Introduction
From abc to the height–conductor ratio
Let E / Q E/\mathbb{Q} E / Q Δ min \Delta_{\min} Δ m i n N N N ε > 0 \varepsilon> 0 ε > 0 C ( ε ) C(\varepsilon) C ( ε )
∣ Δ min ∣ ≤ C ( ε ) N 6 + ε . |\Delta_{\min}| \le C(\varepsilon)N^{6+\varepsilon}. ∣ Δ m i n ∣ ≤ C ( ε ) N 6 + ε . Via the Frey curve construction, Szpiro’s conjecture is equivalent to the abc conