 / d 1 , 2 / d 1 , … } \{1/d_1, 2/d_1, \dots\} { 1/ d 1 , 2/ d 1 , … } { 1 / d 2 , 2 / d 2 , … } \{1/d_2, 2/d_2, \dots\} { 1/ d 2 , 2/ d 2 , … } r = k 1 / d 1 = ( k 2 + 1 ) / d 2 r = k_1 / d_1 = (k_2+1) / d_2 r = k 1 / d 1 = ( k 2 + 1 ) / d 2 k 1 + k 2 = k − 1 k_1 + k_2 = k-1 k 1 + k 2 = k − 1
(5) Choose a minimizing permutation σ \sigma σ
M ( p ℓ − σ ℓ ) = ⌈ w ( p ℓ − σ ℓ ) ⌉ ≥ w ( p ℓ − σ ℓ ) ≥ p w ( ℓ ) − w ( σ ℓ ) , M(p\ell - \sigma\ell) = \left\lceil w(p\ell - \sigma\ell)\right\rceil
\ge w(p\ell - \sigma\ell) \ge p w(\ell) - w(\sigma\ell), M ( p ℓ − σ ℓ ) = ⌈ w ( p ℓ − σ ℓ ) ⌉ ≥ w ( p ℓ − σ ℓ ) ≥ pw ( ℓ ) − w ( σ ℓ ) , so
S p ( I ) = ∑ ℓ ∈ I M ( p ℓ − σ ℓ ) ≥ ( p − 1 ) ∑ ℓ ∈ I w ( ℓ ) = ( p − 1 ) w ( I ) . S^p(\mathcal{I}) = \sum_{\ell \in \mathcal{I}} M(p\ell - \sigma\ell)
\ge (p-1) \sum_{\ell \in \mathcal{I}} w(\ell)
= (p-1) w(\mathcal{I}). S p ( I ) = ℓ ∈ I ∑ M ( p ℓ − σ ℓ ) ≥ ( p − 1 ) ℓ ∈ I ∑ w ( ℓ ) = ( p − 1 ) w ( I ) . If the choice of I \mathcal{I} I S p ( I ) = S p ( k ) S^p(\mathcal{I}) = S^p(k) S p ( I ) = S p ( k ) S p ( k ) = S p ( I ) ≥ ( p − 1 ) w ( I ) ≥ ( p − 1 ) W k S^p(k) = S^p(\mathcal{I}) \ge (p-1)w(\mathcal{I}) \ge (p-1)W_k S p ( k ) = S p ( I ) ≥ ( p − 1 ) w ( I ) ≥ ( p − 1 ) W k
(6) Let I ′ = { − k 1 , … , k 2 } \mathcal{I}' = \{-k_1, \dots, k_2\} I ′ = { − k 1 , … , k 2 } σ ′ \sigma' σ ′ M ( p ℓ − ℓ ) = ⌈ w ( p ℓ − ℓ ) ⌉ < ( p − 1 ) w ( ℓ ) + 1 M(p\ell - \ell) = \lceil w(p\ell - \ell)\rceil
< (p-1)w(\ell) + 1 M ( p ℓ − ℓ ) = ⌈ w ( p ℓ − ℓ )⌉ < ( p − 1 ) w ( ℓ ) + 1
S p ( k ) ≤ ∑ ℓ ∈ I ′ M ( p ℓ − ℓ ) < ( p − 1 ) W k + k . S^p(k) \le \sum_{\ell \in \mathcal{I}'} M(p\ell -
\ell) < (p-1) W_k + k. S p ( k ) ≤ ℓ ∈ I ′ ∑ M ( p ℓ − ℓ ) < ( p − 1 ) W k + k . ◻
Next we introduce a key permutation lemma.
Lemma 2.10 (Permutation lemma). Let 1 ≤ k ≤ d 1 + d 2 + 1 1 \le k \le d_1 + d_2 + 1 1 ≤ k ≤ d 1 + d 2 + 1 p > 2 D p > 2D p > 2 D b ≥ 1 b \ge 1 b ≥ 1 m ∈ Z / b Z m \in \mathbb{Z}/ b\mathbb{Z} m ∈ Z / b Z I m ∈ ( Z k ) \mathcal{I}_m \in \binom{\mathbb{Z}}{k} I m ∈ ( k Z ) I m \mathcal{I}_m I m A = ⨆ m ∈ Z / b Z I m × { m } . \mathcal{A} = \bigsqcup_{m \in \mathbb{Z}/ b\mathbb{Z}} \mathcal{I}_m \times \{m\}. A = ⨆ m ∈ Z / b Z I m × { m } . a = ( ℓ , m ) ∈ A a = (\ell, m) \in \mathcal{A} a = ( ℓ , m ) ∈ A π 1 ( a ) = ℓ \pi_1(a) = \ell π 1 ( a ) = ℓ
Then for any permutation τ : A → A \tau : \mathcal{A} \to \mathcal{A} τ : A → A ∑ a ∈ A M ( p π 1 ( a ) − π 1 ( τ a ) ) ≥ b S p ( k ) , \sum_{a \in \mathcal{A}} M(p \pi_1(a) - \pi_1(\tau a)) \ge b S^p(k), ∑ a ∈ A M ( p π 1 ( a ) − π 1 ( τ a )) ≥ b S p ( k ) , I m \mathcal{I}_m I m m ∈ Z / b Z m \in \mathbb{Z}/ b\mathbb{Z} m ∈ Z / b Z ∑ a ∈ A M ( p π 1 ( a ) − π 1 ( τ a ) ) ≥ ∑ m ∈ Z / b Z S p ( I m ) . \sum_{a \in \mathcal{A}} M(p \pi_1(a) - \pi_1(\tau a)) \ge
\sum_{m \in \mathbb{Z}/ b\mathbb{Z}} S^p(\mathcal{I}_m). ∑ a ∈ A M ( p π 1 ( a ) − π 1 ( τ a )) ≥ ∑ m ∈ Z / b Z S p ( I m ) .
Proof. Write
C ( A , τ ) = ∑ a ∈ A M ( p π 1 ( a ) − π 1 ( τ a ) ) , F d ( t ) = ∑ j = 1 t ⌈ p j − t d ⌉ , C(\mathcal A,\tau)
=\sum_{a\in\mathcal A}
M\bigl(p\pi_1(a)-\pi_1(\tau a)\bigr),
\qquad
F_d(t)=\sum_{j=1}^{t}
\left\lceil\frac{pj-t}{d}\right\rceil, C ( A , τ ) = a ∈ A ∑ M ( p π 1 ( a ) − π 1 ( τ a ) ) , F d ( t ) = j = 1 ∑ t ⌈ d p j − t ⌉ , with F d ( 0 ) = 0 F_d(0)=0 F d ( 0 ) = 0
Weight-minimizing fibers. First suppose that every I m \mathcal{I}_m I m k ≤ d 1 + d 2 + 1 k\le d_1+d_2+1 k ≤ d 1 + d 2 + 1 { − d 1 , … , d 2 } \{-d_1,\ldots,d_2\} { − d 1 , … , d 2 }
If all the fibers are equal to I = { − k 1 , … , k 2 } \mathcal{I}=\{-k_1,\ldots,k_2\} I = { − k 1 , … , k 2 } I \mathcal{I} I Proposition 2.6 therefore give
C ( A , τ ) ≥ b ( F d 1 ( k 1 ) + F d 2 ( k 2 ) ) = b S p ( I ) = ∑ m S p ( I m ) ≥ b S p ( k ) . C(\mathcal A,\tau)
\ge b\bigl(F_{d_1}(k_1)+F_{d_2}(k_2)\bigr)
=bS^p(\mathcal{I})
=\sum_m S^p(\mathcal{I}_m)
\ge bS^p(k). C ( A , τ ) ≥ b ( F d 1 ( k 1 ) + F d 2 ( k 2 ) ) = b S p ( I ) = m ∑ S p ( I m ) ≥ b S p ( k ) . This includes the case k = d 1 + d 2 + 1 k=d_1+d_2+1 k = d 1 + d 2 + 1
Otherwise, Lemma 2.9 shows that the fibers take the two values
I = { − k 1 , … , k 2 } , J = { − k 1 + 1 , … , k 2 + 1 } . \mathcal{I}=\{-k_1,\ldots,k_2\},
\qquad
\mathcal J=\{-k_1+1,\ldots,k_2+1\}. I = { − k 1 , … , k 2 } , J = { − k 1 + 1 , … , k 2 + 1 } . Both values occur, so there are r r r I \mathcal{I} I b − r b-r b − r J \mathcal J J 0 < r < b 0<r<b 0 < r < b 1 ≤ k 1 ≤ d 1 1\le k_1\le d_1 1 ≤ k 1 ≤ d 1 0 ≤ k 2 < d 2 0\le k_2<d_2 0 ≤ k 2 < d 2
Let C − C_- C − C + C_+ C + C ( A , τ ) C(\mathcal A,\tau) C ( A , τ ) 1 ≤ j < k 1 1\le j<k_1 1 ≤ j < k 1 b b b − j -j − j r r r − k 1 -k_1 − k 1 − k 1 + 1 -k_1+1 − k 1 + 1
r ⌈ p j − k 1 d 1 ⌉ + ( b − r ) ⌈ p j − ( k 1 − 1 ) d 1 ⌉ . r\left\lceil\frac{pj-k_1}{d_1}\right\rceil
+(b-r)\left\lceil\frac{pj-(k_1-1)}{d_1}\right\rceil. r ⌈ d 1 p j − k 1 ⌉ + ( b − r ) ⌈ d 1 p j − ( k 1 − 1 ) ⌉ . Adding the contribution of the r r r − k 1 -k_1 − k 1
C − ≥ r F d 1 ( k 1 ) + ( b − r ) F d 1 ( k 1 − 1 ) . C_-
\ge rF_{d_1}(k_1)+(b-r)F_{d_1}(k_1-1). C − ≥ r F d 1 ( k 1 ) + ( b − r ) F d 1 ( k 1 − 1 ) . Similarly, there are only b − r b-r b − r k 2 + 1 k_2+1 k 2 + 1 1 ≤ j ≤ k 2 1\le j\le k_2 1 ≤ j ≤ k 2 r r r b b b j j j k 2 k_2 k 2 b − r b-r b − r k 2 + 1 k_2+1 k 2 + 1
C + ≥ r F d 2 ( k 2 ) + ( b − r ) F d 2 ( k 2 + 1 ) . C_+
\ge rF_{d_2}(k_2)+(b-r)F_{d_2}(k_2+1). C + ≥ r F d 2 ( k 2 ) + ( b − r ) F d 2 ( k 2 + 1 ) . The zero-source contributions are nonnegative. Both I \mathcal{I} I J \mathcal J J Proposition 2.6 , so
C ( A , τ ) ≥ r ( F d 1 ( k 1 ) + F d 2 ( k 2 ) ) + ( b − r ) ( F d 1 ( k 1 − 1 ) + F d 2 ( k 2 + 1 ) ) = r S p ( I ) + ( b − r ) S p ( J ) = ∑ m S p ( I m ) ≥ b S p ( k ) . \begin{align*}
C(\mathcal A,\tau)
&\ge r\bigl(F_{d_1}(k_1)+F_{d_2}(k_2)\bigr)
+(b-r)\bigl(F_{d_1}(k_1-1)+F_{d_2}(k_2+1)\bigr)\\
&=rS^p(\mathcal{I})+(b-r)S^p(\mathcal J)
=\sum_m S^p(\mathcal{I}_m)
\ge bS^p(k).
\end{align*} C ( A , τ ) ≥ r ( F d 1 ( k 1 ) + F d 2 ( k 2 ) ) + ( b − r ) ( F d 1 ( k 1 − 1 ) + F d 2 ( k 2 + 1 ) ) = r S p ( I ) + ( b − r ) S p ( J ) = m ∑ S p ( I m ) ≥ b S p ( k ) . A strict reduction for a non-weight-minimizing fiber. Suppose now that some fiber is not weight-minimizing. Every weight-minimizing k k k k k k I m \mathcal{I}_m I m x x x A \mathcal A A I m \mathcal{I}_m I m z ∉ I m z\notin\mathcal{I}_m z ∈ / I m w ( z ) < w ( x ) w(z)<w(x) w ( z ) < w ( x )
Replace ( x , m ) (x,m) ( x , m ) ( z , m ) (z,m) ( z , m ) A ′ \mathcal A' A ′ ( x , m ) (x,m) ( x , m ) τ \tau τ ( z , m ) (z,m) ( z , m ) ( x , m ) (x,m) ( x , m ) ( z , m ) (z,m) ( z , m ) τ ′ \tau' τ ′ C ( A ′ , τ ′ ) < C ( A , τ ) C(\mathcal A',\tau')<C(\mathcal A,\tau) C ( A ′ , τ ′ ) < C ( A , τ )
The values of w w w D − 1 Z ≥ 0 D^{-1}\mathbb{Z}_{\ge0} D − 1 Z ≥ 0 p > 2 D p>2D p > 2 D p − 1 ≥ 2 D p-1\ge2D p − 1 ≥ 2 D
⌊ ( p − 1 ) w ( x ) ⌋ − ⌈ ( p − 1 ) w ( z ) ⌉ > ( p − 1 ) ( w ( x ) − w ( z ) ) − 2 ≥ 0. \begin{equation}
\lfloor(p-1)w(x)\rfloor-\lceil(p-1)w(z)\rceil
>(p-1)\bigl(w(x)-w(z)\bigr)-2\ge0. \tag*{(2)}
\end{equation} ⌊( p − 1 ) w ( x )⌋ − ⌈( p − 1 ) w ( z )⌉ > ( p − 1 ) ( w ( x ) − w ( z ) ) − 2 ≥ 0. (2) If ( x , m ) (x,m) ( x , m )
C ( A , τ ) − C ( A ′ , τ ′ ) = M ( ( p − 1 ) x ) − M ( ( p − 1 ) z ) = ⌈ ( p − 1 ) w ( x ) ⌉ − ⌈ ( p − 1 ) w ( z ) ⌉ > 0. \begin{align*}
C(\mathcal A,\tau)-C(\mathcal A',\tau')
&=M((p-1)x)-M((p-1)z)\\
&=\lceil(p-1)w(x)\rceil-\lceil(p-1)w(z)\rceil>0.
\end{align*} C ( A , τ ) − C ( A ′ , τ ′ ) = M (( p − 1 ) x ) − M (( p − 1 ) z ) = ⌈( p − 1 ) w ( x )⌉ − ⌈( p − 1 ) w ( z )⌉ > 0. Otherwise, put u = π 1 ( τ − 1 ( x , m ) ) u=\pi_1(\tau^{-1}(x,m)) u = π 1 ( τ − 1 ( x , m )) v = π 1 ( τ ( x , m ) ) v=\pi_1(\tau(x,m)) v = π 1 ( τ ( x , m )) M M M
M ( p u − v ) ≤ M ( p u − x ) + M ( x − v ) . M(pu-v)\le M(pu-x)+M(x-v). M ( p u − v ) ≤ M ( p u − x ) + M ( x − v ) . Therefore
C ( A , τ ) − C ( A ′ , τ ′ ) = M ( p x − v ) + M ( p u − x ) − M ( p u − v ) − M ( ( p − 1 ) z ) ≥ M ( p x − v ) − M ( x − v ) − M ( ( p − 1 ) z ) . \begin{align*}
C(\mathcal A,\tau)-C(\mathcal A',\tau')
&=M(px-v)+M(pu-x)-M(pu-v)-M((p-1)z)\\
&\ge M(px-v)-M(x-v)-M((p-1)z).
\end{align*} C ( A , τ ) − C ( A ′ , τ ′ ) = M ( p x − v ) + M ( p u − x ) − M ( p u − v ) − M (( p − 1 ) z ) ≥ M ( p x − v ) − M ( x − v ) − M (( p − 1 ) z ) . Since w ( v ) ≤ w ( x ) w(v)\le w(x) w ( v ) ≤ w ( x ) x ≠ 0 x\ne0 x = 0 p x − v px-v p x − v x − v x-v x − v x x x x − v = 0 x-v=0 x − v = 0 d = d 1 d=d_1 d = d 1 x < 0 x<0 x < 0 d = d 2 d=d_2 d = d 2 x > 0 x>0 x > 0
M ( p x − v ) − M ( x − v ) = ⌈ ∣ p x − v ∣ d ⌉ − ⌈ ∣ x − v ∣ d ⌉ ≥ ⌊ ∣ p x − v ∣ − ∣ x − v ∣ d ⌋ = ⌊ ( p − 1 ) w ( x ) ⌋ . \begin{align*}
M(px-v)-M(x-v)
&=\left\lceil\frac{|px-v|}{d}\right\rceil
-\left\lceil\frac{|x-v|}{d}\right\rceil\\
&\ge\left\lfloor
\frac{|px-v|-|x-v|}{d}\right\rfloor
=\lfloor(p-1)w(x)\rfloor.
\end{align*} M ( p x − v ) − M ( x − v ) = ⌈ d ∣ p x − v ∣ ⌉ − ⌈ d ∣ x − v ∣ ⌉ ≥ ⌊ d ∣ p x − v ∣ − ∣ x − v ∣ ⌋ = ⌊( p − 1 ) w ( x )⌋ . Together with (2) , this proves the strict decrease in the remaining case as well.
Repeat this operation while a non-weight-minimizing fiber remains. The process terminates because the cost is a nonnegative integer and strictly decreases at each step. The resulting pair ( A ∘ , τ ∘ ) (\mathcal A^\circ,\tau^\circ) ( A ∘ , τ ∘ )
C ( A , τ ) > C ( A ∘ , τ ∘ ) ≥ b S p ( k ) . C(\mathcal A,\tau)
>C(\mathcal A^\circ,\tau^\circ)
\ge bS^p(k). C ( A , τ ) > C ( A ∘ , τ ∘ ) ≥ b S p ( k ) . Thus equality in the asserted lower bound is possible only when every fiber is weight-minimizing; in that case the first part also gives the stronger bound by ∑ m S p ( I m ) \sum_m S^p(\mathcal{I}_m) ∑ m S p ( I m )
We follow up with some important consequences of this lemma.
Proposition 2.11. If 1 ≤ k ≤ d 1 + d 2 + 1 1 \le k \le d_1 + d_2+1 1 ≤ k ≤ d 1 + d 2 + 1 p > 2 D p > 2D p > 2 D S p ( k ) = S p ( I ) = ∑ ℓ = 1 k 1 ⌈ p ℓ − k 1 d 1 ⌉ + ∑ ℓ = 1 k 2 ⌈ p ℓ − k 2 d 2 ⌉ , S^p(k) = S^p(\mathcal{I}) =
\sum_{\ell=1}^{k_1} \left\lceil \frac{p\ell -
k_1}{d_1}\right\rceil
+
\sum_{\ell = 1}^{k_2}
\left\lceil \frac{p\ell-k_2}{d_2} \right\rceil, S p ( k ) = S p ( I ) = ∑ ℓ = 1 k 1 ⌈ d 1 p ℓ − k 1 ⌉ + ∑ ℓ = 1 k 2 ⌈ d 2 p ℓ − k 2 ⌉ , I = { − k 1 , … , k 2 } \mathcal{I}= \{-k_1, \dots, k_2\} I = { − k 1 , … , k 2 }
Proof. Let I ∈ ( Z k ) \mathcal{I}\in \binom{\mathbb{Z}}{k} I ∈ ( k Z ) b = 1 b = 1 b = 1 A = I × { 0 } \mathcal{A} = \mathcal{I}\times \{0\} A = I × { 0 } σ \sigma σ I \mathcal{I} I σ \sigma σ τ : A → A \tau : \mathcal{A} \to \mathcal{A} τ : A → A τ ( ( ℓ , 0 ) ) = ( σ ℓ , 0 ) \tau((\ell, 0)) = (\sigma\ell, 0) τ (( ℓ , 0 )) = ( σ ℓ , 0 ) Lemma 2.10 now implies
∑ a ∈ A M ( p π 1 ( a ) − π 1 ( τ a ) ) = ∑ ℓ ∈ I M ( p ℓ − σ ℓ ) = S p ( k ) \sum_{a \in \mathcal{A}} M(p \pi_1(a) - \pi_1(\tau a))
= \sum_{\ell \in \mathcal{I}} M(p \ell - \sigma\ell)
= S^p(k) a ∈ A ∑ M ( p π 1 ( a ) − π 1 ( τ a )) = ℓ ∈ I ∑ M ( p ℓ − σ ℓ ) = S p ( k ) only if I \mathcal{I} I S p ( k ) = S p ( I ) S^p(k) = S^p(\mathcal{I}) S p ( k ) = S p ( I ) I \mathcal{I} I I \mathcal{I} I Lemma 2.9 (2), so the formula for S p ( k ) = S p ( I ) S^p(k) = S^p(\mathcal{I}) S p ( k ) = S p ( I ) Proposition 2.6 . ◻
Corollary 2.12. Let p , p ′ > 2 D p, p' > 2D p , p ′ > 2 D k ≤ d 1 + d 2 + 1 k \le d_1 + d_2+1 k ≤ d 1 + d 2 + 1 p ≡ p ′ ( m o d d 1 ) p \equiv p' \pmod{d_1} p ≡ p ′ ( mod d 1 ) p ≡ p ′ ( m o d d 2 ) p \equiv p' \pmod{d_2} p ≡ p ′ ( mod d 2 ) S p ( k ) S^p(k) S p ( k ) S p ′ ( k ) S^{p'}(k) S p ′ ( k ) I \mathcal{I} I
Proof. By Proposition 2.11 , every subset achieving S p ( k ) S^p(k) S p ( k ) I = { − k 1 , … , k 2 } \mathcal{I}= \{-k_1, \dots, k_2\} I = { − k 1 , … , k 2 } d 1 ∣ p ′ − p d_1 \mid p' - p d 1 ∣ p ′ − p ⌈ p ′ ℓ − k 1 d 1 ⌉ = ⌈ p ℓ − k 1 d 1 ⌉ + ( p ′ − p ) ℓ d 1 \left\lceil \frac{p'\ell - k_1}{d_1}\right\rceil = \left\lceil \frac{p\ell
- k_1}{d_1}\right\rceil + \frac{(p'-p)\ell}{d_1} ⌈ d 1 p ′ ℓ − k 1 ⌉ = ⌈ d 1 p ℓ − k 1 ⌉ + d 1 ( p ′ − p ) ℓ d 2 d_2 d 2
S p ′ ( I ) = S p ( I ) + ( p ′ − p ) W k . S^{p'}(\mathcal{I}) = S^p(\mathcal{I}) + (p'-p)W_k. S p ′ ( I ) = S p ( I ) + ( p ′ − p ) W k . Each candidate minimizing interval has weight W k W_k W k ( p ′ − p ) W k (p'-p)W_k ( p ′ − p ) W k
We now establish two results needed for characterizing the generic polygon defined in § 5 and comparing it with the Newton polygon of the Dwork operator. Corollary 2.13 establishes the convexity of the sequence ( S p ( k ) ) k = 1 , … , d 1 + d 2 (S^p(k))_{k=1,\dots,d_1+d_2} ( S p ( k ) ) k = 1 , … , d 1 + d 2 Proposition 2.14 extends the minimization problem from permutations of a single set to chains of bijections between different sets. Its lower bound controls the terms in the Cauchy-Binet–type expansion encountered in Proposition 5.1 .
Corollary 2.13 (Convexity of S p ( k ) S^p(k) S p ( k ) 1 < k ≤ d 1 + d 2 1 < k \le d_1 + d_2 1 < k ≤ d 1 + d 2 p > 2 D p > 2D p > 2 D 2 S p ( k ) ≤ S p ( k − 1 ) + S p ( k + 1 ) 2S^p(k) \le S^p(k-1) + S^p(k+1) 2 S p ( k ) ≤ S p ( k − 1 ) + S p ( k + 1 ) I 1 , I 2 ∈ ( Z k ) \mathcal{I}_1, \mathcal{I}_2 \in \binom{\mathbb{Z}}{k} I 1 , I 2 ∈ ( k Z ) S p ( I 1 ) = S p ( I 2 ) = S p ( k ) S^p(\mathcal{I}_1) = S^p(\mathcal{I}_2) = S^p(k) S p ( I 1 ) = S p ( I 2 ) = S p ( k )
Consequently, for every k ∈ { 1 , … , d 1 + d 2 − 1 } k \in \{1, \dots, d_1+d_2-1\} k ∈ { 1 , … , d 1 + d 2 − 1 } ( k , S p ( k ) p − 1 ) \left(k, \frac{S^p(k)}{p-1}\right) ( k , p − 1 S p ( k ) ) G L ( d 1 , d 2 , p ) \mathop{\mathrm{GL}}(d_1,d_2,p) GL ( d 1 , d 2 , p ) k ∈ V p k \in V_p k ∈ V p G L ( d 1 , d 2 , p ) \mathop{\mathrm{GL}}(d_1,d_2,p) GL ( d 1 , d 2 , p ) k k k I ∈ ( Z k ) \mathcal{I}\in \binom{\mathbb{Z}}{k} I ∈ ( k Z ) S p ( k ) = S p ( I ) S^p(k) = S^p(\mathcal{I}) S p ( k ) = S p ( I )
Proof. For a positive integer d < p / 2 d < p/2 d < p /2 F d ( n ) = ∑ ℓ = 1 n ⌈ p ℓ − n d ⌉ F_d(n) = \sum_{\ell = 1}^n \left\lceil \frac{p\ell - n}{d}\right\rceil F d ( n ) = ∑ ℓ = 1 n ⌈ d p ℓ − n ⌉ Δ d ( j ) = F d ( j ) − F d ( j − 1 ) \Delta_d(j) = F_d(j) - F_d(j-1) Δ d ( j ) = F d ( j ) − F d ( j − 1 ) j ≥ 1 j \ge 1 j ≥ 1 Δ d ( 0 ) = 0 \Delta_d(0) = 0 Δ d ( 0 ) = 0 j ≤ d j \le d j ≤ d
Δ d ( j ) = ⌈ ( p − 1 ) j d ⌉ − # { 1 ≤ i < j ∣ p i ≡ j ( m o d d ) } , \Delta_d(j) = \left\lceil \frac{(p-1)j}{d} \right\rceil
- \# \left\{1 \le i < j \mid pi \equiv j \pmod{d}\right\}, Δ d ( j ) = ⌈ d ( p − 1 ) j ⌉ − # { 1 ≤ i < j ∣ p i ≡ j ( mod d ) } , and the last term is always either 0 0 0 1 1 1 p p p d d d Δ d ( j ) \Delta_d(j) Δ d ( j ) j < d j < d j < d
Δ d ( j + 1 ) − Δ d ( j ) ≥ ⌊ p − 1 d ⌋ − 1 ≥ 1. \begin{equation}
\Delta_d(j+1) - \Delta_d(j) \ge
\left\lfloor\frac{p-1}{d}\right\rfloor - 1 \ge 1. \tag*{(3)}
\end{equation} Δ d ( j + 1 ) − Δ d ( j ) ≥ ⌊ d p − 1 ⌋ − 1 ≥ 1. (3) Consider the multiset
U ≔ { Δ d 1 ( j ) ∣ j = 1 , 2 , … , d 1 } ⊔ { Δ d 2 ( j ) ∣ j = 1 , 2 , … , d 2 } . \mathbf{U} \coloneqq
\left\{\Delta_{d_1}(j) \mid j=1,2, \dots, d_1 \right\}
\sqcup
\left\{\Delta_{d_2}(j) \mid j=1,2, \dots, d_2 \right\}. U : = { Δ d 1 ( j ) ∣ j = 1 , 2 , … , d 1 } ⊔ { Δ d 2 ( j ) ∣ j = 1 , 2 , … , d 2 } . Sort the finite elements as u 1 ≤ u 2 ≤ ⋯ ≤ u d 1 + d 2 u_1 \le u_2 \le \dots \le u_{d_1+d_2} u 1 ≤ u 2 ≤ ⋯ ≤ u d 1 + d 2 Proposition 2.11 that S p ( k ) = u 1 + ⋯ + u k − 1 S^p(k) = u_1 + \dots + u_{k-1} S p ( k ) = u 1 + ⋯ + u k − 1
S p ( k + 1 ) − S p ( k ) = u k ≥ u k − 1 = S p ( k ) − S p ( k − 1 ) , S^p(k+1) - S^p(k) = u_k \ge u_{k-1} = S^p(k) - S^p(k-1), S p ( k + 1 ) − S p ( k ) = u k ≥ u k − 1 = S p ( k ) − S p ( k − 1 ) , with equality if and only if u k = u k − 1 u_k = u_{k-1} u k = u k − 1 p > 2 D p > 2D p > 2 D ( Δ d 1 ( j ) ) j = 1 d 1 (\Delta_{d_1}(j))_{j=1}^{d_1} ( Δ d 1 ( j ) ) j = 1 d 1 ( Δ d 2 ( j ) ) j = 1 d 2 (\Delta_{d_2}(j))_{j=1}^{d_2} ( Δ d 2 ( j ) ) j = 1 d 2 (3) . Therefore u k − 1 = u k u_{k-1} = u_k u k − 1 = u k k 1 k_1 k 1 k 2 k_2 k 2 k k k Δ d 1 ( k 1 ) = Δ d 2 ( k 2 ) = u k − 1 = u k \Delta_{d_1}(k_1) = \Delta_{d_2}(k_2) = u_{k-1} = u_k Δ d 1 ( k 1 ) = Δ d 2 ( k 2 ) = u k − 1 = u k
∑ ℓ = 1 k 1 ⌈ p ℓ − k 1 d 1 ⌉ + ∑ ℓ = 1 k 2 − 1 ⌈ p ℓ − ( k 2 − 1 ) d 2 ⌉ = ∑ ℓ = 1 k 1 − 1 ⌈ p ℓ − ( k 1 − 1 ) d 1 ⌉ + ∑ ℓ = 1 k 2 ⌈ p ℓ − k 2 d 2 ⌉ . \sum_{\ell=1}^{k_1} \left\lceil \frac{p\ell - k_1}{d_1}\right\rceil
+ \sum_{\ell=1}^{k_2-1} \left\lceil \frac{p\ell -
(k_2-1)}{d_2}\right\rceil
= \sum_{\ell=1}^{k_1-1} \left\lceil \frac{p\ell -
(k_1-1)}{d_1}\right\rceil +
\sum_{\ell=1}^{k_2} \left\lceil \frac{p\ell - k_2}{d_2}\right\rceil. ℓ = 1 ∑ k 1 ⌈ d 1 p ℓ − k 1 ⌉ + ℓ = 1 ∑ k 2 − 1 ⌈ d 2 p ℓ − ( k 2 − 1 ) ⌉ = ℓ = 1 ∑ k 1 − 1 ⌈ d 1 p ℓ − ( k 1 − 1 ) ⌉ + ℓ = 1 ∑ k 2 ⌈ d 2 p ℓ − k 2 ⌉ . By Proposition 2.6 , the lefthand side is equal to S p ( I 1 ) S^p(\mathcal{I}_1) S p ( I 1 ) I 1 = { − k 1 , … , k 2 − 1 } \mathcal{I}_1 = \{-k_1, \dots, k_2-1\} I 1 = { − k 1 , … , k 2 − 1 } S p ( I 2 ) S^p(\mathcal{I}_2) S p ( I 2 ) I 2 = { − k 1 + 1 , … , k 2 } \mathcal{I}_2 = \{-k_1+1, \dots, k_2\} I 2 = { − k 1 + 1 , … , k 2 } S p ( k ) = u 1 + ⋯ + u k − 1 = u 1 + ⋯ + u k − 2 + u k S^p(k) = u_1 + \dots + u_{k-1}
= u_1 + \dots + u_{k-2} + u_k S p ( k ) = u 1 + ⋯ + u k − 1 = u 1 + ⋯ + u k − 2 + u k
The rest of the corollary is immediate for all k > 1 k>1 k > 1 k = 1 k=1 k = 1 S p ( 1 ) S^p(1) S p ( 1 ) { 0 } \{0\} { 0 }
Proposition 2.14 (Cyclic cost bound). Let 1 ≤ k ≤ d 1 + d 2 + 1 1 \le k \le d_1 + d_2+1 1 ≤ k ≤ d 1 + d 2 + 1 p > 2 D p > 2D p > 2 D I , J ∈ ( Z k ) \mathcal{I}, \mathcal{J} \in \binom{\mathbb{Z}}{k} I , J ∈ ( k Z ) S p ( I , J ) ≔ min σ ∑ ℓ ∈ I M ( p ℓ − σ ℓ ) , S^p(\mathcal{I}, \mathcal{J}) \coloneqq \min_\sigma
\sum_{\ell \in \mathcal{I}} M(p\ell - \sigma \ell), S p ( I , J ) : = min σ ∑ ℓ ∈ I M ( p ℓ − σ ℓ ) , min σ \min_\sigma min σ σ : I → J \sigma : \mathcal{I}\to \mathcal{J} σ : I → J
Let b ∈ Z ≥ 1 b \in \mathbb{Z}_{\ge 1} b ∈ Z ≥ 1 ( I 0 , … , I b − 1 ) ∈ ( Z k ) Z / b Z (\mathcal{I}_0, \dots, \mathcal{I}_{b-1}) \in \binom{\mathbb{Z}}{k}^{\mathbb{Z}/ b\mathbb{Z}} ( I 0 , … , I b − 1 ) ∈ ( k Z ) Z / b Z ∑ m ∈ Z / b Z S p ( I m + 1 , I m ) ≥ b S p ( k ) . \sum_{m \in \mathbb{Z}/ b\mathbb{Z}} S^p(\mathcal{I}_{m+1}, \mathcal{I}_m) \ge bS^p(k). ∑ m ∈ Z / b Z S p ( I m + 1 , I m ) ≥ b S p ( k ) . I \mathcal{I} I ( Z k ) \binom{\mathbb{Z}}{k} ( k Z ) S p ( I ) = S p ( k ) S^p(\mathcal{I}) = S^p(k) S p ( I ) = S p ( k ) ( I 0 , … , I b − 1 ) = ( I , … , I ) (\mathcal{I}_0, \dots, \mathcal{I}_{b-1}) = (\mathcal{I}, \dots, \mathcal{I}) ( I 0 , … , I b − 1 ) = ( I , … , I ) b b b
Proof. Form the disjoint union
A = ⨆ m ∈ Z / b Z I m × { m } . \mathcal{A} = \bigsqcup_{m \in \mathbb{Z}/ b\mathbb{Z}} \mathcal{I}_m \times \{m\}. A = m ∈ Z / b Z ⨆ I m × { m } . For each m m m σ m : I m → I m − 1 \sigma_m : \mathcal{I}_m \to \mathcal{I}_{m-1} σ m : I m → I m − 1 τ : A → A \tau : \mathcal{A} \to \mathcal{A} τ : A → A τ ( ℓ , m ) = ( σ m ( ℓ ) , m − 1 ) \tau(\ell, m) = (\sigma_m(\ell), m-1) τ ( ℓ , m ) = ( σ m ( ℓ ) , m − 1 )
∑ m ∈ Z / b Z S p ( I m + 1 , I m ) = ∑ a ∈ A M ( p π 1 ( a ) − π 1 ( τ a ) ) ≥ b S p ( k ) \sum_{m \in \mathbb{Z}/ b\mathbb{Z}} S^p(\mathcal{I}_{m+1}, \mathcal{I}_m)
= \sum_{a \in \mathcal{A}} M(p\pi_1(a) - \pi_1(\tau a))
\ge b S^p(k) m ∈ Z / b Z ∑ S p ( I m + 1 , I m ) = a ∈ A ∑ M ( p π 1 ( a ) − π 1 ( τ a )) ≥ b S p ( k ) by Lemma 2.10 , with equality only if I m \mathcal{I}_m I m m m m
∑ m ∈ Z / b Z S p ( I m + 1 , I m ) ≥ ∑ m ∈ Z / b Z S p ( I m ) . \sum_{m \in \mathbb{Z}/ b\mathbb{Z}} S^p(\mathcal{I}_{m+1}, \mathcal{I}_m)
\ge \sum_{m \in \mathbb{Z}/ b\mathbb{Z}} S^p(\mathcal{I}_m). m ∈ Z / b Z ∑ S p ( I m + 1 , I m ) ≥ m ∈ Z / b Z ∑ S p ( I m ) . If there is only one I ∈ ( Z k ) \mathcal{I}\in \binom{\mathbb{Z}}{k} I ∈ ( k Z ) S p ( I ) S^p(\mathcal{I}) S p ( I ) I m = I \mathcal{I}_m = \mathcal{I} I m = I m m m I m = I \mathcal{I}_m = \mathcal{I} I m = I m m m S p ( I m + 1 , I m ) = S p ( I ) = S p ( k ) S^p(\mathcal{I}_{m+1}, \mathcal{I}_m) = S^p(\mathcal{I}) = S^p(k) S p ( I m + 1 , I m ) = S p ( I ) = S p ( k ) Proposition 2.11 , so the desired equality holds. ◻
A total ordering on solution vectors
The minimum costs computed above identify the terms that will ultimately contribute to the minimum valuation in the coefficients of the Fredholm determinant in § 4 . However, there is the potential for term cancellation in the determinant expansion. In order to address this, we must identify a term in the expansion that cannot cancel. We will do this by developing a lexicographic-type ordering on the monomial vectors, and showing that there is a unique term that is minimal under this ordering.
Recall that the representations in Definition 2.1 are nonnegative integer vectors in Z ≥ 0 I 0 \mathbb{Z}_{\ge 0}^{I_0} Z ≥ 0 I 0
Definition 2.15. Denote by O \mathcal{O} O Z ≥ 0 I 0 \mathbb{Z}_{\ge 0}^{I_0} Z ≥ 0 I 0 i ∈ I 0 i \in I_0 i ∈ I 0 e i e_i e i e i = ( δ i j ) j ∈ I 0 e_i = (\delta_{ij})_{j \in I_0} e i = ( δ ij ) j ∈ I 0 δ \delta δ
e 1 < O e 2 < O ⋯ < O e d 2 < O e − 1 < O e − 2 < O ⋯ < O e − d 1 . e_1 <_\mathcal{O}e_2 <_\mathcal{O}\cdots <_\mathcal{O}e_{d_2} <_\mathcal{O}
e_{-1} <_\mathcal{O}e_{-2} <_\mathcal{O}\cdots <_\mathcal{O}e_{-d_1}. e 1 < O e 2 < O ⋯ < O e d 2 < O e − 1 < O e − 2 < O ⋯ < O e − d 1 . Then, given two vectors s = ( s i ) i ∈ I 0 \mathbf{s} = (s_i)_{i\in I_0} s = ( s i ) i ∈ I 0 t = ( t i ) i ∈ I 0 \mathbf{t}= (t_i)_{i \in I_0} t = ( t i ) i ∈ I 0 s < O t \mathbf{s} <_\mathcal{O}\mathbf{t} s < O t
∣ s ∣ < ∣ t ∣ |\mathbf{s}| < |\mathbf{t}| ∣ s ∣ < ∣ t ∣
∣ s ∣ = ∣ t ∣ |\mathbf{s}| = |\mathbf{t}| ∣ s ∣ = ∣ t ∣ s i > t i s_i > t_i s i > t i e i e_i e i s i ≠ t i s_i \ne t_i s i = t i
We first identify the minimal solution for a particular N N N
Lemma 2.16 (Minimal solution under O \mathcal{O} O N N N S o l ( N ) \mathop{\mathrm{Sol}}(N) Sol ( N ) O \mathcal{O} O t + \mathbf{t}^+ t + Lemma 2.2 .
Proof. Let t ∈ S o l ( N ) \mathbf{t}\in \mathop{\mathrm{Sol}}(N) t ∈ Sol ( N ) ∣ N ∣ = q d + r |N| = qd+r ∣ N ∣ = q d + r d = d 1 d = d_1 d = d 1 N < 0 N<0 N < 0 d = d 2 d=d_2 d = d 2 N ≥ 0 N\ge0 N ≥ 0 r ∈ { 0 , … , d − 1 } r \in \{0,\dots,d-1\} r ∈ { 0 , … , d − 1 } N = 0 N=0 N = 0 r = 0 r=0 r = 0 t + \mathbf{t}^+ t + N ≠ 0 N \ne 0 N = 0 r > 0 r>0 r > 0 ε = s g n ( N ) \varepsilon = \mathop{\mathrm{sgn}}(N) ε = sgn ( N ) ∣ t ∣ = q + 1 |\mathbf{t}| = q+1 ∣ t ∣ = q + 1 j ∈ I 0 j \in I_0 j ∈ I 0 s g n ( j ) ≠ ε \mathop{\mathrm{sgn}}(j) \ne \varepsilon sgn ( j ) = ε t j = 0 t_j = 0 t j = 0 Lemma 2.2 , so suppose s g n ( j ) = ε \mathop{\mathrm{sgn}}(j) = \varepsilon sgn ( j ) = ε ∣ j ∣ < r |j| < r ∣ j ∣ < r t j = 0 t_j = 0 t j = 0 ∣ N ∣ = ∑ i ∈ I 0 ∣ i ∣ t i < q d + r = ∣ N ∣ |N| = \sum_{i \in I_0} |i| t_i < qd+r = |N| ∣ N ∣ = ∑ i ∈ I 0 ∣ i ∣ t i < q d + r = ∣ N ∣ ∣ j ∣ = r |j| = r ∣ j ∣ = r t j ≤ 1 t_j \le 1 t j ≤ 1 2 r + ( q − 1 ) d < q d + r = ∣ N ∣ 2r + (q-1)d < qd+r = |N| 2 r + ( q − 1 ) d < q d + r = ∣ N ∣ t j = 1 t_j = 1 t j = 1 ± d \pm d ± d t = t + \mathbf{t}= \mathbf{t}^+ t = t +
The greedy solution t + \mathbf{t}^+ t + t ε r + = 1 t^+_{\varepsilon r} = 1 t ε r + = 1 e ε r e_{\varepsilon r} e ε r 1 1 1 0 0 0 t + \mathbf{t}^+ t + O \mathcal{O} O
Lemma 2.17 (Minimal permutation under O \mathcal{O} O − d 1 ≤ − k 1 ≤ 0 ≤ k 2 ≤ d 2 -d_1 \le -k_1 \le 0 \le k_2 \le d_2 − d 1 ≤ − k 1 ≤ 0 ≤ k 2 ≤ d 2 I = { − k 1 , … , k 2 } \mathcal{I}= \{-k_1, \dots, k_2\} I = { − k 1 , … , k 2 } t + ( N ) \mathbf{t}^+(N) t + ( N ) S o l ( N ) \mathop{\mathrm{Sol}}(N) Sol ( N ) Lemma 2.2 . If p p p d 1 d_1 d 1 d 2 d_2 d 2 τ \tau τ ∑ ℓ ∈ I t + ( p ℓ − τ ℓ ) \sum_{\ell \in \mathcal{I}} \mathbf{t}^+(p\ell - \tau\ell) ∑ ℓ ∈ I t + ( p ℓ − τ ℓ ) O \mathcal{O} O
Proof. We define τ \tau τ Proposition 2.6 . Starting with ℓ = k 2 \ell = k_2 ℓ = k 2 I ℓ = { 1 , … , k 2 } ∖ { τ − 1 ( j ) ∣ j ≥ ℓ + 1 } I_\ell = \{1, \dots, k_2\} \setminus\{\tau^{-1}(j) \mid j \ge
\ell+1\} I ℓ = { 1 , … , k 2 } ∖ { τ − 1 ( j ) ∣ j ≥ ℓ + 1 } τ − 1 ( ℓ ) = i \tau^{-1}(\ell) = i τ − 1 ( ℓ ) = i i i i I ℓ I_\ell I ℓ [ p i − ℓ ] d 2 [pi-\ell]_{d_2} [ p i − ℓ ] d 2 ℓ \ell ℓ ( k 1 , d 1 ) (k_1, d_1) ( k 1 , d 1 ) ( k 2 , d 2 ) (k_2, d_2) ( k 2 , d 2 ) τ ′ \tau' τ ′ { 1 , … , k 1 } \{1, \dots, k_1\} { 1 , … , k 1 } τ ′ − 1 ( − ℓ ) = i \tau'^{-1}(-\ell) = i τ ′ − 1 ( − ℓ ) = i i i i { 1 , … , k 1 } ∖ { τ ′ − 1 ( j ) ∣ j ≥ − ℓ + 1 } \{1, \dots, k_1\}
\setminus\{\tau'^{-1}(j)\mid j \ge -\ell+1\} { 1 , … , k 1 } ∖ { τ ′ − 1 ( j ) ∣ j ≥ − ℓ + 1 } [ p i + ℓ ] d 1 [pi+\ell]_{d_1} [ p i + ℓ ] d 1 τ ( ℓ ) = − τ ′ ( − ℓ ) \tau(\ell) = -\tau'(-\ell) τ ( ℓ ) = − τ ′ ( − ℓ ) τ ( 0 ) = 0 \tau(0) = 0 τ ( 0 ) = 0
Note that in the proof of Proposition 2.6 , we already established that τ − 1 \tau^{-1} τ − 1 τ ∈ Σ p ( I ) \tau \in \mathbf{\Sigma}^p(\mathcal{I}) τ ∈ Σ p ( I )
Let σ \sigma σ
∑ ℓ ∈ I t + ( p ℓ − σ ℓ ) ≤ O ∑ ℓ ∈ I t + ( p ℓ − τ ℓ ) . \begin{equation}
\sum_{\ell \in \mathcal{I}} \mathbf{t}^+(p\ell - \sigma\ell)
\le_{\mathcal{O}} \sum_{\ell \in \mathcal{I}} \mathbf{t}^+(p\ell - \tau\ell). \tag*{(4)}
\end{equation} ℓ ∈ I ∑ t + ( p ℓ − σ ℓ ) ≤ O ℓ ∈ I ∑ t + ( p ℓ − τ ℓ ) . (4) We will show that σ = τ \sigma = \tau σ = τ σ − 1 \sigma^{-1} σ − 1 τ − 1 \tau^{-1} τ − 1 ℓ ∈ { 1 , … , k 2 } \ell \in \{1, \dots, k_2\} ℓ ∈ { 1 , … , k 2 } σ − 1 ( j ) = τ − 1 ( j ) \sigma^{-1}(j) = \tau^{-1}(j) σ − 1 ( j ) = τ − 1 ( j ) j > ℓ j > \ell j > ℓ
Let z = τ − 1 ( ℓ ) z = \tau^{-1}(\ell) z = τ − 1 ( ℓ ) r = [ p z − ℓ ] d 2 r = [pz - \ell]_{d_2} r = [ p z − ℓ ] d 2 r = 0 r = 0 r = 0 σ − 1 ( ℓ ) = z \sigma^{-1}(\ell) = z σ − 1 ( ℓ ) = z σ z < ℓ \sigma z < \ell σ z < ℓ ⌈ p z − σ z d 2 ⌉ > ⌈ p z − k 2 d 2 ⌉ \left\lceil \frac{pz - \sigma z}{d_2}\right\rceil >
\left\lceil \frac{pz - k_2}{d_2}\right\rceil ⌈ d 2 p z − σ z ⌉ > ⌈ d 2 p z − k 2 ⌉ Proposition 2.6 . So assume r > 0 r > 0 r > 0
Let y ≥ 1 y \ge 1 y ≥ 1 σ y ≤ ℓ \sigma y \le \ell σ y ≤ ℓ [ p y − σ y ] d 2 = 0 [py-\sigma y]_{d_2} = 0 [ p y − σ y ] d 2 = 0 r ′ = d 2 r' = d_2 r ′ = d 2 r ′ = [ p y − σ y ] d 2 r' = [py - \sigma y]_{d_2} r ′ = [ p y − σ y ] d 2 r ′ > r r' > r r ′ > r σ − 1 ( ℓ ) = τ − 1 ( ℓ ) \sigma^{-1}(\ell) = \tau^{-1}(\ell) σ − 1 ( ℓ ) = τ − 1 ( ℓ )
Note that if y = τ − 1 ( j ) y = \tau^{-1}(j) y = τ − 1 ( j ) j ≥ ℓ + 1 j \ge \ell+1 j ≥ ℓ + 1 y = σ − 1 ( j ) y = \sigma^{-1}(j) y = σ − 1 ( j ) σ y ≤ ℓ \sigma y \le \ell σ y ≤ ℓ y ∈ I ℓ y \in I_\ell y ∈ I ℓ z = τ − 1 ( ℓ ) z = \tau^{-1}(\ell) z = τ − 1 ( ℓ )
[ p y − ℓ ] d 2 ≥ [ p z − ℓ ] d 2 = r \begin{equation}
[py - \ell]_{d_2} \ge [pz - \ell]_{d_2} = r \tag*{(5)}
\end{equation} [ p y − ℓ ] d 2 ≥ [ p z − ℓ ] d 2 = r (5) with equality if and only if y = z y = z y = z O \mathcal{O} O τ ∈ Σ p ( I ) \tau \in \mathbf{\Sigma}^p(\mathcal{I}) τ ∈ Σ p ( I ) (4) that σ ∈ Σ p ( I ) \sigma \in \mathbf{\Sigma}^p(\mathcal{I}) σ ∈ Σ p ( I ) Proposition 2.6 , ⌈ p y − σ y d 2 ⌉ = ⌈ p y − k 2 d 2 ⌉ \left\lceil \frac{py - \sigma y}{d_2}\right\rceil =
\left\lceil \frac{py - k_2}{d_2}\right\rceil ⌈ d 2 p y − σ y ⌉ = ⌈ d 2 p y − k 2 ⌉ [ p y − j ] d 2 ≠ 0 [py-j]_{d_2} \ne 0 [ p y − j ] d 2 = 0 j ∈ { σ y + 1 , … , k 2 } j \in \{\sigma y + 1, \dots, k_2\} j ∈ { σ y + 1 , … , k 2 } [ p y − σ y ] d 2 = 0 [py-\sigma y]_{d_2} = 0 [ p y − σ y ] d 2 = 0 [ p y − σ y ] d 2 ≥ [ p y − ℓ ] d 2 [py - \sigma y]_{d_2} \ge [py - \ell]_{d_2} [ p y − σ y ] d 2 ≥ [ p y − ℓ ] d 2 σ y = ℓ \sigma y = \ell σ y = ℓ (5) , we obtain r ′ ≥ r r' \ge r r ′ ≥ r y = z y = z y = z σ y = ℓ \sigma y = \ell σ y = ℓ y = z y=z y = z σ y = ℓ \sigma y = \ell σ y = ℓ σ − 1 ( ℓ ) = τ − 1 ( ℓ ) \sigma^{-1}(\ell) =
\tau^{-1}(\ell) σ − 1 ( ℓ ) = τ − 1 ( ℓ )
If r ′ ≠ d 2 r' \ne d_2 r ′ = d 2 e r ′ e_{r'} e r ′ t + ( p y − σ y ) \mathbf{t}^+(py - \sigma y) t + ( p y − σ y ) 1 1 1 e d 2 e_{d_2} e d 2 r ′ = d 2 r' = d_2 r ′ = d 2 t + ( p y − σ y ) \mathbf{t}^+(py-\sigma y) t + ( p y − σ y ) e d 2 e_{d_2} e d 2 1 ≤ s ≤ r 1 \le s \le r 1 ≤ s ≤ r e s e_s e s ∑ { y > 0 ∣ σ y ≤ ℓ } t + ( p y − σ y ) \sum_{\{y>0 \mid \sigma y \le \ell\}} \mathbf{t}^+(py - \sigma y) ∑ { y > 0 ∣ σ y ≤ ℓ } t + ( p y − σ y ) 0 0 0 s = r s = r s = r σ − 1 ( ℓ ) = τ − 1 ( ℓ ) \sigma^{-1}(\ell) = \tau^{-1}(\ell) σ − 1 ( ℓ ) = τ − 1 ( ℓ ) (4) , this must be the case.
A symmetric argument shows that σ = τ \sigma = \tau σ = τ I \mathcal{I} I τ \tau τ O \mathcal{O} O
Dwork theory and exponential-sum L L L
In this section, we lay the groundwork for future sections by defining the Dwork operator with respect to a Laurent polynomial f f f L L L f f f [ 0 , d 1 + d 2 ] [0,d_1+d_2] [ 0 , d 1 + d 2 ] { s i ∣ i ∈ Z } \{s^i \mid i \in \mathbb{Z}\} { s i ∣ i ∈ Z }
The Dwork operator
The power series
E p ( s ) ≔ exp ( ∑ m = 0 ∞ s p m p m ) = ∑ i = 0 ∞ β i s i \begin{equation}
E_p(s) \coloneqq \exp \left( \sum_{m=0}^\infty \frac{s^{p^m}}{p^m} \right)
= \sum_{i=0}^\infty \beta_i s^i \tag*{(6)}
\end{equation} E p ( s ) : = exp ( m = 0 ∑ ∞ p m s p m ) = i = 0 ∑ ∞ β i s i (6) is the p p p ( Q ∩ Z p ) [ [ s ] ] (\mathbb{Q}\cap \mathbb{Z}_p)[[s]] ( Q ∩ Z p ) [[ s ]] γ \gamma γ log ( E p ( s ) ) \log(E_p(s)) log ( E p ( s )) p p p v p ( γ ) = 1 p − 1 v_p(\gamma) = \frac{1}{p-1} v p ( γ ) = p − 1 1 E p ( γ ) E_p(\gamma) E p ( γ ) p p p ζ p \zeta_p ζ p Z p [ ζ p ] = Z p [ γ ] \mathbb{Z}_p[\zeta_p] = \mathbb{Z}_p[\gamma] Z p [ ζ p ] = Z p [ γ ]
Let f = f ( x ) = ∑ i = − d 1 d 2 a i x i ∈ F q [ x ± 1 ] f = f(x) = \sum_{i=-d_1}^{d_2} a_i x^i \in \mathbb{F}_q[x^{\pm1}] f = f ( x ) = ∑ i = − d 1 d 2 a i x i ∈ F q [ x ± 1 ] a − d 1 a d 2 ≠ 0 a_{-d_1}a_{d_2} \ne 0 a − d 1 a d 2 = 0 f f f d 2 d_2 d 2 d 1 d_1 d 1 p ∤ d 1 d 2 p \nmid d_1d_2 p ∤ d 1 d 2 m ≥ 1 m \ge 1 m ≥ 1 m m m S m ∗ ( f ) S_m^*(f) S m ∗ ( f )
S m ∗ ( f ) ≔ ∑ z ∈ F q m × ζ p T r F q m / F p ( f ( z ) ) , \begin{equation}
S_m^*(f) \coloneqq \sum_{z \in \mathbb{F}_{q^m}^\times}
\zeta_p^{\mathop{\mathrm{Tr}}_{\mathbb{F}_{q^m}/\mathbb{F}_p}(f(z))}, \tag*{(7)}
\end{equation} S m ∗ ( f ) : = z ∈ F q m × ∑ ζ p Tr F q m / F p ( f ( z )) , (7) and the L L L f f f
L ∗ ( f ; s ) = exp ( ∑ m = 1 ∞ S m ∗ ( f ) s m m ) . \begin{equation}
L^*(f; s) = \exp \left(\sum_{m=1}^\infty S^*_m(f) \frac{s^m}{m}\right). \tag*{(8)}
\end{equation} L ∗ ( f ; s ) = exp ( m = 1 ∑ ∞ S m ∗ ( f ) m s m ) . (8) It is a well known corollary of the Weil Conjecture for curves that the L L L 1 + s Z p [ ζ p ] [ s ] 1 + s\mathbb{Z}_p[\zeta_p][s] 1 + s Z p [ ζ p ] [ s ] d 1 + d 2 d_1 + d_2 d 1 + d 2 [14 Theorem 1.14] . Moreover, the valuations of the reciprocal roots of L ∗ ( f ; s ) L^*(f; s) L ∗ ( f ; s ) S S S ℓ \ell ℓ λ \lambda λ q q q L ∗ ( f ; s ) L^*(f; s) L ∗ ( f ; s ) S ′ S' S ′ ℓ \ell ℓ 1 − λ 1 - \lambda 1 − λ L ∗ ∈ 1 + s Z p [ ζ p ] [ s ] L^* \in 1 + s \mathbb{Z}_p[\zeta_p][s] L ∗ ∈ 1 + s Z p [ ζ p ] [ s ] λ \lambda λ 0 0 0 1 1 1
To the Laurent polynomial f f f
F ( s ) = ∏ i ∈ I 0 E p ( a ^ i γ s i ) = ∑ n = − ∞ ∞ G n s n , \begin{equation}
F(s) = \prod_{i\in I_0} E_p(\widehat{a}_i \gamma s^i)
= \sum_{n=-\infty}^\infty G_n s^n, \tag*{(9)}
\end{equation} F ( s ) = i ∈ I 0 ∏ E p ( a i γ s i ) = n = − ∞ ∑ ∞ G n s n , (9) where a ^ \widehat{a} a a ∈ F q a \in \mathbb{F}_q a ∈ F q Q q \mathbb{Q}_q Q q b b b Q p \mathbb{Q}_p Q p
Let w : Z → Q ≥ 0 w : \mathbb{Z}\to \mathbb{Q}_{\ge 0} w : Z → Q ≥ 0
w(i) =
\begin{cases*}
\frac{-i}{d_1} & if $i \le 0$ \\
\frac{i}{d_2} & if $i > 0$.
\end{cases*}
Then F ( s ) F(s) F ( s )
H ≔ { ∑ j = − ∞ ∞ B j s j ∈ Q q ( ζ p ) [ [ s ± 1 ] ] ∣ v p ( B j ) ≥ w ( j ) p − 1 + O ( 1 ) } . \mathcal{H} \coloneqq \left\{\sum_{j=-\infty}^\infty B_j s^j
\in \mathbb{Q}_q(\zeta_p)[[s^{\pm 1}]]
\:\Big|\:
v_p(B_j) \ge \frac{w(j)}{p-1} + O(1)\right\}. H : = { j = − ∞ ∑ ∞ B j s j ∈ Q q ( ζ p ) [[ s ± 1 ]] v p ( B j ) ≥ p − 1 w ( j ) + O ( 1 ) } . When equipped with the norm
∣ ∑ j B j s j ∣ = sup j ∈ Z ( ∣ B j ∣ p p w ( j ) / ( p − 1 ) ) , \left|
\sum_j B_j s^j
\right| =
\sup_{j \in \mathbb{Z}}\left(|B_j|_p p^{w(j)/(p-1)}\right), j ∑ B j s j = j ∈ Z sup ( ∣ B j ∣ p p w ( j ) / ( p − 1 ) ) , H \mathcal{H} H Q q ( ζ p ) \mathbb{Q}_q(\zeta_p) Q q ( ζ p ) ξ ∈ Q q ( ζ p ) [ [ s ± 1 ] ] \xi \in \mathbb{Q}_q(\zeta_p)[[s^{\pm1}]] ξ ∈ Q q ( ζ p ) [[ s ± 1 ]] ∣ ξ ∣ < ∞ |\xi| < \infty ∣ ξ ∣ < ∞
Let τ \tau τ F q \mathbb{F}_q F q Q q \mathbb{Q}_q Q q [17 ] , is given first by setting α 0 = τ − 1 ∘ ψ p ∘ F , \alpha_0 = \tau^{-1} \circ \psi_p \circ F, α 0 = τ − 1 ∘ ψ p ∘ F , ψ p ( ∑ i B i s i ) = ∑ i B p i s i \psi_p \left(\sum_i B_i s^i\right) = \sum_i B_{pi} s^i ψ p ( ∑ i B i s i ) = ∑ i B p i s i F ( H ) = F ⋅ H F(H) = F \cdot H F ( H ) = F ⋅ H
α = α 0 b . \begin{equation}
\alpha = \alpha_0^b. \tag*{(10)}
\end{equation} α = α 0 b . (10) Following the theory of Serre in [22 ] , the operator α \alpha α H \mathcal{H} H
We omit the proof of the following proposition as it is well known, see [17 ] .
Proposition 3.2. Let f f f F q \mathbb{F}_q F q d 2 d_2 d 2 d 1 d_1 d 1 p ∤ d 1 d 2 p\nmid d_1d_2 p ∤ d 1 d 2 L ∗ ( f ; s ) L^*(f; s) L ∗ ( f ; s ) α \alpha α (8) and (10) , respectively. Then N P q ( L ∗ ( f ; s ) ) = N P q ( det ( 1 − α s ) ) ∣ [ 0 , d 1 + d 2 ] . \mathop{\mathrm{NP}}_q(L^*(f; s)) = \mathop{\mathrm{NP}}_q(\det(1 - \alpha s))\big|_{[0,d_1+d_2]}. NP q ( L ∗ ( f ; s )) = NP q ( det ( 1 − α s )) [ 0 , d 1 + d 2 ] .
There exists a matrix expression for α \alpha α B = s Z = { s i ∣ i ∈ Z } \mathcal{B}= s^{\mathbb{Z}} = \{s^i \mid i \in \mathbb{Z}\} B = s Z = { s i ∣ i ∈ Z } G = M a t B ( ψ p ∘ F ) \mathcal{G}= \mathop{\mathrm{Mat}}_{\mathcal{B}}(\psi_p \circ F) G = Mat B ( ψ p ∘ F ) ψ p ∘ F \psi_p \circ F ψ p ∘ F B \mathcal{B} B
G = ( G p i − j ) i , j ∈ Z . \begin{equation}
\mathcal{G}= (G_{pi-j})_{i,j \in \mathbb{Z}}. \tag*{(11)}
\end{equation} G = ( G p i − j ) i , j ∈ Z . (11) Write
[ G , τ , b ] = G τ b − 1 G τ b − 2 ⋯ G τ G . \begin{equation}
[\mathcal{G}, \tau, b] = \mathcal{G}^{\tau^{b-1}} \mathcal{G}^{\tau^{b-2}} \cdots \mathcal{G}^\tau \mathcal{G}. \tag*{(12)}
\end{equation} [ G , τ , b ] = G τ b − 1 G τ b − 2 ⋯ G τ G . (12) Let M a t B ( α 0 ) \mathop{\mathrm{Mat}}_{\mathcal{B}}(\alpha_0) Mat B ( α 0 ) i i i α 0 ( s i ) \alpha_0(s^i) α 0 ( s i ) G τ − 1 = M a t B ( α 0 ) \mathcal{G}^{\tau^{-1}} = \mathop{\mathrm{Mat}}_{\mathcal{B}}(\alpha_0) G τ − 1 = Mat B ( α 0 ) τ b = 1 \tau^b = 1 τ b = 1
M a t B ( α ) = M a t B ( α 0 b ) = G τ − 1 G τ − 2 ⋯ G τ − b = [ G , τ , b ] . \begin{equation}
\mathop{\mathrm{Mat}}_{\mathcal{B}}(\alpha) =
\mathop{\mathrm{Mat}}_{\mathcal{B}}(\alpha_0^b) = \mathcal{G}^{\tau^{-1}} \mathcal{G}^{\tau^{-2}} \cdots
\mathcal{G}^{\tau^{-b}} = [\mathcal{G}, \tau, b]. \tag*{(13)}
\end{equation} Mat B ( α ) = Mat B ( α 0 b ) = G τ − 1 G τ − 2 ⋯ G τ − b = [ G , τ , b ] . (13) With (13) , it follows from Proposition 3.2 that
N P q ( L ∗ ( f ; s ) ) = N P q ( det ( 1 − [ G , τ , b ] s ) ) ∣ [ 0 , d 1 + d 2 ] . \begin{equation}
\mathop{\mathrm{NP}}_q(L^*(f; s)) = \mathop{\mathrm{NP}}_q(\det(1 - [\mathcal{G}, \tau, b]s))|_{[0,d_1+d_2]}. \tag*{(14)}
\end{equation} NP q ( L ∗ ( f ; s )) = NP q ( det ( 1 − [ G , τ , b ] s )) ∣ [ 0 , d 1 + d 2 ] . (14) In a later section (see § 5 ), we will prove that the valuations of the interior vertices of det ( 1 − G s ) \det(1 - \mathcal{G}s) det ( 1 − G s ) det ( 1 − [ G , τ , b ] s ) \det(1 - [\mathcal{G}, \tau, b]s) det ( 1 − [ G , τ , b ] s )
Proposition 3.3. Then det ( 1 − G s ) = 1 + C 1 s + C 2 s 2 + … \det(1 - \mathcal{G}s) = 1 + C_1 s + C_2 s^2 + \dots det ( 1 − G s ) = 1 + C 1 s + C 2 s 2 + … C k = ( − 1 ) k ∑ ( I , σ ) s g n ( σ ) ∏ ℓ ∈ I G p ℓ − σ ℓ , C_k = (-1)^k \sum_{(\mathcal{I}, \sigma)} \mathop{\mathrm{sgn}}(\sigma)
\prod_{\ell \in \mathcal{I}} G_{p\ell - \sigma \ell}, C k = ( − 1 ) k ∑ ( I , σ ) sgn ( σ ) ∏ ℓ ∈ I G p ℓ − σ ℓ , ∑ ( I , σ ) \sum_{(\mathcal{I}, \sigma)} ∑ ( I , σ ) ( I , σ ) (\mathcal{I}, \sigma) ( I , σ ) I ∈ ( Z k ) \mathcal{I}\in \binom{\mathbb{Z}}{k} I ∈ ( k Z ) σ \sigma σ I \mathcal{I} I
Rank-ℓ \ell ℓ
We introduce the L L L F p ℓ \mathbb{F}_{p^\ell} F p ℓ
The function χ 1 : F p → Q ‾ p × \chi_1 : \mathbb{F}_p \to \overline{\mathbb{Q}}_p^\times χ 1 : F p → Q p × χ 1 ( z ) = ζ p z \chi_1(z) =
\zeta_p^{z} χ 1 ( z ) = ζ p z F p \mathbb{F}_p F p (7) and (8) correspond to the exponential sums and L L L ℓ ∈ Z ≥ 1 \ell \in \mathbb{Z}_{\ge 1} ℓ ∈ Z ≥ 1 b b b χ ℓ : F p ℓ → Q ‾ p × \chi_\ell : \mathbb{F}_{p^\ell} \to \overline{\mathbb{Q}}_p^\times χ ℓ : F p ℓ → Q p × (7) and (8) by defining the m m m f f f F q \mathbb{F}_q F q χ ℓ \chi_\ell χ ℓ
S m ∗ ( f , χ ℓ ) = ∑ z ∈ F q m × χ ℓ ( T r F q m / F p ℓ ( f ( z ) ) ) S_m^*(f, \chi_\ell) = \sum_{z \in \mathbb{F}_{q^m}^\times}
\chi_\ell(\mathop{\mathrm{Tr}}_{\mathbb{F}_{q^m} / \mathbb{F}_{p^\ell}}(f(z))) S m ∗ ( f , χ ℓ ) = z ∈ F q m × ∑ χ ℓ ( Tr F q m / F p ℓ ( f ( z ))) and its L L L
L ∗ ( f , χ ℓ ; s ) = exp ( ∑ m = 1 ∞ S m ∗ ( f , χ ℓ ) s m m ) . L^*(f, \chi_\ell; s) =
\exp \left(\sum_{m=1}^\infty S_m^*(f, \chi_\ell) \frac{s^m}{m}\right). L ∗ ( f , χ ℓ ; s ) = exp ( m = 1 ∑ ∞ S m ∗ ( f , χ ℓ ) m s m ) . This is the rank-ℓ \ell ℓ L L L f f f 1 1 1 L L L
Proposition 3.4. Let ℓ ∣ b \ell \mid b ℓ ∣ b χ ℓ : F p ℓ → Q ‾ p × \chi_\ell : \mathbb{F}_{p^\ell} \to \overline{\mathbb{Q}}_p^\times χ ℓ : F p ℓ → Q p × α ∈ F p ℓ × \alpha \in \mathbb{F}_{p^\ell}^\times α ∈ F p ℓ × f ∈ F q [ x ± 1 ] f \in \mathbb{F}_q[x^{\pm 1}] f ∈ F q [ x ± 1 ] L ∗ ( f , χ ℓ ; s ) = L ∗ ( α f ; s ) . L^*(f, \chi_\ell; s) = L^*(\alpha f; s). L ∗ ( f , χ ℓ ; s ) = L ∗ ( α f ; s ) .
See the discussion above [26 Proposition 2.3] for a proof of the statement for f ∈ F q [ x ] f \in \mathbb{F}_q[x] f ∈ F q [ x ]
On the other hand, the rank-ℓ \ell ℓ L L L X f , ℓ : y p ℓ − y = f X_{f, \ell} : y^{p^\ell} - y = f X f , ℓ : y p ℓ − y = f
Proposition 3.5. Let ℓ ∣ b \ell \mid b ℓ ∣ b f ∈ F q [ x ± 1 ] f \in \mathbb{F}_q[x^{\pm 1}] f ∈ F q [ x ± 1 ] 0 0 0 ∞ \infty ∞ p p p X f , ℓ : y p ℓ − y = f ( x ) X_{f, \ell} : y^{p^\ell} - y = f(x) X f , ℓ : y p ℓ − y = f ( x ) ℓ \ell ℓ f f f Z ( X f , ℓ / F q ; s ) = ∏ λ ∈ F p ℓ × L ∗ ( λ f ; s ) ( 1 − s ) ( 1 − q s ) . Z(X_{f, \ell} / \mathbb{F}_q; s) =
\frac{\prod_{\lambda \in \mathbb{F}_{p^\ell}^\times}
L^*(\lambda f; s)}{(1-s)(1-qs)}. Z ( X f , ℓ / F q ; s ) = ( 1 − s ) ( 1 − q s ) ∏ λ ∈ F p ℓ × L ∗ ( λ f ; s ) .
T T T C C C Liu and Wan developed T T T p p p g ∈ F q [ x ± 1 ] g \in \mathbb{F}_q[x^{\pm 1}] g ∈ F q [ x ± 1 ]
S m ∗ ( g ; T ) ≔ ∑ x ∈ F q m × ( 1 + T ) T r Q q m / Q p ( g ^ ( x ^ ) ) S_m^*(g ; T) \coloneqq \sum_{x \in \mathbb{F}_{q^m}^\times}
(1 + T)^{\mathop{\mathrm{Tr}}_{\mathbb{Q}_{q^m} / \mathbb{Q}_p}(\widehat{g}(\widehat{x}))} S m ∗ ( g ; T ) : = x ∈ F q m × ∑ ( 1 + T ) Tr Q q m / Q p ( g ( x )) and
C ∗ ( g ; T , s ) ≔ exp ( − ∑ m = 1 ∞ S m ∗ ( g ; T ) q m − 1 s m m ) . C^*(g; T, s) \coloneqq \exp \left(-\sum_{m = 1}^\infty
\frac{S_m^*(g ; T)}{q^m-1} \frac{s^m}{m} \right). C ∗ ( g ; T , s ) : = exp ( − m = 1 ∑ ∞ q m − 1 S m ∗ ( g ; T ) m s m ) . Here, Q q m / Q p \mathbb{Q}_{q^m} / \mathbb{Q}_p Q q m / Q p F q m \mathbb{F}_{q^m} F q m g ^ \widehat{g} g x ^ \widehat{x} x g ^ \widehat{g} g
L ∗ ( g ; T , s ) ≔ C ∗ ( g ; T , s ) C ∗ ( g ; T , q s ) . L^*(g; T, s) \coloneqq \frac{C^*(g; T, s)}{C^*(g; T, qs)}. L ∗ ( g ; T , s ) : = C ∗ ( g ; T , q s ) C ∗ ( g ; T , s ) . For each k ≥ 1 k \ge 1 k ≥ 1 ζ p k \zeta_{p^k} ζ p k p k p^k p k ζ p 1 \zeta_{p^1} ζ p 1 ζ p = E p ( γ ) \zeta_p = E_p(\gamma) ζ p = E p ( γ ) π k = ζ p k − 1 \pi_k = \zeta_{p^k}-1 π k = ζ p k − 1 T T T π k \pi_k π k q q q
o r d T ( T ) = 1 , o r d π k ( π k ) = 1 , o r d q ( q ) = 1 , o r d π m ( q ) = b p m − 1 ( p − 1 ) . \mathop{\mathrm{ord}}_T(T) = 1, \quad \mathop{\mathrm{ord}}_{\pi_k}(\pi_k) = 1, \quad \mathop{\mathrm{ord}}_q(q) = 1, \quad \mathop{\mathrm{ord}}_{\pi_m}(q)=bp^{m-1}(p-1). ord T ( T ) = 1 , ord π k ( π k ) = 1 , ord q ( q ) = 1 , ord π m ( q ) = b p m − 1 ( p − 1 ) . The T T T π k \pi_k π k p p p q q q T T T o r d T ( p ) = 0 \mathop{\mathrm{ord}}_T(p)=0 ord T ( p ) = 0 k = 1 k = 1 k = 1
L ∗ ( g ; T , s ) ∣ T = π 1 = L ∗ ( g ; s ) . \begin{equation}
L^*(g; T, s)|_{T=\pi_1} = L^*(g; s). \tag*{(15)}
\end{equation} L ∗ ( g ; T , s ) ∣ T = π 1 = L ∗ ( g ; s ) . (15) In § 3.1 , we introduced the splitting function F ( s ) F(s) F ( s ) G \mathcal{G} G det ( 1 − G s ) \det(1 - \mathcal{G}s) det ( 1 − G s )
F(s) = \sum_{n \in \mathbb{Z}} G_n s^n,
\quad
\mathcal{G}= (G_{pi-j})_{i, j \in \mathbb{Z}},
\mbox{ and }
\det(1 - \mathcal{G}s)= 1 + \sum_{k=1}^\infty C_k s^k.
(See equation (9) , (11) , and Proposition 3.3 , respectively.) These coefficients G n G_n G n C k C_k C k Q q ( ζ p ) \mathbb{Q}_q(\zeta_p) Q q ( ζ p )
In this section, we replace the specific choice of coefficients a i ∈ F q a_i \in \mathbb{F}_q a i ∈ F q A i A_i A i G n G_n G n F ( s ) F(s) F ( s ) G n \mathbf{G}_n G n C k C_k C k C k \mathbf{C}_k C k G n \mathbf{G}_n G n C k \mathbf{C}_k C k ( Z p ∩ Q ) [ γ ] (\mathbb{Z}_p \cap \mathbb{Q})[\gamma] ( Z p ∩ Q ) [ γ ] γ \gamma γ p p p E p ( s ) E_p(s) E p ( s ) ( p − 1 ) − 1 (p-1)^{-1} ( p − 1 ) − 1 p p p § 2 .
Expressions in the coefficient space
Inherit all notation from § 3.1 . In this first subsection, we develop the domain space by replacing the coefficients of f f f d 1 d_1 d 1 d 2 d_2 d 2 f f f d 1 d_1 d 1 d 2 d_2 d 2 p p p
Define free variables A i A_i A i i ∈ I 0 i \in I_0 i ∈ I 0 A = ( A i ) i ∈ I 0 A = (A_i)_{i \in I_0} A = ( A i ) i ∈ I 0
F A ( s ) = ∏ i ∈ I 0 E p ( A i γ s i ) = ∑ n = − ∞ ∞ G n s n . \begin{equation}
F_A(s) = \prod_{i \in I_0} E_p(A_i \gamma s^i)
= \sum_{n=-\infty}^\infty \mathbf{G}_n s^n. \tag*{(16)}
\end{equation} F A ( s ) = i ∈ I 0 ∏ E p ( A i γ s i ) = n = − ∞ ∑ ∞ G n s n . (16) It follows from Proposition 4.1 below that each G n = G n ( A ) \mathbf{G}_n = \mathbf{G}_n(A) G n = G n ( A ) Z p [ γ ] < A > \mathbb{Z}_p[\gamma]\left<A\right> Z p [ γ ] ⟨ A ⟩ Q [ γ ] \mathbb{Q}[\gamma] Q [ γ ] f = ∑ i ∈ I 0 a i x i f = \sum_{i \in I_0}a_i x^i f = ∑ i ∈ I 0 a i x i F q \mathbb{F}_q F q
G n ( a ^ ) = G n \mathbf{G}_n(\widehat{a}) = G_n G n ( a ) = G n from § 3.1 , where a ^ = ( a ^ i ) i ∈ I 0 \widehat{a} = (\widehat{a}_i)_{i \in I_0} a = ( a i ) i ∈ I 0
To describe G n \mathbf{G}_n G n Definition 2.1 from § 2.1 . A representation of an integer n n n t = ( t i ) i ∈ I 0 ∈ Z ≥ 0 I 0 \mathbf{t}= (t_i)_{i \in I_0} \in \mathbb{Z}_{\ge 0}^{I_0} t = ( t i ) i ∈ I 0 ∈ Z ≥ 0 I 0 n = ∑ i ∈ I 0 t i i n = \sum_{i \in I_0} t_i i n = ∑ i ∈ I 0 t i i M ( n ) M(n) M ( n ) ∣ t ∣ = ∑ i t i |\mathbf{t}| = \sum_i t_i ∣ t ∣ = ∑ i t i n n n n n n n n n
Let T ( n ) \mathcal{T}(n) T ( n ) n n n S o l ( n ) \mathop{\mathrm{Sol}}(n) Sol ( n ) n n n
Proposition 4.1. Let E p ( s ) = ∑ m = 0 ∞ β m s m E_p(s) = \sum_{m=0}^\infty \beta_m s^m E p ( s ) = ∑ m = 0 ∞ β m s m n ∈ Z n \in \mathbb{Z} n ∈ Z G n = ∑ t ∈ T ( n ) ( ∏ i ∈ I 0 β t i ) γ ∣ t ∣ A t , \mathbf{G}_n = \sum_{\mathbf{t}\in \mathcal{T}(n)}
\left(\prod_{i \in I_0} \beta_{t_i}\right)
\gamma^{|\mathbf{t}|} A^\mathbf{t}, G n = ∑ t ∈ T ( n ) ( ∏ i ∈ I 0 β t i ) γ ∣ t ∣ A t , A t A^\mathbf{t} A t ∏ i ∈ I 0 A i t i \prod_{i\in I_0} A_i^{t_i} ∏ i ∈ I 0 A i t i v p ( G n ) ≥ M ( n ) p − 1 v_p(\mathbf{G}_n) \ge \frac{M(n)}{p-1} v p ( G n ) ≥ p − 1 M ( n ) − p d 1 < n < p d 2 -pd_1 < n < pd_2 − p d 1 < n < p d 2 v p ( G n ) = M ( n ) p − 1 v_p(\mathbf{G}_n) = \frac{M(n)}{p-1} v p ( G n ) = p − 1 M ( n )
Proof. The expression for G n \mathbf{G}_n G n (16) . The inequality v p ( G n ) ≥ M ( n ) p − 1 v_p(\mathbf{G}_n) \ge \frac{M(n)}{p-1} v p ( G n ) ≥ p − 1 M ( n ) ∏ i ∈ I 0 β t i \prod_{i \in I_0} \beta_{t_i} ∏ i ∈ I 0 β t i p p p
Suppose − p d 1 < n < p d 2 -pd_1 < n < pd_2 − p d 1 < n < p d 2 n > 0 n > 0 n > 0 t + ∈ S o l ( n ) \mathbf{t}^+ \in \mathop{\mathrm{Sol}}(n) t + ∈ Sol ( n ) Lemma 2.2 . Since n < p d 2 n < pd_2 n < p d 2 t d 2 + = ⌊ n / d 2 ⌋ < p t^+_{d_2} = \left\lfloor n/d_2\right\rfloor < p t d 2 + = ⌊ n / d 2 ⌋ < p t j + < p t^+_j < p t j + < p j ∈ I 0 j \in I_0 j ∈ I 0 ∏ i ∈ I 0 β t i + \prod_{i \in I_0} \beta_{t^+_i} ∏ i ∈ I 0 β t i + p p p β j = 1 j ! ∈ Z p × \beta_j = \frac{1}{j!} \in \mathbb{Z}_p^\times β j = j ! 1 ∈ Z p × j < p j < p j < p A t + A^{\mathbf{t}^+} A t + G n \mathbf{G}_n G n M ( n ) p − 1 \frac{M(n)}{p-1} p − 1 M ( n ) v p ( G n ) = M ( n ) p − 1 v_p(\mathbf{G}_n) = \frac{M(n)}{p-1} v p ( G n ) = p − 1 M ( n )
If n < 0 n < 0 n < 0 t + \mathbf{t}^+ t + p p p A t + A^{\mathbf{t}^+} A t + γ M ( n ) \gamma^{M(n)} γ M ( n ) p p p v p ( G n ) = M ( n ) p − 1 v_p(\mathbf{G}_n) = \frac{M(n)}{p-1} v p ( G n ) = p − 1 M ( n ) n = 0 n=0 n = 0 G 0 \mathbf{G}_0 G 0 1 1 1
Next we extend the coefficients C k C_k C k Proposition 3.3 into our coefficient space. For every integer k ≥ 1 k \ge 1 k ≥ 1
C k ( A ) ≔ ( − 1 ) k ∑ ( I , σ ) s g n ( σ ) ∏ ℓ ∈ I G p ℓ − σ ℓ ( A ) , \begin{equation}
\mathbf{C}_k(A) \coloneqq (-1)^k \sum_{(\mathcal{I}, \sigma)} \mathop{\mathrm{sgn}}(\sigma)
\prod_{\ell \in \mathcal{I}} \mathbf{G}_{p\ell - \sigma\ell}(A), \tag*{(17)}
\end{equation} C k ( A ) : = ( − 1 ) k ( I , σ ) ∑ sgn ( σ ) ℓ ∈ I ∏ G p ℓ − σ ℓ ( A ) , (17) where ∑ ( I , σ ) \sum_{(\mathcal{I}, \sigma)} ∑ ( I , σ ) ( I , σ ) (\mathcal{I}, \sigma) ( I , σ ) I ∈ ( Z k ) \mathcal{I}\in \binom{\mathbb{Z}}{k} I ∈ ( k Z ) σ \sigma σ I \mathcal{I} I C k \mathbf{C}_k C k Z p [ γ ] < A > \mathbb{Z}_p[\gamma]\left<A\right> Z p [ γ ] ⟨ A ⟩ Q [ γ ] \mathbb{Q}[\gamma] Q [ γ ] f = ∑ i ∈ I 0 a i x i f = \sum_{i \in I_0} a_i x^i f = ∑ i ∈ I 0 a i x i
C k ( a ^ ) = C k , \mathbf{C}_k(\widehat{a}) = C_k, C k ( a ) = C k , where C k C_k C k Proposition 3.3 , and a ^ = ( a ^ i ) i ∈ I 0 \widehat{a} = (\widehat{a}_i)_{i \in I_0} a = ( a i ) i ∈ I 0
Our goal is to establish a precise value for v p ( C k ) v_p(\mathbf{C}_k) v p ( C k ) Corollary 4.4 .
Isolating terms of minimal valuation
We next isolate the terms in C k \mathbf{C}_k C k n ∈ Z n \in \mathbb{Z} n ∈ Z Proposition 4.1 we may express G n \mathbf{G}_n G n γ \gamma γ
G n = ∑ v = 0 ∞ G n , v γ v , \mathbf{G}_n = \sum_{v = 0}^\infty \mathbf{G}_{n, v} \gamma^v, G n = v = 0 ∑ ∞ G n , v γ v , where
G n , v = ∑ t ∈ T ( n ) ∣ t ∣ = v ( ∏ i ∈ I 0 β t i ) A t . \mathbf{G}_{n, v} = \sum_{
\substack{\mathbf{t}\in \mathcal{T}(n)\\ |\mathbf{t}|=v}}
\left(\prod_{i\in I_0} \beta_{t_i}\right) A^{\mathbf{t}}. G n , v = t ∈ T ( n ) ∣ t ∣ = v ∑ ( i ∈ I 0 ∏ β t i ) A t . Note that G n , v = 0 \mathbf{G}_{n, v} = 0 G n , v = 0 v < M ( n ) v < M(n) v < M ( n )
G n ∘ ≔ G n , M ( n ) = ∑ t ∈ S o l ( n ) ( ∏ i ∈ I 0 β t i ) A t . \begin{equation}
\mathbf{G}_n^\circ \coloneqq \mathbf{G}_{n, M(n)}
= \sum_{\mathbf{t}\in \mathop{\mathrm{Sol}}(n)}
\left(\prod_{i\in I_0} \beta_{t_i}\right) A^{\mathbf{t}}. \tag*{(18)}
\end{equation} G n ∘ : = G n , M ( n ) = t ∈ Sol ( n ) ∑ ( i ∈ I 0 ∏ β t i ) A t . (18) Note that S o l ( n ) \mathop{\mathrm{Sol}}(n) Sol ( n ) ( t i ) i ∈ I 0 (t_i)_{i\in I_0} ( t i ) i ∈ I 0 G n ∘ \mathbf{G}_n^\circ G n ∘ M ( n ) M(n) M ( n ) ( Z p ∩ Q ) [ A ] (\mathbb{Z}_p \cap \mathbb{Q})[A] ( Z p ∩ Q ) [ A ] β t i \beta_{t_i} β t i
For each k ≥ 1 k \ge 1 k ≥ 1 C k \mathbf{C}_k C k γ \gamma γ
C k = ∑ v = 0 ∞ C k , v γ v , \begin{equation}
\mathbf{C}_k = \sum_{v=0}^\infty \mathbf{C}_{k, v} \gamma^v, \tag*{(19)}
\end{equation} C k = v = 0 ∑ ∞ C k , v γ v , (19) where C k , v \mathbf{C}_{k, v} C k , v C k \mathbf{C}_k C k v v v γ v \gamma^v γ v
Here we recall the notation from Definition 2.4 — namely S p ( k ) S^p(k) S p ( k ) ∑ ℓ ∈ I M ( p ℓ − σ ℓ ) \sum_{\ell \in \mathcal{I}} M(p\ell - \sigma\ell) ∑ ℓ ∈ I M ( p ℓ − σ ℓ ) I \mathcal{I} I σ \sigma σ (17) . Let Σ k p = { ( I , σ ) ∣ ∑ ℓ ∈ I M ( p ℓ − σ ℓ ) = S p ( k ) } \mathbf{\Sigma}_k^p = \left\{(\mathcal{I}, \sigma) \mid \sum_{\ell \in \mathcal{I}} M(p\ell -
\sigma\ell) = S^p(k) \right\} Σ k p = { ( I , σ ) ∣ ∑ ℓ ∈ I M ( p ℓ − σ ℓ ) = S p ( k ) } G n ∘ \mathbf{G}_n^\circ G n ∘
C k ∘ ≔ ( − 1 ) k ∑ ( I , σ ) ∈ Σ k p s g n ( σ ) ∏ ℓ ∈ I G p ℓ − σ ℓ ∘ . \begin{equation}
\mathbf{C}_k^\circ \coloneqq
(-1)^k \sum_{(\mathcal{I}, \sigma) \in \mathbf{\Sigma}_k^p}
\mathop{\mathrm{sgn}}(\sigma)
\prod_{\ell \in \mathcal{I}} \mathbf{G}_{p\ell - \sigma\ell}^\circ. \tag*{(20)}
\end{equation} C k ∘ : = ( − 1 ) k ( I , σ ) ∈ Σ k p ∑ sgn ( σ ) ℓ ∈ I ∏ G p ℓ − σ ℓ ∘ . (20) By unraveling the expressions of C k , v \mathbf{C}_{k,v} C k , v G n \mathbf{G}_n G n C k , v = 0 \mathbf{C}_{k, v} = 0 C k , v = 0 v < S p ( k ) v < S^p(k) v < S p ( k ) C k , S p ( k ) = C k ∘ \mathbf{C}_{k, S^p(k)} = \mathbf{C}_k^\circ C k , S p ( k ) = C k ∘ C k ∘ \mathbf{C}_k^\circ C k ∘ S p ( k ) S^p(k) S p ( k ) ( Z p ∩ Q ) [ A ] (\mathbb{Z}_p \cap \mathbb{Q})[A] ( Z p ∩ Q ) [ A ]
We summarize this observation with a proposition.
Proposition 4.2. Suppose k ≥ 1 k \ge 1 k ≥ 1 p > 2 D p > 2D p > 2 D v p ( C k ) ≥ ( p − 1 ) − 1 S p ( k ) v_p(\mathbf{C}_k) \ge (p-1)^{-1} S^p(k) v p ( C k ) ≥ ( p − 1 ) − 1 S p ( k ) C k ∘ \mathbf{C}_k^\circ C k ∘ p p p v p ( C k ) = ( p − 1 ) − 1 S p ( k ) v_p(\mathbf{C}_k) = (p-1)^{-1} S^p(k) v p ( C k ) = ( p − 1 ) − 1 S p ( k )
We can therefore establish a formula for v p ( C k ) v_p(\mathbf{C}_k) v p ( C k ) C k ∘ \mathbf{C}_k^{\circ} C k ∘ p p p § 2 to isolate a nonzero term.
A nonzero term in C k ∘ \mathbf{C}_k^\circ C k ∘
Assume p > 2 D p > 2D p > 2 D G L ( d 1 , d 2 , p ) \mathop{\mathrm{GL}}(d_1,d_2,p) GL ( d 1 , d 2 , p ) Definition 2.5 , along with the set V p V_p V p Corollary 2.13 , an internal point k k k V p V_p V p I ∈ ( Z k ) \mathcal{I}\in \binom{\mathbb{Z}}{k} I ∈ ( k Z ) S p ( k ) = S p ( I ) S^p(k) = S^p(\mathcal{I}) S p ( k ) = S p ( I )
When k ∈ V p k \in V_p k ∈ V p C k ∘ \mathbf{C}_k^\circ C k ∘ (20) can be expressed as
C k ∘ = ( − 1 ) k ∑ σ ∈ Σ p ( I ) s g n ( σ ) ∏ ℓ ∈ I G p ℓ − σ ℓ ∘ , \begin{equation}
\mathbf{C}_k^\circ = (-1)^k \sum_{\sigma \in \mathbf{\Sigma}^p(\mathcal{I})}
\mathop{\mathrm{sgn}}(\sigma) \prod_{\ell \in \mathcal{I}} \mathbf{G}_{p\ell - \sigma\ell}^\circ, \tag*{(21)}
\end{equation} C k ∘ = ( − 1 ) k σ ∈ Σ p ( I ) ∑ sgn ( σ ) ℓ ∈ I ∏ G p ℓ − σ ℓ ∘ , (21) where Σ p ( I ) = { σ ∈ S y m ( I ) ∣ ∑ ℓ M ( p ℓ − σ ℓ ) = S p ( I ) = S p ( k ) } \mathbf{\Sigma}^p(\mathcal{I}) = \{\sigma \in \mathop{\mathrm{Sym}}(\mathcal{I}) \mid \sum_\ell M(p\ell -
\sigma\ell) = S^p(\mathcal{I}) = S^p(k)\} Σ p ( I ) = { σ ∈ Sym ( I ) ∣ ∑ ℓ M ( p ℓ − σ ℓ ) = S p ( I ) = S p ( k )}
Recall the total ordering O \mathcal{O} O Z ≥ 0 I 0 \mathbb{Z}_{\ge 0}^{I_0} Z ≥ 0 I 0 § 2.3 . This ordering compares the sum of the coordinates first, then follows a reverse lexicographic order with coordinates ordered (from smallest to largest) 1 , … , d 2 1, \dots, d_2 1 , … , d 2 − 1 , … , − d 1 -1, \dots, -d_1 − 1 , … , − d 1 Definition 2.15 for the complete formal definition. Extend O \mathcal{O} O Z p < A > \mathbb{Z}_p\left<A\right> Z p ⟨ A ⟩
Proposition 4.3. If p > 2 D p>2D p > 2 D k ∈ V p k \in V_p k ∈ V p C k ∘ ≢ 0 ( m o d p ) \mathbf{C}_k^\circ \not \equiv 0 \pmod{p} C k ∘ ≡ 0 ( mod p )
Proof. Since k ∈ V p k \in V_p k ∈ V p I ∈ ( Z k ) \mathcal{I}\in \binom{\mathbb{Z}}{k} I ∈ ( k Z ) C k ∘ \mathbf{C}_k^\circ C k ∘ (21) . Let S o l ( I , σ ) = ∏ ℓ ∈ I S o l ( p ℓ − σ ℓ ) \mathop{\mathrm{Sol}}(\mathcal{I}, \sigma) = \prod_{\ell \in \mathcal{I}} \mathop{\mathrm{Sol}}(p\ell - \sigma\ell) Sol ( I , σ ) = ∏ ℓ ∈ I Sol ( p ℓ − σ ℓ ) Lemma 2.16 ,
min ( t ℓ ) ∈ S o l ( I , σ ) ∑ ℓ ∈ I t ℓ = ∑ ℓ ∈ I t ℓ + , \min_{(\mathbf{t}_\ell) \in \mathop{\mathrm{Sol}}(\mathcal{I}, \sigma)}
\sum_{\ell \in \mathcal{I}} \mathbf{t}_\ell = \sum_{\ell \in \mathcal{I}} \mathbf{t}_\ell^+, ( t ℓ ) ∈ Sol ( I , σ ) min ℓ ∈ I ∑ t ℓ = ℓ ∈ I ∑ t ℓ + , where t ℓ + \mathbf{t}_\ell^+ t ℓ + p ℓ − σ ℓ p\ell - \sigma\ell p ℓ − σ ℓ Lemma 2.2 , and the minimum is under the ordering O \mathcal{O} O ( t ℓ + ) ℓ ∈ I (\mathbf{t}_\ell^+)_{\ell \in \mathcal{I}} ( t ℓ + ) ℓ ∈ I
By Proposition 2.11 , I \mathcal{I} I { − k 1 , … , k 2 } \{-k_1, \dots, k_2\} { − k 1 , … , k 2 } − d 1 ≤ − k 1 ≤ 0 ≤ k 2 ≤ d 2 -d_1 \le -k_1 \le 0 \le k_2 \le d_2 − d 1 ≤ − k 1 ≤ 0 ≤ k 2 ≤ d 2 Lemma 2.17 , there is a unique permutation τ \tau τ
T O ≔ ∑ ℓ ∈ I t + ( p ℓ − τ ℓ ) \mathbf{T}_{\mathcal{O}} \coloneqq \sum_{\ell \in \mathcal{I}} \mathbf{t}^+(p\ell - \tau\ell) T O : = ℓ ∈ I ∑ t + ( p ℓ − τ ℓ ) is minimal under O \mathcal{O} O τ ∈ Σ p ( I ) \tau \in \mathbf{\Sigma}^p(\mathcal{I}) τ ∈ Σ p ( I ) A T O A^{\mathbf{T}_{\mathcal{O}}} A T O C k ∘ \mathbf{C}_k^\circ C k ∘
D O ≔ ( − 1 ) k s g n ( τ ) ∏ ℓ ∈ I ∏ i ∈ I 0 β ( t + ( p ℓ − τ ℓ ) ) i . \begin{equation}
D_{\mathcal{O}} \coloneqq
(-1)^k\mathop{\mathrm{sgn}}(\tau) \prod_{\ell \in \mathcal{I}}
\prod_{i\in I_0} \beta_{(\mathbf{t}^+(p\ell - \tau\ell))_i}. \tag*{(22)}
\end{equation} D O : = ( − 1 ) k sgn ( τ ) ℓ ∈ I ∏ i ∈ I 0 ∏ β ( t + ( p ℓ − τ ℓ ) ) i . (22) For ease of notation, fix ℓ ∈ I \ell \in \mathcal{I} ℓ ∈ I t + ( p ℓ − τ ℓ ) = ( t i ) i \mathbf{t}^+(p\ell - \tau\ell) = (t_i)_i t + ( p ℓ − τ ℓ ) = ( t i ) i i = − d 1 i = -d_1 i = − d 1 ℓ < 0 \ell < 0 ℓ < 0 t i = ⌊ p ℓ − τ ℓ − d 1 ⌋ ≤ ⌊ ( p − 1 ) d 1 d 1 ⌋ = p − 1 t_i = \left\lfloor \frac{p\ell - \tau\ell}{-d_1}\right\rfloor
\le \left\lfloor \frac{(p-1)d_1}{d_1}\right\rfloor = p-1 t i = ⌊ − d 1 p ℓ − τ ℓ ⌋ ≤ ⌊ d 1 ( p − 1 ) d 1 ⌋ = p − 1 β t i \beta_{t_i} β t i p p p β m \beta_m β m x m x^m x m E p ( x ) E_p(x) E p ( x ) t i ≤ p − 1 t_i \le p-1 t i ≤ p − 1 i = d 2 i=d_2 i = d 2 ℓ > 0 \ell > 0 ℓ > 0 t i = 0 t_i = 0 t i = 0 t i = 1 t_i = 1 t i = 1 β 0 = β 1 = 1 \beta_0 = \beta_1 = 1 β 0 = β 1 = 1 D O D_\mathcal{O} D O p p p
Corollary 4.4. Suppose p > 2 D p > 2D p > 2 D k ∈ V p k \in V_p k ∈ V p z = ( z i ) i ∈ I 0 z = (z_i)_{i \in I_0} z = ( z i ) i ∈ I 0 Q ‾ p \overline{\mathbb{Q}}_p Q p v p ( C k ( z ) ) = S p ( k ) p − 1 ⟺ C k ∘ ( z ) ≢ 0 ( m o d m Q ‾ p ) , v_p(\mathbf{C}_k(z)) = \frac{S^p(k)}{p-1} \iff
\mathbf{C}_k^\circ(z) \not\equiv 0 \pmod{\mathfrak{m}_{\overline{\mathbb{Q}}_p}}, v p ( C k ( z )) = p − 1 S p ( k ) ⟺ C k ∘ ( z ) ≡ 0 ( mod m Q p ) , m Q ‾ p \mathfrak{m}_{\overline{\mathbb{Q}}_p} m Q p O Q ‾ p {\mathcal O}_{\overline{\mathbb{Q}}_p} O Q p
Proof. By Proposition 4.3 , C k ∘ ≢ 0 ( m o d p ) \mathbf{C}_k^\circ \not\equiv 0 \pmod{p} C k ∘ ≡ 0 ( mod p ) C k ∘ ≠ 0 \mathbf{C}_k^\circ\ne 0 C k ∘ = 0 Proposition 4.2 , it now follows that v p ( C k ) = ( p − 1 ) − 1 S p ( k ) v_p(\mathbf{C}_k) = (p-1)^{-1}S^p(k) v p ( C k ) = ( p − 1 ) − 1 S p ( k ) C k ∘ \mathbf{C}_k^\circ C k ∘
A p p p
When a particular Laurent polynomial f = ∑ i ∈ I 0 a i x i ∈ F q [ x ± 1 ] f = \sum_{i \in I_0} a_i x^i \in \mathbb{F}_q[x^{\pm1}] f = ∑ i ∈ I 0 a i x i ∈ F q [ x ± 1 ] a ^ = ( a ^ i ) i ∈ I 0 \widehat{a} = (\widehat{a}_i)_{i \in I_0} a = ( a i ) i ∈ I 0 A A A C k ( a ^ ) \mathbf{C}_k(\widehat{a}) C k ( a ) C k C_k C k s k s^k s k det ( 1 − G s ) \det(1 - \mathcal{G}s) det ( 1 − G s ) f f f
G = ( G p i − j ) i , j ∈ Z = ( G p i − j ( a ^ ) ) i , j ∈ Z . \mathcal{G}= (G_{pi-j})_{i,j \in \mathbb{Z}} = (\mathbf{G}_{pi-j}(\widehat{a}))_{i, j \in \mathbb{Z}}. G = ( G p i − j ) i , j ∈ Z = ( G p i − j ( a ) ) i , j ∈ Z . Therefore, by Corollary 4.4 , v p ( C k ) ≥ S p ( k ) p − 1 v_p(C_k) \ge \frac{S^p(k)}{p-1} v p ( C k ) ≥ p − 1 S p ( k ) C k ∘ ‾ ( a ) ≠ 0 \overline{\mathbf{C}_k^\circ}(a) \ne 0 C k ∘ ( a ) = 0
Recall the polygon G L ( d 1 , d 2 , p ) \mathop{\mathrm{GL}}(d_1,d_2,p) GL ( d 1 , d 2 , p ) Definition 2.5 . Consider the product
C p ( A ) = ∏ k ∈ V p C k ∘ ( A ) . \begin{equation}
\mathbf{C}^p(A) = \prod_{k \in V_p} \mathbf{C}_k^\circ(A). \tag*{(23)}
\end{equation} C p ( A ) = k ∈ V p ∏ C k ∘ ( A ) . (23) Since C k ∘ \mathbf{C}_k^\circ C k ∘ p p p k ∈ V p k\in V_p k ∈ V p C p ( A ) \mathbf{C}^p(A) C p ( A ) Q ∩ Z p \mathbb{Q}\cap \mathbb{Z}_p Q ∩ Z p C p ‾ ( a ) ≠ 0 \overline{\mathbf{C}^p}(a) \ne 0 C p ( a ) = 0 v p ( C k ) = S p ( k ) p − 1 v_p(C_k) = \frac{S^p(k)}{p-1} v p ( C k ) = p − 1 S p ( k ) k ∈ V p k \in V_p k ∈ V p § 5 , we will show that these are precisely the conditions needed to achieve the generic Newton polygon for L ∗ ( f ; s ) L^*(f; s) L ∗ ( f ; s )
In fact, we can already prove more about C p \mathbf{C}^p C p p p p C p \mathbf{C}^p C p p p p
Lemma 4.5. Let r ∈ ( Z / D Z ) × r \in (\mathbb{Z}/ D\mathbb{Z})^\times r ∈ ( Z / D Z ) × H ( r ) ( A ) ∈ Q [ A ] H^{(r)}(A) \in \mathbb{Q}[A] H ( r ) ( A ) ∈ Q [ A ] p > max { 2 D , d 1 2 , d 2 2 } p > \max\{2D, d_1^2, d_2^2\} p > max { 2 D , d 1 2 , d 2 2 } p ≡ r ( m o d D ) p \equiv r \pmod{D} p ≡ r ( mod D ) H ( r ) ∈ ( Z p ∩ Q ) [ A ] H^{(r)}\in(\mathbb{Z}_p\cap\mathbb{Q})[A] H ( r ) ∈ ( Z p ∩ Q ) [ A ] C p ( A ) ≡ U p ( A ) ⋅ H ( r ) ( A ) ( m o d p ) , \mathbf{C}^p(A) \equiv U_p(A) \cdot H^{(r)}(A) \pmod{p}, C p ( A ) ≡ U p ( A ) ⋅ H ( r ) ( A ) ( mod p ) , U p ( A ) = u p A − d 1 e p , − A d 2 e p , + , u p ∈ Z p × ∩ Q , e p , ± ∈ Z ≥ 0 . U_p(A) = u_p A_{-d_1}^{e_{p,-}} A_{d_2}^{e_{p,+}},
\qquad u_p \in \mathbb{Z}_p^\times \cap \mathbb{Q},
\quad e_{p,\pm} \in \mathbb{Z}_{\ge 0}. U p ( A ) = u p A − d 1 e p , − A d 2 e p , + , u p ∈ Z p × ∩ Q , e p , ± ∈ Z ≥ 0 .
Proof. We first record a uniform form of the minimal-representation expansion. Let ( z ) v = z ( z − 1 ) ⋯ ( z − v + 1 ) (z)_v = z(z-1)\cdots(z-v+1) ( z ) v = z ( z − 1 ) ⋯ ( z − v + 1 ) z z z v v v ( z ) 0 = 1 (z)_0 = 1 ( z ) 0 = 1
Let Δ \Delta Δ d d d 0 ≤ Δ < d 0 \le \Delta < d 0 ≤ Δ < d v v v n n n T v ( n ) \mathcal{T}_{v}(n) T v ( n ) u = ( u 1 , … , u v ) ∈ Z ≥ 0 v \mathbf{u} = (u_1, \dots, u_v) \in \mathbb{Z}_{\ge 0}^v u = ( u 1 , … , u v ) ∈ Z ≥ 0 v ∑ i = 1 v u i i = n \sum_{i=1}^v u_i i = n ∑ i = 1 v u i i = n T 0 ( 0 ) = { ( ) } \mathcal{T}_0(0) = \{()\} T 0 ( 0 ) = {( )} T 0 ( n ) = ∅ \mathcal{T}_0(n) = \varnothing T 0 ( n ) = ∅ n n n
P d , Δ ( z ; B 1 , … , B d ) ≔ ∑ u ∈ T d − 1 ( Δ ) ( z ) ∣ u ∣ ∏ i = 1 d − 1 u d − i ! B d d − 1 − ∣ u ∣ ∏ i = 1 d − 1 B i u d − i . P_{d, \Delta}(z; B_1, \dots, B_d)
\coloneqq
\sum_{\mathbf{u} \in \mathcal{T}_{d-1}(\Delta)}
\frac{(z)_{|\mathbf{u}|}}{\prod_{i=1}^{d-1}u_{d-i}!}
B_d^{d-1-|\mathbf{u}|}\prod_{i=1}^{d-1} B_i^{u_{d-i}}. P d , Δ ( z ; B 1 , … , B d ) : = u ∈ T d − 1 ( Δ ) ∑ ∏ i = 1 d − 1 u d − i ! ( z ) ∣ u ∣ B d d − 1 − ∣ u ∣ i = 1 ∏ d − 1 B i u d − i . Because 0 ≤ Δ < d 0 \le \Delta < d 0 ≤ Δ < d P d , Δ P_{d, \Delta} P d , Δ d − 1 d-1 d − 1 B 1 , … , B d B_1, \dots, B_d B 1 , … , B d
Choose N N N d − 1 ≤ M ( N ) < p d-1 \le M(N) < p d − 1 ≤ M ( N ) < p d = d 1 d = d_1 d = d 1 N < 0 N<0 N < 0 d = d 2 d = d_2 d = d 2 N > 0 N>0 N > 0 t = ( t i ) ∈ S o l ( N ) \mathbf{t}= (t_i) \in \mathop{\mathrm{Sol}}(N) t = ( t i ) ∈ Sol ( N ) β t i \beta_{t_i} β t i s t i s^{t_i} s t i E p ( s ) E_p(s) E p ( s ) ( t i ) ! − 1 (t_i)!^{-1} ( t i ) ! − 1 Lemma 2.3 with (18) , we obtain
G N ∘ ( A ) = A ε d M ( N ) − d + 1 M ( N ) ! P d , Δ N ( M ( N ) ; A ε 1 , … , A ε d ) , \begin{equation}
\mathbf{G}_N^\circ(A) =
\frac{A_{\varepsilon d}^{M(N)-d+1}}{M(N)!}
P_{d, \Delta_N}(M(N); A_{\varepsilon 1}, \dots, A_{\varepsilon d}), \tag*{(24)}
\end{equation} G N ∘ ( A ) = M ( N )! A ε d M ( N ) − d + 1 P d , Δ N ( M ( N ) ; A ε 1 , … , A ε d ) , (24) where Δ N = d M ( N ) − ∣ N ∣ \Delta_N = dM(N) - |N| Δ N = d M ( N ) − ∣ N ∣ ε = s g n ( N ) \varepsilon = \mathop{\mathrm{sgn}}(N) ε = sgn ( N )
Let r ∈ ( Z / D Z ) × r \in (\mathbb{Z}/ D\mathbb{Z})^\times r ∈ ( Z / D Z ) × i ∈ { 1 , 2 } i\in\{1,2\} i ∈ { 1 , 2 } r i r_i r i r r r ( Z / d i Z ) × (\mathbb{Z}/ d_i\mathbb{Z})^\times ( Z / d i Z ) × p > max { 2 D , d 1 2 , d 2 2 } p > \max\{2D, d_1^2, d_2^2\} p > max { 2 D , d 1 2 , d 2 2 } r r r D D D Corollary 2.12 , the set V p V_p V p r r r I = { − k 1 , … , k 2 } \mathcal{I}= \{-k_1, \dots, k_2\} I = { − k 1 , … , k 2 } k ∈ V p k \in V_p k ∈ V p V p V_p V p V r V_r V r k k k I = { − k 1 , … , k 2 } \mathcal{I}= \{-k_1, \dots, k_2\} I = { − k 1 , … , k 2 } ℓ ∈ I \ell \in \mathcal{I} ℓ ∈ I
(\varepsilon_\ell, \delta_\ell, \kappa_\ell, \rho_\ell) =
\begin{cases*}
(\phantom{-}1, d_2, k_2, r_2) & if $\ell > 0$ \\
(-1, d_1, k_1, r_1) & if $\ell < 0$.
\end{cases*}
Now set
e ℓ = [ κ ℓ − ρ ℓ ⋅ ∣ ℓ ∣ ] δ ℓ , m ℓ = e ℓ + p ∣ ℓ ∣ − κ ℓ δ ℓ , c ℓ = e ℓ − κ ℓ δ ℓ . e_\ell = [\kappa_\ell - \rho_\ell\cdot|\ell|]_{\delta_\ell}, \quad
m_\ell = \frac{e_\ell + p|\ell|-\kappa_\ell}{\delta_\ell}, \quad
c_\ell = \frac{e_\ell - \kappa_\ell}{\delta_\ell}. e ℓ = [ κ ℓ − ρ ℓ ⋅ ∣ ℓ ∣ ] δ ℓ , m ℓ = δ ℓ e ℓ + p ∣ ℓ ∣ − κ ℓ , c ℓ = δ ℓ e ℓ − κ ℓ . Note that e ℓ e_\ell e ℓ m ℓ m_\ell m ℓ c ℓ ∈ Z p ∩ Q c_\ell \in \mathbb{Z}_p \cap \mathbb{Q} c ℓ ∈ Z p ∩ Q m ℓ ≡ c ℓ ( m o d p ) m_\ell \equiv c_\ell \pmod{p} m ℓ ≡ c ℓ ( mod p ) p p p ∣ ℓ ∣ ≤ κ ℓ ≤ δ ℓ |\ell| \le \kappa_\ell \le \delta_\ell ∣ ℓ ∣ ≤ κ ℓ ≤ δ ℓ δ ℓ − 1 ≤ m ℓ < p \delta_\ell-1 \le m_\ell < p δ ℓ − 1 ≤ m ℓ < p
Note that, as argued in Remark 2.7 , the set Σ p ( I ) \mathbf{\Sigma}^p(\mathcal{I}) Σ p ( I ) r r r Σ r ( I ) \mathbf{\Sigma}^r(\mathcal{I}) Σ r ( I ) Σ p ( I ) \mathbf{\Sigma}^p(\mathcal{I}) Σ p ( I ) σ ∈ Σ r ( I ) \sigma \in \mathbf{\Sigma}^r(\mathcal{I}) σ ∈ Σ r ( I ) Proposition 2.6 , M ( p ℓ − σ ℓ ) = ⌈ p ∣ ℓ ∣ − κ ℓ δ ℓ ⌉ = m ℓ M(p\ell - \sigma\ell) =
\left\lceil \frac{p|\ell| - \kappa_\ell}{\delta_\ell}
\right\rceil = m_\ell M ( p ℓ − σ ℓ ) = ⌈ δ ℓ p ∣ ℓ ∣ − κ ℓ ⌉ = m ℓ
Δ ℓ , σ = δ ℓ m ℓ − ∣ p ℓ − σ ℓ ∣ = e ℓ − κ ℓ + ε ℓ σ ℓ ∈ { 0 , … , δ ℓ − 1 } . \Delta_{\ell, \sigma} = \delta_\ell m_\ell - |p\ell - \sigma\ell|
= e_\ell - \kappa_\ell + \varepsilon_\ell
\sigma\ell \in \{0, \dots, \delta_\ell-1\}. Δ ℓ , σ = δ ℓ m ℓ − ∣ p ℓ − σ ℓ ∣ = e ℓ − κ ℓ + ε ℓ σ ℓ ∈ { 0 , … , δ ℓ − 1 } . Notice that Δ ℓ , σ \Delta_{\ell, \sigma} Δ ℓ , σ p p p Δ p ℓ − σ ℓ = Δ ℓ , σ \Delta_{p\ell - \sigma\ell} = \Delta_{\ell, \sigma} Δ p ℓ − σ ℓ = Δ ℓ , σ
Since δ ℓ − 1 ≤ m ℓ < p \delta_\ell-1 \le m_\ell < p δ ℓ − 1 ≤ m ℓ < p (24) to (20) and reduce modulo p p p
C k ∘ ( A ) ≡ U p , k ( A ) H k ( r ) ( A ) ( m o d p ) , \mathbf{C}_k^\circ(A) \equiv U_{p, k}(A) H_k^{(r)}(A) \pmod{p}, C k ∘ ( A ) ≡ U p , k ( A ) H k ( r ) ( A ) ( mod p ) , where
U p , k ( A ) = ∏ ℓ ∈ I ∖ { 0 } A ε ℓ δ ℓ m ℓ − δ ℓ + 1 m ℓ ! U_{p, k}(A) = \prod_{\ell \in \mathcal{I}\setminus \{0\}}
\frac{A_{\varepsilon_\ell \delta_\ell}^{m_\ell - \delta_\ell + 1}}
{m_\ell!} U p , k ( A ) = ℓ ∈ I ∖ { 0 } ∏ m ℓ ! A ε ℓ δ ℓ m ℓ − δ ℓ + 1 and
H k ( r ) ( A ) = ( − 1 ) k ∑ σ ∈ Σ r ( I ) s g n ( σ ) ∏ ℓ ∈ I ∖ { 0 } P δ ℓ , Δ ℓ , σ ( c ℓ , A ε ℓ 1 , … , A ε ℓ δ ℓ ) . H_k^{(r)}(A) =
(-1)^k \sum_{\sigma \in \mathbf{\Sigma}^{r}(\mathcal{I})}
\mathop{\mathrm{sgn}}(\sigma) \prod_{\ell \in \mathcal{I}\setminus\{0\}}
P_{\delta_\ell, \Delta_{\ell, \sigma}}(c_\ell, A_{\varepsilon_\ell 1},
\dots, A_{\varepsilon_\ell \delta_\ell}). H k ( r ) ( A ) = ( − 1 ) k σ ∈ Σ r ( I ) ∑ sgn ( σ ) ℓ ∈ I ∖ { 0 } ∏ P δ ℓ , Δ ℓ , σ ( c ℓ , A ε ℓ 1 , … , A ε ℓ δ ℓ ) . We see that H k ( r ) ( A ) H_k^{(r)}(A) H k ( r ) ( A ) p p p k 1 ( d 1 − 1 ) + k 2 ( d 2 − 1 ) k_1(d_1-1) + k_2(d_2-1) k 1 ( d 1 − 1 ) + k 2 ( d 2 − 1 ) p p p U p , k ( A ) U_{p, k}(A) U p , k ( A ) A − d 1 A_{-d_1} A − d 1 A d 2 A_{d_2} A d 2 p p p U p ( A ) = ∏ k ∈ V r U p , k ( A ) U_p(A) = \prod_{k \in V_r} U_{p, k}(A) U p ( A ) = ∏ k ∈ V r U p , k ( A ) H ( r ) ( A ) = ∏ k ∈ V r H k ( r ) ( A ) H^{(r)}(A) = \prod_{k \in V_r} H_k^{(r)}(A) H ( r ) ( A ) = ∏ k ∈ V r H k ( r ) ( A )
C p ( A ) ≡ U p ( A ) ⋅ H ( r ) ( A ) ( m o d p ) . \mathbf{C}^p(A) \equiv U_p(A) \cdot H^{(r)}(A) \pmod{p}. C p ( A ) ≡ U p ( A ) ⋅ H ( r ) ( A ) ( mod p ) . Note that this congruence itself implies that H ( r ) H^{(r)} H ( r ) C p \mathbf{C}^p C p p p p Corollary 4.4 . Therefore the lemma is proven. ◻
Example 4.7. We shall compute H ( r ) = H 1 ( r ) H 3 ( r ) H 5 ( r ) H^{(r)} = H^{(r)}_1 H^{(r)}_3 H^{(r)}_5 H ( r ) = H 1 ( r ) H 3 ( r ) H 5 ( r ) Remark 4.6 . For k = 1 k=1 k = 1 I = { 0 } \mathcal{I}= \{0\} I = { 0 } H 1 ( r ) = − 1 H_1^{(r)} = -1 H 1 ( r ) = − 1
For k = 3 k=3 k = 3 I = { − 1 , 0 , 1 } \mathcal{I}= \{-1, 0, 1\} I = { − 1 , 0 , 1 } c − 1 = c 1 = 1 3 c_{-1} = c_1 = \frac{1}{3} c − 1 = c 1 = 3 1 σ 1 = i d \sigma_1 = \mathop{\mathrm{id}} σ 1 = id σ 2 = ( − 1 1 ) \sigma_2 = (-1\;\;1) σ 2 = ( − 1 1 ) Δ − 1 , σ 1 = Δ 1 , σ 1 = 2 \Delta_{-1, \sigma_1} = \Delta_{1, \sigma_1} = 2 Δ − 1 , σ 1 = Δ 1 , σ 1 = 2 Δ − 1 , σ 2 = Δ 1 , σ 2 = 0 \Delta_{-1, \sigma_2} = \Delta_{1, \sigma_2} = 0 Δ − 1 , σ 2 = Δ 1 , σ 2 = 0
H 3 ( r ) = − P 3 , 2 ( 1 3 , A − 1 , A − 2 , A − 3 ) P 3 , 2 ( 1 3 , A 1 , A 2 , A 3 ) + P 3 , 0 ( 1 3 , A − 1 , A − 2 , A − 3 ) P 3 , 0 ( 1 3 , A 1 , A 2 , A 3 ) = − ( − 1 9 A − 2 2 + 1 3 A − 1 A − 3 ) ( − 1 9 A 2 2 + 1 3 A 1 A 3 ) + A − 3 2 A 3 2 . \begin{align*}
H_3^{(r)} &= -P_{3,2}(\frac{1}{3},A_{-1},A_{-2},A_{-3})
P_{3,2}(\frac{1}{3},A_1,A_2,A_3) \\
&\quad +
P_{3,0}(\frac{1}{3}, A_{-1},A_{-2},A_{-3})
P_{3,0}(\frac{1}{3}, A_1,A_2,A_3) \\
&= -\left(-\frac{1}{9}A_{-2}^2 + \frac{1}{3}A_{-1}A_{-3}\right)
\left(-\frac{1}{9}A_2^2 + \frac{1}{3}A_{1}A_{3}\right) \\
&\quad + A_{-3}^2 A_3^2.
\end{align*} H 3 ( r ) = − P 3 , 2 ( 3 1 , A − 1 , A − 2 , A − 3 ) P 3 , 2 ( 3 1 , A 1 , A 2 , A 3 ) + P 3 , 0 ( 3 1 , A − 1 , A − 2 , A − 3 ) P 3 , 0 ( 3 1 , A 1 , A 2 , A 3 ) = − ( − 9 1 A − 2 2 + 3 1 A − 1 A − 3 ) ( − 9 1 A 2 2 + 3 1 A 1 A 3 ) + A − 3 2 A 3 2 . For k = 5 k=5 k = 5 σ = ( 1 2 ) ( − 1 − 2 ) \sigma = (1\;\;2)(-1\;\;-2) σ = ( 1 2 ) ( − 1 − 2 ) Δ ℓ , σ = 0 \Delta_{\ell, \sigma} = 0 Δ ℓ , σ = 0 ℓ ∈ I ∖ { 0 } \ell \in \mathcal{I}\setminus\{0\} ℓ ∈ I ∖ { 0 } c c c P 3 , 0 ( c , B 1 , B 2 , B 3 ) = B 3 2 P_{3, 0}(c, B_1, B_2, B_3) = B_3^2 P 3 , 0 ( c , B 1 , B 2 , B 3 ) = B 3 2
H 5 ( r ) = − A − 3 4 A 3 4 . H_5^{(r)} = -A_{-3}^4 A_3^4. H 5 ( r ) = − A − 3 4 A 3 4 . This yields
H ( r ) = 1 81 A − 3 4 A 3 4 ( 81 A − 3 2 A 3 2 − ( 3 A − 1 A − 3 − A − 2 2 ) ( 3 A 1 A 3 − A 2 2 ) ) . H^{(r)} = \frac{1}{81} A_{-3}^4A_3^4
\left(81 A_{-3}^2 A_3^2 -
\left(3A_{-1}A_{-3}-A_{-2}^2\right)(3A_1A_3 - A_2^2)\right). H ( r ) = 81 1 A − 3 4 A 3 4 ( 81 A − 3 2 A 3 2 − ( 3 A − 1 A − 3 − A − 2 2 ) ( 3 A 1 A 3 − A 2 2 ) ) . Genericity of the Fredholm determinant for field extensions
In this brief section, we prove a proposition that effectively reduces the generic condition to the base field case.
Recall the value S p ( k ) S^p(k) S p ( k ) § 2.2 . Also recall the combinatorial cost polygon G L ( d 1 , d 2 , p ) \mathop{\mathrm{GL}}(d_1,d_2,p) GL ( d 1 , d 2 , p ) Definition 2.5 . When p > 2 D p > 2D p > 2 D Corollary 2.13 that k ∈ V p k \in V_p k ∈ V p S p ( I ) = S p ( k ) S^p(\mathcal{I}) = S^p(k) S p ( I ) = S p ( k ) I ∈ ( Z k ) \mathcal{I}\in \binom{\mathbb{Z}}{k} I ∈ ( k Z )
We argued in § 4.4 that for sufficiently large p p p G L ( d 1 , d 2 , p ) \mathop{\mathrm{GL}}(d_1,d_2,p) GL ( d 1 , d 2 , p ) det ( 1 − G s ) \det(1 - \mathcal{G}s) det ( 1 − G s ) § 3.1 . We wish to establish G L ( d 1 , d 2 , p ) \mathop{\mathrm{GL}}(d_1,d_2,p) GL ( d 1 , d 2 , p ) L L L L ∗ ( f ; s ) L^*(f; s) L ∗ ( f ; s ) (14) connects the Newton polygon of L ∗ ( f ; s ) L^*(f; s) L ∗ ( f ; s ) [ G , τ , b ] = G τ b − 1 ⋯ G τ G [\mathcal{G}, \tau, b] = \mathcal{G}^{\tau^{b-1}} \cdots \mathcal{G}^\tau \mathcal{G} [ G , τ , b ] = G τ b − 1 ⋯ G τ G (12) , where τ \tau τ [ G , τ , b ] [\mathcal{G}, \tau, b] [ G , τ , b ] G \mathcal{G} G G L ( d 1 , d 2 , p ) \mathop{\mathrm{GL}}(d_1,d_2,p) GL ( d 1 , d 2 , p )
Proposition 5.1. Assume p > 2 D p > 2D p > 2 D q = p b q = p^b q = p b τ \tau τ F q \mathbb{F}_q F q Q q ( ζ p ) \mathbb{Q}_q(\zeta_p) Q q ( ζ p ) ζ p \zeta_p ζ p a = ( a i ) i ∈ I 0 ∈ F q I 0 a = (a_i)_{i \in I_0} \in \mathbb{F}_q^{I_0} a = ( a i ) i ∈ I 0 ∈ F q I 0 I 0 = { − d 1 , … , d 2 } ∖ { 0 } I_0 = \{-d_1, \dots, d_2\} \setminus \{0\} I 0 = { − d 1 , … , d 2 } ∖ { 0 } a − d 1 a d 2 ≠ 0 a_{-d_1}a_{d_2} \ne 0 a − d 1 a d 2 = 0 N ∈ Z N \in \mathbb{Z} N ∈ Z G N = G N ( a ^ ) G_N = {\mathbf{G}}_N(\widehat{a}) G N = G N ( a ) G = ( G p i − j ) i , j ∈ Z \mathcal{G}= (G_{pi-j})_{i,j \in \mathbb{Z}} G = ( G p i − j ) i , j ∈ Z [ G , τ , b ] = G τ b − 1 ⋯ G τ G [\mathcal{G}, \tau, b] = \mathcal{G}^{\tau^{b-1}} \cdots \mathcal{G}^\tau \mathcal{G} [ G , τ , b ] = G τ b − 1 ⋯ G τ G
Then N P q ( det ( 1 − [ G , τ , b ] s ) ) ∣ [ 0 , d 1 + d 2 ] ≥ G L ( d 1 , d 2 , p ) . \mathop{\mathrm{NP}}_q(\det(1 - [\mathcal{G}, \tau, b]s))\big|_{[0,d_1+d_2]} \ge
\mathop{\mathrm{GL}}(d_1,d_2,p). NP q ( det ( 1 − [ G , τ , b ] s )) [ 0 , d 1 + d 2 ] ≥ GL ( d 1 , d 2 , p ) . v p ( C k ) = S p ( k ) p − 1 v_p(C_k) = \frac{S^p(k)}{p-1} v p ( C k ) = p − 1 S p ( k ) k ∈ V p k \in V_p k ∈ V p
Proof. A Cauchy-Binet–type argument produces the following result.
Lemma 5.2 (Cauchy-Binet). For every m ∈ Z / b Z m \in \mathbb{Z}/ b\mathbb{Z} m ∈ Z / b Z A m A_m A m { e i } i ∈ Z \{e_i\}_{i \in \mathbb{Z}} { e i } i ∈ Z P = A b − 1 A b − 2 ⋯ A 0 P = A_{b-1}A_{b-2}\cdots A_0 P = A b − 1 A b − 2 ⋯ A 0 det ( 1 − P s ) = 1 + P 1 s + P 2 s 2 + … . \det(1 - Ps) = 1 + P_1s + P_2s^2 + \dots. det ( 1 − P s ) = 1 + P 1 s + P 2 s 2 + … . k ≥ 1 k \ge 1 k ≥ 1 P k = ( − 1 ) k ∑ I 0 , … , I b − 1 ∈ ( Z k ) ∏ m ∈ Z / b Z det ( ( A m ) I m , I m − 1 ) , P_k = (-1)^k \sum_{\mathcal{I}_0, \dots, \mathcal{I}_{b-1} \in \binom{\mathbb{Z}}{k}}
\prod_{m \in \mathbb{Z}/ b\mathbb{Z}} \det((A_m)_{\mathcal{I}_m, \mathcal{I}_{m-1}}), P k = ( − 1 ) k ∑ I 0 , … , I b − 1 ∈ ( k Z ) ∏ m ∈ Z / b Z det (( A m ) I m , I m − 1 ) , ( A m ) I m , I m − 1 (A_m)_{\mathcal{I}_m, \mathcal{I}_{m-1}} ( A m ) I m , I m − 1 A m A_m A m I m \mathcal{I}_m I m I m − 1 \mathcal{I}_{m-1} I m − 1
The proof involves a routine argument using finite truncations, leveraging the convergence of the Fredholm expansion of the product P P P
Assume 1 ≤ k ≤ d 1 + d 2 1 \le k \le d_1+d_2 1 ≤ k ≤ d 1 + d 2 D k D_k D k s k s^k s k det ( 1 − [ G , τ , b ] s ) \det(1-[\mathcal{G}, \tau, b]s) det ( 1 − [ G , τ , b ] s ) Lemma 5.2 , we have that
v p ( D k ) ≥ min I 0 , … , I b − 1 ∑ m ∈ Z / b Z v p ( det ( G I m , I m − 1 τ m ) ) , v_p(D_k) \ge \min_{\mathcal{I}_0, \dots, \mathcal{I}_{b-1}}
\sum_{m \in \mathbb{Z}/ b\mathbb{Z}} v_p(\det(\mathcal{G}^{\tau^m}_{\mathcal{I}_m, \mathcal{I}_{m-1}})), v p ( D k ) ≥ I 0 , … , I b − 1 min m ∈ Z / b Z ∑ v p ( det ( G I m , I m − 1 τ m )) , with equality if there is a unique tuple ( I m ) m ∈ Z / b Z (\mathcal{I}_m)_{m \in \mathbb{Z}/b\mathbb{Z}} ( I m ) m ∈ Z / b Z v p ( det ( N ) ) = v p ( det ( N τ i ) ) v_p(\det(N)) = v_p(\det(N^{\tau^{i}})) v p ( det ( N )) = v p ( det ( N τ i )) N N N v p v_p v p k × k k \times k k × k G \mathcal{G} G I , J ∈ ( Z k ) \mathcal{I}, \mathcal{J} \in \binom{\mathbb{Z}}{k} I , J ∈ ( k Z ) Δ [ I , J ] = v p ( det ( G I , J ) ) \Delta[\mathcal{I}, \mathcal{J}] = v_p(\det(\mathcal{G}_{\mathcal{I}, \mathcal{J}})) Δ [ I , J ] = v p ( det ( G I , J ))
Choose any tuple ( I m ) ∈ ( Z k ) Z / b Z (\mathcal{I}_m) \in \binom{\mathbb{Z}}{k}^{\mathbb{Z}/ b\mathbb{Z}} ( I m ) ∈ ( k Z ) Z / b Z
det ( G I m , I m − 1 ) = ∑ σ s g n ( σ ) ∏ ℓ ∈ I m G p ℓ − σ ℓ , \det(\mathcal{G}_{\mathcal{I}_m, \mathcal{I}_{m-1}}) =
\sum_{\sigma}\mathop{\mathrm{sgn}}(\sigma) \prod_{\ell \in \mathcal{I}_m}
G_{p\ell - \sigma\ell}, det ( G I m , I m − 1 ) = σ ∑ sgn ( σ ) ℓ ∈ I m ∏ G p ℓ − σ ℓ , where ∑ σ \sum_{\sigma} ∑ σ σ : I m → I m − 1 \sigma : \mathcal{I}_m \to \mathcal{I}_{m-1} σ : I m → I m − 1 Proposition 4.1 that
Δ [ I m , I m − 1 ] ≥ min σ ∑ ℓ ∈ I m M ( p ℓ − σ ℓ ) p − 1 = S p ( I m , I m − 1 ) p − 1 , \begin{equation}
\Delta[\mathcal{I}_m, \mathcal{I}_{m-1}]
\ge \min_\sigma \sum_{\ell \in \mathcal{I}_{m}}
\frac{M(p\ell-\sigma\ell)}{p-1}
= \frac{S^p(\mathcal{I}_{m}, \mathcal{I}_{m-1})}{p-1}, \tag*{(25)}
\end{equation} Δ [ I m , I m − 1 ] ≥ σ min ℓ ∈ I m ∑ p − 1 M ( p ℓ − σ ℓ ) = p − 1 S p ( I m , I m − 1 ) , (25) where S p ( I m , I m − 1 ) S^p(\mathcal{I}_m, \mathcal{I}_{m-1}) S p ( I m , I m − 1 ) ∑ ℓ ∈ I m M ( p ℓ − σ ℓ ) \sum_{\ell \in \mathcal{I}_m} M(p\ell - \sigma\ell) ∑ ℓ ∈ I m M ( p ℓ − σ ℓ ) σ \sigma σ I m → I m − 1 \mathcal{I}_m \to \mathcal{I}_{m-1} I m → I m − 1 Proposition 2.14 that v p ( D k ) ≥ ( b S p ( k ) ) / ( p − 1 ) v_p(D_k) \ge (bS^p(k))/(p-1) v p ( D k ) ≥ ( b S p ( k )) / ( p − 1 )
v q ( D k ) ≥ S p ( k ) p − 1 . v_q(D_k)
\ge \frac{S^p(k)}{p-1}. v q ( D k ) ≥ p − 1 S p ( k ) . These coefficient bounds suffice to establish the claimed inequality for the full Newton polygon — that is, N P q ( det ( 1 − [ G , τ , b ] s ) ) ∣ [ 0 , d 1 + d 2 ] ≥ G L ( d 1 , d 2 , p ) \mathop{\mathrm{NP}}_q(\det(1 - [\mathcal{G}, \tau, b]s))|_{[0,d_1+d_2]} \ge \mathop{\mathrm{GL}}(d_1,d_2,p) NP q ( det ( 1 − [ G , τ , b ] s )) ∣ [ 0 , d 1 + d 2 ] ≥ GL ( d 1 , d 2 , p ) Proposition 3.2 and (14) , the Newton polygon of det ( 1 − [ G , τ , b ] s ) \det(1-[\mathcal{G},\tau,b]s) det ( 1 − [ G , τ , b ] s ) [ 0 , d 1 + d 2 ] [0,d_1+d_2] [ 0 , d 1 + d 2 ] L ∗ ( f ; s ) L^*(f;s) L ∗ ( f ; s ) [ 0 , 1 ] [0,1] [ 0 , 1 ] ( d 1 + d 2 , ( d 1 + d 2 ) / 2 ) (d_1+d_2,(d_1+d_2)/2) ( d 1 + d 2 , ( d 1 + d 2 ) /2 ) G L ( d 1 , d 2 , p ) \mathop{\mathrm{GL}}(d_1,d_2,p) GL ( d 1 , d 2 , p ) 1 1 1 Corollary 2.13 . Thus, coefficients of degree greater than d 1 + d 2 d_1+d_2 d 1 + d 2 G L ( d 1 , d 2 , p ) \mathop{\mathrm{GL}}(d_1,d_2,p) GL ( d 1 , d 2 , p )
Let k ∈ V p k \in V_p k ∈ V p Corollary 2.13 , there must be a unique I ∈ ( Z k ) \mathcal{I}\in \binom{\mathbb{Z}}{k} I ∈ ( k Z ) S p ( I ) = S p ( k ) S^p(\mathcal{I}) = S^p(k) S p ( I ) = S p ( k ) (25) and Proposition 2.14 ,
∑ m ∈ Z / b Z Δ [ I m , I m − 1 ] > b S p ( k ) p − 1 \sum_{m \in \mathbb{Z}/ b\mathbb{Z}} \Delta[\mathcal{I}_m, \mathcal{I}_{m-1}]
> \frac{bS^p(k)}{p-1} m ∈ Z / b Z ∑ Δ [ I m , I m − 1 ] > p − 1 b S p ( k ) if I m ≠ I \mathcal{I}_m \ne \mathcal{I} I m = I m ∈ Z / b Z m \in \mathbb{Z}/ b\mathbb{Z} m ∈ Z / b Z
Δ [ I ′ , I ′ ] > S p ( k ) p − 1 \Delta[\mathcal{I}', \mathcal{I}'] > \frac{S^p(k)}{p-1} Δ [ I ′ , I ′ ] > p − 1 S p ( k ) if I ′ ≠ I \mathcal{I}' \ne \mathcal{I} I ′ = I
v_p(D_k) \ge \min_{\mathcal{I}_0, \dots, \mathcal{I}_{b-1}}
\sum_m \Delta[\mathcal{I}_m, \mathcal{I}_{m-1}]
\quad \mbox{and} \quad
v_p(C_k) \ge \min_{\mathcal{I}'} \Delta[\mathcal{I}', \mathcal{I}']
with equalities if the minima are achieved uniquely, we have
v q ( D k ) = S p ( k ) p − 1 ⟺ Δ [ I , I ] = S p ( k ) p − 1 ⟺ v p ( C k ) = S p ( k ) p − 1 . v_q(D_k) = \frac{S^p(k)}{p-1} \iff \Delta[\mathcal{I}, \mathcal{I}] = \frac{S^p(k)}{p-1}
\iff v_p(C_k) = \frac{S^p(k)}{p-1}. v q ( D k ) = p − 1 S p ( k ) ⟺ Δ [ I , I ] = p − 1 S p ( k ) ⟺ v p ( C k ) = p − 1 S p ( k ) . If
N P q ( det ( 1 − [ G , τ , b ] s ) ) ∣ [ 0 , d 1 + d 2 ] = G L ( d 1 , d 2 , p ) , \mathop{\mathrm{NP}}_q(\det(1 - [\mathcal{G}, \tau, b]s))|_{[0, d_1+d_2]} = \mathop{\mathrm{GL}}(d_1,d_2,p), NP q ( det ( 1 − [ G , τ , b ] s )) ∣ [ 0 , d 1 + d 2 ] = GL ( d 1 , d 2 , p ) , then v q ( D k ) = S p ( k ) p − 1 v_q(D_k) = \frac{S^p(k)}{p-1} v q ( D k ) = p − 1 S p ( k ) v p ( C k ) = S p ( k ) p − 1 v_p(C_k) = \frac{S^p(k)}{p-1} v p ( C k ) = p − 1 S p ( k ) k ∈ V p k \in V_p k ∈ V p v p ( C k ) = S p ( k ) p − 1 v_p(C_k) = \frac{S^p(k)}{p-1} v p ( C k ) = p − 1 S p ( k ) k ∈ V p k \in V_p k ∈ V p N P q ( det ( 1 − [ G , τ , b ] s ) ) \mathop{\mathrm{NP}}_q(\det(1 - [\mathcal{G}, \tau, b]s)) NP q ( det ( 1 − [ G , τ , b ] s )) G L ( d 1 , d 2 , p ) \mathop{\mathrm{GL}}(d_1,d_2,p) GL ( d 1 , d 2 , p ) k ∈ V p k \in V_p k ∈ V p k = 0 k=0 k = 0 ( 0 , 0 ) (0,0) ( 0 , 0 ) k = d 1 + d 2 k=d_1+d_2 k = d 1 + d 2 S p ( d 1 + d 2 ) = ( p − 1 ) ( d 1 + d 2 ) 2 S^p(d_1 + d_2) = \frac{(p-1)(d_1+d_2)}{2} S p ( d 1 + d 2 ) = 2 ( p − 1 ) ( d 1 + d 2 ) I = { − d 1 , … , d 2 − 1 } \mathcal{I}= \{-d_1, \dots, d_2-1\} I = { − d 1 , … , d 2 − 1 } σ \sigma σ I \mathcal{I} I p ℓ ≡ σ ℓ ( m o d d 1 ) p\ell \equiv \sigma\ell \pmod{d_1} p ℓ ≡ σ ℓ ( mod d 1 ) ℓ < 0 \ell < 0 ℓ < 0 p ℓ ≡ σ ℓ ( m o d d 2 ) p\ell \equiv \sigma\ell \pmod{d_2} p ℓ ≡ σ ℓ ( mod d 2 ) ℓ > 0 \ell > 0 ℓ > 0 σ 0 = 0 \sigma 0 = 0 σ 0 = 0 p p p d 1 d 2 d_1d_2 d 1 d 2 σ \sigma σ
S p ( k ) ≤ S p ( I ) ≤ p − 1 d 1 ( 1 + ⋯ + d 1 ) + p − 1 d 2 ( 1 + ⋯ + d 2 − 1 ) = ( p − 1 ) ( d 1 + d 2 ) 2 . S^p(k) \le S^p(\mathcal{I}) \le \frac{p-1}{d_1}(1 + \dots + d_1)
+ \frac{p-1}{d_2}(1 + \dots + d_2-1) = \frac{(p-1)(d_1+d_2)}{2}. S p ( k ) ≤ S p ( I ) ≤ d 1 p − 1 ( 1 + ⋯ + d 1 ) + d 2 p − 1 ( 1 + ⋯ + d 2 − 1 ) = 2 ( p − 1 ) ( d 1 + d 2 ) . This is also a lower bound for S p ( k ) S^p(k) S p ( k ) W d 1 + d 2 = w ( I ) = d 1 + d 2 2 W_{d_1+d_2} = w(\mathcal{I}) = \frac{d_1 + d_2}{2} W d 1 + d 2 = w ( I ) = 2 d 1 + d 2 S p ( k ) ≥ ( p − 1 ) W d 1 + d 2 S^p(k) \ge (p-1)W_{d_1+d_2} S p ( k ) ≥ ( p − 1 ) W d 1 + d 2 Lemma 2.9 (5). Thus G L ( d 1 , d 2 , p ) ( d 1 + d 2 ) = d 1 + d 2 2 \mathop{\mathrm{GL}}(d_1,d_2,p)(d_1+d_2) = \frac{d_1+d_2}{2} GL ( d 1 , d 2 , p ) ( d 1 + d 2 ) = 2 d 1 + d 2
On the other hand, by geometry, the L L L L ∗ ( f ; s ) L^*(f; s) L ∗ ( f ; s ) ( d 1 + d 2 , d 1 + d 2 2 ) \left(d_1+d_2, \frac{d_1+d_2}{2}\right) ( d 1 + d 2 , 2 d 1 + d 2 ) q q q d 1 + d 2 d_1+d_2 d 1 + d 2 λ \lambda λ 1 − λ 1-\lambda 1 − λ (14) , the same endpoint lies on N P q ( det ( 1 − [ G , τ , b ] s ) ) ∣ [ 0 , d 1 + d 2 ] \mathop{\mathrm{NP}}_q(\det(1 - [\mathcal{G}, \tau, b]s))|_{[0,d_1+d_2]} NP q ( det ( 1 − [ G , τ , b ] s )) ∣ [ 0 , d 1 + d 2 ]
Main results
Here, we combine the results of S:ck_formula,S:extension_theorem to establish the generic condition for the L L L
§ 5 established the condition for genericity for the Newton polygon of the Fredholm determinant det ( 1 − [ G , τ , b ] s ) \det(1 - [\mathcal{G}, \tau, b]s) det ( 1 − [ G , τ , b ] s ) [ 0 , d 1 + d 2 ] [0, d_1+d_2] [ 0 , d 1 + d 2 ] [ G , τ , b ] [\mathcal{G}, \tau, b] [ G , τ , b ] f f f (14) , Proposition 5.1 also establishes the generic condition for L ∗ ( f ; s ) L^*(f; s) L ∗ ( f ; s ) L ∗ ( f ; s ) L^*(f; s) L ∗ ( f ; s ) C k C_k C k det ( 1 − G s ) = 1 + C 1 s + … \det(1 - \mathcal{G}s) = 1 + C_1s + \dots det ( 1 − G s ) = 1 + C 1 s + … G L ( d 1 , d 2 , p ) \mathop{\mathrm{GL}}(d_1,d_2,p) GL ( d 1 , d 2 , p )
Proof of (?)
Recall the expression C p \mathbf{C}^p C p (23) . This was proven to be a nonzero homogeneous polynomial in ( Z p ∩ Q ) [ A ] (\mathbb{Z}_p \cap \mathbb{Q})[A] ( Z p ∩ Q ) [ A ] C p \mathbf{C}^p C p L ∗ ( f ; s ) L^*(f; s) L ∗ ( f ; s )
Proposition 6.1. Let f = ∑ i ∈ I 0 a i x i ∈ F q [ x ± 1 ] f = \sum_{i \in I_0} a_i x^i \in \mathbb{F}_{q}[x^{\pm1}] f = ∑ i ∈ I 0 a i x i ∈ F q [ x ± 1 ] a − d 1 a d 2 ≠ 0. a_{-d_1} a_{d_2} \ne 0. a − d 1 a d 2 = 0. p > 2 D p > 2D p > 2 D C p ‾ \overline{\mathbf{C}^p} C p C p \mathbf{C}^p C p F ‾ p \overline{\mathbb{F}}_p F p N P q ( L ∗ ( f ; s ) ) ≥ G L ( d 1 , d 2 , p ) , \mathop{\mathrm{NP}}_q(L^*(f; s)) \ge \mathop{\mathrm{GL}}(d_1,d_2,p), NP q ( L ∗ ( f ; s )) ≥ GL ( d 1 , d 2 , p ) , C p ‾ ( a ) ≠ 0 \overline{\mathbf{C}^p}(a) \ne 0 C p ( a ) = 0
Proof. By combining (14) , and Proposition 5.1 , we immediately obtain
N P q ( L ∗ ( f ; s ) ) ≥ G L ( d 1 , d 2 , p ) , \mathop{\mathrm{NP}}_q(L^*(f; s)) \ge \mathop{\mathrm{GL}}(d_1,d_2,p), NP q ( L ∗ ( f ; s )) ≥ GL ( d 1 , d 2 , p ) , with equality if and only if v p ( C k ) = S p ( k ) p − 1 v_p(C_k) = \frac{S^p(k)}{p-1} v p ( C k ) = p − 1 S p ( k ) k ∈ V p k \in V_p k ∈ V p C p ‾ ( a ) ≠ 0 \overline{\mathbf{C}^p}(a) \ne 0 C p ( a ) = 0 k ∈ V p k \in V_p k ∈ V p C k C_k C k s k s^k s k det ( 1 − G s ) \det(1 - \mathcal{G}s) det ( 1 − G s ) C k ( a ^ ) \mathbf{C}_k(\widehat{a}) C k ( a ) a ^ \widehat{a} a a i a_i a i Corollary 4.4 that
v p ( C k ) = S p ( k ) p − 1 ⟺ C k ∘ ( a ^ ) ≢ 0 ( m o d m Q ‾ p ) ⟺ C k ∘ ‾ ( a ) ≠ 0. v_p(C_k) = \frac{S^p(k)}{p-1}
\iff
\mathbf{C}_k^\circ(\widehat{a}) \not\equiv 0 \pmod{\mathfrak{m}_{\overline{\mathbb{Q}}_p}}
\iff
\overline{\mathbf{C}_k^\circ}(a) \ne 0. v p ( C k ) = p − 1 S p ( k ) ⟺ C k ∘ ( a ) ≡ 0 ( mod m Q p ) ⟺ C k ∘ ( a ) = 0. Thus v p ( C k ) = S p ( k ) p − 1 v_p(C_k) = \frac{S^p(k)}{p-1} v p ( C k ) = p − 1 S p ( k ) k ∈ V p k \in V_p k ∈ V p C p ‾ ( a ) = ∏ k ∈ V p C k ∘ ‾ ( a ) ≠ 0 \overline{\mathbf{C}^p}(a) = \prod_{k \in V_p}
\overline{\mathbf{C}_k^\circ}(a) \ne 0 C p ( a ) = ∏ k ∈ V p C k ∘ ( a ) = 0
We now proceed to proving our main results.
Theorem 6.2. For any a ∈ A Q I ( Q ‾ ) a \in \mathbb{A}^I_\mathbb{Q}(\overline{\mathbb{Q}}) a ∈ A Q I ( Q ) f a f_a f a ∑ i ∈ I a i x i \sum_{i \in I} a_i x^i ∑ i ∈ I a i x i a a a U ⊆ A Q I \mathcal{U} \subseteq \mathbb{A}^I_{\mathbb{Q}} U ⊆ A Q I Q \mathbb{Q} Q a ∈ A Q I ( Q ‾ ) a \in \mathbb{A}^I_\mathbb{Q}(\overline{\mathbb{Q}}) a ∈ A Q I ( Q ) a − d 1 a d 2 ≠ 0 a_{-d_1}a_{d_2} \ne 0 a − d 1 a d 2 = 0
a ∈ U ( Q ‾ ) a \in \mathcal{U}(\overline{\mathbb{Q}}) a ∈ U ( Q )
There is a constant N a N_a N a p > N a p > N_a p > N a K / Q ( a i ∣ i ∈ I ) K / \mathbb{Q}(a_i \mid i \in I) K / Q ( a i ∣ i ∈ I ) p \mathfrak{p} p K K K p p p f ‾ a = f a ( m o d p ) \overline{f}_a = f_a \pmod{\mathfrak{p}} f a = f a ( mod p ) ℓ \ell ℓ [ k p : F p ] [k_\mathfrak{p}: \mathbb{F}_p] [ k p : F p ] χ ℓ : F p ℓ → Q ‾ p × \chi_\ell : \mathbb{F}_{p^\ell} \to \overline{\mathbb{Q}}_p^\times χ ℓ : F p ℓ → Q p × L L L L ∗ ( f ‾ a , χ ℓ ; s ) L^*(\overline{f}_a, \chi_\ell; s) L ∗ ( f a , χ ℓ ; s )
Proof. Let
H(A) = A_{-d_1}A_{d_2} \prod_{r \in (\mathbb{Z}/ D\mathbb{Z})^\times} H^{(r)}(A)
\quad \mbox{ and } \quad \mathcal{U} = D(H).
As defined, this is a polynomial in the variables indexed by I 0 I_0 I 0 I I I A 0 A_0 A 0 H H H Q \mathbb{Q} Q U \mathcal{U} U Q \mathbb{Q} Q
Let a ∈ U ( Q ‾ ) a \in \mathcal{U}(\overline{\mathbb{Q}}) a ∈ U ( Q ) H ( r ) ( a ) ≠ 0 H^{(r)}(a) \ne 0 H ( r ) ( a ) = 0 r r r E = Q ( a i ∣ i ∈ I ) E = \mathbb{Q}(a_i \mid i \in I) E = Q ( a i ∣ i ∈ I )
The set of rational primes below places of E E E a i a_i a i i ∈ I i \in I i ∈ I
The set of rational primes below places of E E E a − d 1 a_{-d_1} a − d 1 a d 2 a_{d_2} a d 2
The set of rational primes below places of E E E H ( r ) ( a ) H^{(r)}(a) H ( r ) ( a ) r ∈ ( Z / D Z ) × r \in (\mathbb{Z}/ D\mathbb{Z})^\times r ∈ ( Z / D Z ) ×
The set of rational primes p p p p ≤ max { 2 D , d 1 2 , d 2 2 } p \le \max \{ 2D, d_1^2, d_2^2\} p ≤ max { 2 D , d 1 2 , d 2 2 }
Let N a N_a N a p > N a p > N_a p > N a K / E K / E K / E p \mathfrak{p} p p p p f ‾ a ↦ f ‾ a − a ‾ 0 \overline{f}_a \mapsto \overline{f}_a - \overline{a}_0 f a ↦ f a − a 0 Remark 3.1 , we may assume that a ‾ 0 = 0 \overline{a}_0 = 0 a 0 = 0 H ( r ) ‾ ( a ‾ ) \overline{H^{(r)}}(\overline{a}) H ( r ) ( a ) k p k_\mathfrak{p} k p r ∈ ( Z / D Z ) × r \in (\mathbb{Z}/ D\mathbb{Z})^\times r ∈ ( Z / D Z ) × r r r p m o d D p\bmod D p mod D Lemma 4.5 gives that
C p ‾ ( a ‾ ) = U p ‾ ( a ‾ ) H ( r ) ‾ ( a ‾ ) ≠ 0. \overline{\mathbf{C}^p}(\overline{a}) =
\overline{U_p}(\overline{a}) \overline{H^{(r)}}(\overline{a}) \ne 0. C p ( a ) = U p ( a ) H ( r ) ( a ) = 0. We thus have by Proposition 6.1 that L ∗ ( f ‾ a ; s ) L^*(\overline{f}_a; s) L ∗ ( f a ; s )
Now, if χ ℓ : F p ℓ → Q ‾ p × \chi_\ell : \mathbb{F}_{p^\ell} \to \overline{\mathbb{Q}}_p^\times χ ℓ : F p ℓ → Q p × Proposition 3.4 , there is an α ∈ F p ℓ × \alpha \in \mathbb{F}_{p^\ell}^\times α ∈ F p ℓ × L ∗ ( f ‾ a , χ ℓ ; s ) = L ∗ ( α f ‾ a , s ) L^*(\overline{f}_a, \chi_\ell; s)
= L^*(\alpha \overline{f}_a, s) L ∗ ( f a , χ ℓ ; s ) = L ∗ ( α f a , s ) C p ‾ ( a ˉ ) ≠ 0 \overline{\mathbf{C}^p}(\bar{a}) \ne 0 C p ( a ˉ ) = 0 C p \mathbf{C}^p C p C p ‾ ( α a ‾ ) ≠ 0 \overline{\mathbf{C}^p}(\alpha \overline{a}) \ne 0 C p ( α a ) = 0 k p k_{\mathfrak{p}} k p Proposition 6.1 we conclude that L ∗ ( α f ‾ a , s ) L^*(\alpha \overline{f}_a, s) L ∗ ( α f a , s )
Conversely, suppose a ∉ U ( Q ‾ ) a \not\in \mathcal{U}(\overline{\mathbb{Q}}) a ∈ U ( Q ) a − d 1 a d 2 ≠ 0 a_{-d_1}a_{d_2} \ne 0 a − d 1 a d 2 = 0 a − d 1 a d 2 ≢ 0 ( m o d p ) a_{-d_1}a_{d_2} \not\equiv 0 \pmod{p} a − d 1 a d 2 ≡ 0 ( mod p ) p p p H ( r ) ( a ) = 0 H^{(r)}(a) = 0 H ( r ) ( a ) = 0 r ∈ ( Z / D Z ) × r \in (\mathbb{Z}/ D\mathbb{Z})^\times r ∈ ( Z / D Z ) × p p p p ′ p' p ′ p ′ p' p ′ p ′ \mathfrak{p}' p ′ p ′ p' p ′ H ( r ) ( a ) = 0 H^{(r)}(a) = 0 H ( r ) ( a ) = 0 Lemma 4.5 that C p ′ ‾ ( a ‾ ) = 0 \overline{\mathbf{C}^{p'}}(\overline{a}) = 0 C p ′ ( a ) = 0 Proposition 6.1 , the L L L L ∗ ( f ‾ a ; s ) L^*(\overline{f}_{a}; s) L ∗ ( f a ; s ) N a N_a N a
Theorem 6.3 ((?) ). With I I I f a f_a f a U \mathcal{U} U a − d 1 a d 2 ≠ 0 a_{-d_1}a_{d_2} \ne 0 a − d 1 a d 2 = 0
a ∈ U ( Q ‾ ) a \in \mathcal{U}(\overline{\mathbb{Q}}) a ∈ U ( Q )
There is a constant N a N_a N a p > N a p > N_a p > N a K / Q ( a i ∣ i ∈ I ) K / \mathbb{Q}(a_i \mid i \in I) K / Q ( a i ∣ i ∈ I ) p \mathfrak{p} p K K K p p p ℓ \ell ℓ [ k p : F p ] [k_\mathfrak{p}: \mathbb{F}_p] [ k p : F p ] f ‾ a = f a ( m o d p ) \overline{f}_a = f_a \pmod{\mathfrak{p}} f a = f a ( mod p ) X f ‾ a , ℓ X_{\overline{f}_a, \ell} X f a , ℓ
Proof. Let a ∈ U ( Q ‾ ) a \in \mathcal{U}(\overline{\mathbb{Q}}) a ∈ U ( Q ) N a N_a N a Theorem 6.2 holds. Choose p > N a p > N_a p > N a K / Q ( a i ∣ i ∈ I ) K / \mathbb{Q}(a_i \mid i \in I) K / Q ( a i ∣ i ∈ I ) p \mathfrak{p} p p p p ℓ \ell ℓ [ k p : F p ] [k_\mathfrak{p}: \mathbb{F}_p] [ k p : F p ] Theorem 6.2 implies that L ∗ ( f ‾ a ; s ) L^*(\overline{f}_a; s) L ∗ ( f a ; s ) C p \mathbf{C}^p C p p \mathfrak{p} p L ∗ ( λ f ‾ a ; s ) L^*(\lambda \overline{f}_a; s) L ∗ ( λ f a ; s ) λ ∈ F p ℓ × \lambda \in \mathbb{F}_{p^\ell}^\times λ ∈ F p ℓ × Proposition 3.5 , the numerator of the zeta function for X f ‾ a , ℓ X_{\overline{f}_a, \ell} X f a , ℓ
∏ λ ∈ F p ℓ × L ∗ ( λ f ‾ a ; s ) . \prod_{\lambda \in \mathbb{F}_{p^\ell}^\times}
L^*(\lambda \overline{f}_a; s). λ ∈ F p ℓ × ∏ L ∗ ( λ f a ; s ) . The Newton slopes of this numerator are equal to the multiset union of the slopes of each factor L ∗ ( λ f ‾ a ; s ) L^*(\lambda \overline{f}_a; s) L ∗ ( λ f a ; s ) X f ‾ a , ℓ X_{\overline{f}_a, \ell} X f a , ℓ L ∗ ( λ f ‾ a ; s ) L^*(\lambda \overline{f}_a; s) L ∗ ( λ f a ; s ) λ ∈ F p ℓ × \lambda \in \mathbb{F}_{p^\ell}^\times λ ∈ F p ℓ ×
Conversely, If a ∉ U ( Q ‾ ) a\notin\mathcal U(\overline{\mathbb Q}) a ∈ / U ( Q ) Theorem 6.2 produces arbitrarily large primes and reductions for which L ∗ ( f ‾ a ; s ) L^*(\overline f_a;s) L ∗ ( f a ; s ) ℓ = 1 \ell=1 ℓ = 1 X f ‾ a , 1 X_{\overline f_a,1} X f a , 1 N a N_a N a
Recall notations from § 3.3 . In Proposition 6.4 below, we establish an interpolation criterion under the hypothesis p ≥ 2 D p\ge2D p ≥ 2 D [19 ] ; its global consequence is Theorem 6.5 . For Δ = [ − d 1 , d 2 ] \Delta=[-d_1,d_2] Δ = [ − d 1 , d 2 ] p Δ p_\Delta p Δ [19 Definition 1.6] .
Proposition 6.4 (Interpolation criterion). Suppose p ≥ 2 D p\ge 2D p ≥ 2 D q = p b q=p^b q = p b h = d 1 + d 2 h=d_1+d_2 h = d 1 + d 2 H ( y ) ∈ F p [ y u ∣ u ∈ I ] H(y) \in \mathbb{F}_p[y_u \mid u \in I] H ( y ) ∈ F p [ y u ∣ u ∈ I ] f = ∑ i ∈ I a i x i ∈ F q [ x ± 1 ] f = \sum_{i \in I} a_i x^i \in \mathbb{F}_q[x^{\pm 1}] f = ∑ i ∈ I a i x i ∈ F q [ x ± 1 ] a − d 1 a d 2 ≠ 0 a_{-d_1}a_{d_2} \ne 0 a − d 1 a d 2 = 0
H ( a ) ≠ 0 H(a) \ne 0 H ( a ) = 0 a = ( a i ) i ∈ I a = (a_i)_{i \in I} a = ( a i ) i ∈ I
L ∗ ( f ; T , s ) ∣ T = π 1 L^*(f; T, s)|_{T=\pi_1} L ∗ ( f ; T , s ) ∣ T = π 1
C ∗ ( f ; T , s ) ∣ T = π 1 C^*(f; T, s)|_{T=\pi_1} C ∗ ( f ; T , s ) ∣ T = π 1
C ∗ ( f ; T , s ) ∣ T = π m C^*(f; T, s)|_{T=\pi_m} C ∗ ( f ; T , s ) ∣ T = π m m ≥ 1 m\ge 1 m ≥ 1
L ∗ ( f ; T , s ) ∣ T = π m L^*(f; T, s)|_{T=\pi_m} L ∗ ( f ; T , s ) ∣ T = π m m ≥ 1 m\ge 1 m ≥ 1
Here the specialized polygons are taken with respect to the π m \pi_m π m b p Δ b\,p_\Delta b p Δ C C C b p Δ ∣ [ 0 , p m − 1 h ] b\,p_\Delta|_{[0,p^{m-1}h]} b p Δ ∣ [ 0 , p m − 1 h ] L L L
Each of these equivalent conditions also implies N P T ( C ∗ ( f ; T , s ) ) = b p Δ , \mathop{\mathrm{NP}}_T(C^*(f;T,s))=b\,p_\Delta, NP T ( C ∗ ( f ; T , s )) = b p Δ , T T T C C C
Proof. First suppose p > 2 D p>2D p > 2 D [19 ] . For a positive integer d d d D D D
r d ( n ) = 1 + n d − ⌈ n d ⌉ , s d ( n ) = r d ( p n ) , g d ( n ) = ⌈ p n d ⌉ − ⌈ n d ⌉ . r_d(n)=1+\frac{n}{d}-\left\lceil\frac{n}{d}\right\rceil,
\qquad
s_d(n)=r_d(pn),
\qquad
g_d(n)=\left\lceil\frac{pn}{d}\right\rceil
-\left\lceil\frac{n}{d}\right\rceil. r d ( n ) = 1 + d n − ⌈ d n ⌉ , s d ( n ) = r d ( p n ) , g d ( n ) = ⌈ d p n ⌉ − ⌈ d n ⌉ . Thus r d ( n ) , s d ( n ) ∈ ( 0 , 1 ] r_d(n),s_d(n)\in(0,1] r d ( n ) , s d ( n ) ∈ ( 0 , 1 ]
⌈ p i − j d ⌉ = ⌈ p i d ⌉ − ⌈ j d ⌉ + 1 { r d ( j ) < s d ( i ) } . \left\lceil\frac{pi-j}{d}\right\rceil
=
\left\lceil\frac{pi}{d}\right\rceil
-\left\lceil\frac{j}{d}\right\rceil
+\mathbf{1}_{\{r_d(j)<s_d(i)\}}. ⌈ d p i − j ⌉ = ⌈ d p i ⌉ − ⌈ d j ⌉ + 1 { r d ( j ) < s d ( i )} . If A ⊆ Z ≥ 0 × { 1 , … , b } A\subseteq\mathbb{Z}_{\ge0}\times\{1,\ldots,b\} A ⊆ Z ≥ 0 × { 1 , … , b } n ( a ) n(a) n ( a ) a a a τ \tau τ A A A
Φ d ( A , τ ) = ∑ a ∈ A ( g d ( n ( a ) ) + 1 { r d ( n ( τ a ) ) < s d ( n ( a ) ) } ) . \Phi_d(A,\tau)=
\sum_{a\in A}
\left(g_d(n(a))+
\mathbf{1}_{\{r_d(n(\tau a))<s_d(n(a))\}}\right). Φ d ( A , τ ) = a ∈ A ∑ ( g d ( n ( a )) + 1 { r d ( n ( τ a )) < s d ( n ( a ))} ) . Suppose A A A n n n m < n m<n m < n n n n m m m τ \tau τ ( A ′ , τ ′ ) (A',\tau') ( A ′ , τ ′ )
Φ d ( A , τ ) ≥ Φ d ( A ′ , τ ′ ) + 1. \Phi_d(A,\tau)\ge \Phi_d(A',\tau')+1. Φ d ( A , τ ) ≥ Φ d ( A ′ , τ ′ ) + 1. Every adjacent increment of g d g_d g d 2 2 2 d ∤ k d\nmid k d ∤ k ⌊ p / d ⌋ ≥ 2 \lfloor p/d\rfloor\ge2 ⌊ p / d ⌋ ≥ 2 d ∣ k d\mid k d ∣ k ⌈ p / d ⌉ − 1 ≥ 2 \lceil p/d\rceil-1\ge2 ⌈ p / d ⌉ − 1 ≥ 2 g d ( n ) − g d ( m ) ≥ 3 g_d(n)-g_d(m)\ge3 g d ( n ) − g d ( m ) ≥ 3 2 2 2 g d ( n ) − g d ( m ) = 2 g_d(n)-g_d(m)=2 g d ( n ) − g d ( m ) = 2 n = m + 1 n=m+1 n = m + 1 d ∣ m d\mid m d ∣ m r d ( m ) = 1 ≥ r d ( n ) r_d(m)=1\ge r_d(n) r d ( m ) = 1 ≥ r d ( n ) d ∤ m d\nmid m d ∤ m ⌈ p n / d ⌉ − ⌈ p m / d ⌉ = 2 \lceil pn/d\rceil-\lceil pm/d\rceil=2 ⌈ p n / d ⌉ − ⌈ p m / d ⌉ = 2 s d ( n ) − s d ( m ) = p / d − 2 > 0 s_d(n)-s_d(m)=p/d-2>0 s d ( n ) − s d ( m ) = p / d − 2 > 0 1 1 1 g d g_d g d 2 2 2
Repeated replacements reduce A A A [19 Lemma 3.3] uses only ( p , d ) = 1 (p,d)=1 ( p , d ) = 1 p > 2 d p>2d p > 2 d b b b
ω d ( n ) = ⌈ ( p − 1 ) n d ⌉ − ϵ d ( n ) , n ≥ 1 , \omega_d(n)=
\left\lceil\frac{(p-1)n}{d}\right\rceil-\epsilon_d(n),
\qquad n\ge1, ω d ( n ) = ⌈ d ( p − 1 ) n ⌉ − ϵ d ( n ) , n ≥ 1 , where ϵ d ( n ) = 1 \epsilon_d(n)=1 ϵ d ( n ) = 1 1 ≤ u < [ n ] d 1\le u<[n]_d 1 ≤ u < [ n ] d p u ≡ n ( m o d d ) pu\equiv n\pmod d p u ≡ n ( mod d ) ϵ d ( n ) = 0 \epsilon_d(n)=0 ϵ d ( n ) = 0 p Δ p_\Delta p Δ 0 0 0 ω d 1 ( n ) , ω d 2 ( n ) \omega_{d_1}(n),\omega_{d_2}(n) ω d 1 ( n ) , ω d 2 ( n ) n ≥ 1 n\ge1 n ≥ 1
To pass to two signs, let A ⊆ Z × { 1 , … , b } A\subseteq\mathbb{Z}\times\{1,\ldots,b\} A ⊆ Z × { 1 , … , b } b k bk bk τ \tau τ τ 0 \tau_0 τ 0 i i i
⌈ p ∣ i ∣ − ∣ n ( τ 0 a ) ∣ d ( i ) ⌉ , d ( i ) = { d 1 , i < 0 , d 2 , i > 0. \left\lceil
\frac{p|i|-|n(\tau_0 a)|}{d(i)}
\right\rceil,
\qquad
d(i)=
\begin{cases}
d_1,&i<0,\\
d_2,&i>0.
\end{cases} ⌈ d ( i ) p ∣ i ∣ − ∣ n ( τ 0 a ) ∣ ⌉ , d ( i ) = { d 1 , d 2 , i < 0 , i > 0. Indeed, this follows from the corresponding signed ceiling when the original arrow preserves sign; otherwise the original cost is at least ⌈ p ∣ i ∣ / d ( i ) ⌉ \lceil p|i|/d(i)\rceil ⌈ p ∣ i ∣/ d ( i )⌉ b k bk bk b b b
∑ a ∈ A M ( p n ( a ) − n ( τ a ) ) ≥ b p Δ ( k ) . \sum_{a\in A}M\bigl(pn(a)-n(\tau a)\bigr)
\ge b\,p_\Delta(k). a ∈ A ∑ M ( p n ( a ) − n ( τ a ) ) ≥ b p Δ ( k ) . This holds for every k ≥ 1 k\ge1 k ≥ 1 T T T [19 Theorems 2.1–2.2 and the proof of Theorem 4.1] , therefore give
N P T ( C ∗ ( f ; T , s ) ) ≥ b p Δ . \begin{equation}
\mathop{\mathrm{NP}}_T(C^*(f;T,s))\ge b\,p_\Delta. \tag*{(26)}
\end{equation} NP T ( C ∗ ( f ; T , s )) ≥ b p Δ . (26) The slope calculation in the proof of Corollary 2.13 identifies the first h = d 1 + d 2 h=d_1+d_2 h = d 1 + d 2
p Δ ∣ [ 0 , h ] = ( p − 1 ) G L ( d 1 , d 2 , p ) . p_\Delta|_{[0,h]}=(p-1)\mathop{\mathrm{GL}}(d_1,d_2,p). p Δ ∣ [ 0 , h ] = ( p − 1 ) GL ( d 1 , d 2 , p ) . Moreover, ω d ( n + d ) = ω d ( n ) + ( p − 1 ) \omega_d(n+d)=\omega_d(n)+(p-1) ω d ( n + d ) = ω d ( n ) + ( p − 1 ) p Δ p_\Delta p Δ h h h j ( p − 1 ) j(p-1) j ( p − 1 ) j ≥ 0 j\ge0 j ≥ 0
A constant term does not affect any of the polygons in the statement. More precisely, for f 0 = f − a 0 f_0=f-a_0 f 0 = f − a 0 c = T r Q q / Q p ( a ^ 0 ) c=\mathop{\mathrm{Tr}}_{\mathbb{Q}_q/\mathbb{Q}_p}(\widehat a_0) c = Tr Q q / Q p ( a 0 ) f 0 f_0 f 0 f f f s s s ( 1 + T ) c s (1+T)^c s ( 1 + T ) c s C ∗ C^* C ∗ L ∗ L^* L ∗ a 0 = 0 a_0=0 a 0 = 0 Proposition 6.1 , we may take H = C p ‾ H=\overline{\mathbf{C}^p} H = C p a 0 a_0 a 0
H ( a ) ≠ 0 ⟺ N P π 1 ( L ∗ ( f ; T , s ) ∣ T = π 1 ) = b p Δ ∣ [ 0 , h ] . H(a)\ne0
\iff
\mathop{\mathrm{NP}}_{\pi_1}\bigl(L^*(f;T,s)|_{T=\pi_1}\bigr)
=b\,p_\Delta|_{[0,h]}. H ( a ) = 0 ⟺ NP π 1 ( L ∗ ( f ; T , s ) ∣ T = π 1 ) = b p Δ ∣ [ 0 , h ] . The product identity
C ∗ ( f ; π 1 , s ) = ∏ j ≥ 0 L ∗ ( f ; π 1 , q j s ) , C^*(f;\pi_1,s)=
\prod_{j\ge0}L^*(f;\pi_1,q^js), C ∗ ( f ; π 1 , s ) = j ≥ 0 ∏ L ∗ ( f ; π 1 , q j s ) , where specialization is indicated by writing π 1 \pi_1 π 1 T T T N P π 1 ( C ∗ ( f ; π 1 , s ) ) = b p Δ \mathop{\mathrm{NP}}_{\pi_1}(C^*(f;\pi_1,s))=b\,p_\Delta NP π 1 ( C ∗ ( f ; π 1 , s )) = b p Δ
Write C ∗ ( f ; T , s ) = ∑ i ≥ 0 c i ( T ) s i C^*(f;T,s)=\sum_{i\ge0}c_i(T)s^i C ∗ ( f ; T , s ) = ∑ i ≥ 0 c i ( T ) s i Z p [ [ T ] ] \mathbb{Z}_p[[T]] Z p [[ T ]]
N P π m ( C ∗ ( f ; π m , s ) ) ≥ N P T ( C ∗ ( f ; T , s ) ) ≥ b p Δ . \mathop{\mathrm{NP}}_{\pi_m}(C^*(f;\pi_m,s))
\ge \mathop{\mathrm{NP}}_T(C^*(f;T,s))\ge b\,p_\Delta. NP π m ( C ∗ ( f ; π m , s )) ≥ NP T ( C ∗ ( f ; T , s )) ≥ b p Δ . Equality at π 1 \pi_1 π 1 ( i , e ) (i,e) ( i , e ) c i ( T ) = T e u i ( T ) c_i(T)=T^e u_i(T) c i ( T ) = T e u i ( T ) u i ( T ) ∈ Z p [ [ T ] ] × u_i(T)\in\mathbb{Z}_p[[T]]^\times u i ( T ) ∈ Z p [[ T ] ] × T = π m T=\pi_m T = π m C C C T T T C C C π 1 \pi_1 π 1
Finally,
C ∗ ( f ; π m , s ) = ∏ j ≥ 0 L ∗ ( f ; π m , q j s ) , o r d π m ( q ) = b p m − 1 ( p − 1 ) . C^*(f;\pi_m,s)=
\prod_{j\ge0}L^*(f;\pi_m,q^js),
\qquad
\mathop{\mathrm{ord}}_{\pi_m}(q)=bp^{m-1}(p-1). C ∗ ( f ; π m , s ) = j ≥ 0 ∏ L ∗ ( f ; π m , q j s ) , ord π m ( q ) = b p m − 1 ( p − 1 ) . The specialized L L L p m − 1 h p^{m-1}h p m − 1 h [19 Lemma 6.3] , shows that the C C C b p Δ b\,p_\Delta b p Δ L L L [ 0 , p m − 1 h ] [0,p^{m-1}h] [ 0 , p m − 1 h ] C C C L L L C C C L L L b p Δ b\,p_\Delta b p Δ H ( a ) ≠ 0 H(a)\ne0 H ( a ) = 0 T T T p > 2 D p>2D p > 2 D
If p = 2 D p=2D p = 2 D p = 2 p=2 p = 2 d 1 = d 2 = D = 1 d_1=d_2=D=1 d 1 = d 2 = D = 1 H = 1 H=1 H = 1 L L L q − 1 q-1 q − 1 { 1 , − 1 } \{1,-1\} { 1 , − 1 } q q q q q q 0 , 1 0,1 0 , 1 D = 1 D=1 D = 1 T T T [20 ] is exactly b p Δ b\,p_\Delta b p Δ
Theorem 6.5 (Corollary 1.2 ). Let I I I f a f_a f a U \mathcal{U} U Theorem 6.2 , and suppose a − d 1 a d 2 ≠ 0 a_{-d_1}a_{d_2} \ne 0 a − d 1 a d 2 = 0
a ∈ U ( Q ‾ ) a \in \mathcal{U}(\overline{\mathbb{Q}}) a ∈ U ( Q )
There is a constant N a N_a N a p > N a p > N_a p > N a K / Q ( a i ∣ i ∈ I ) K / \mathbb{Q}(a_i \mid i \in I) K / Q ( a i ∣ i ∈ I ) p \mathfrak{p} p K K K p p p m ≥ 1 m \ge 1 m ≥ 1 C ∗ ( f ‾ a ; T , s ) C^*(\overline{f}_a; T, s) C ∗ ( f a ; T , s ) C ∗ ( f ‾ a ; T , s ) ∣ T = π m C^*(\overline{f}_a; T, s)|_{T = \pi_m} C ∗ ( f a ; T , s ) ∣ T = π m L ∗ ( f ‾ a ; T , s ) ∣ T = π m L^*(\overline{f}_a; T, s)|_{T = \pi_m} L ∗ ( f a ; T , s ) ∣ T = π m
Proof. Suppose a ∈ U ( Q ‾ ) a\in\mathcal{U}(\overline{\mathbb{Q}}) a ∈ U ( Q ) Theorem 6.2 , after enlarging N a N_a N a N a ≥ 2 D N_a\ge2D N a ≥ 2 D L L L L ∗ ( f ‾ a ; s ) L^*(\overline f_a;s) L ∗ ( f a ; s ) (15) and Proposition 6.4 then give genericity of all the specialized C C C L L L T T T C C C
Conversely, condition (2), with m = 1 m=1 m = 1 L ∗ ( f ‾ a ; s ) L^*(\overline f_a;s) L ∗ ( f a ; s ) (15) . The converse argument in the proof of Theorem 6.2 therefore gives a ∈ U ( Q ‾ ) a\in\mathcal{U}(\overline{\mathbb{Q}}) a ∈ U ( Q )
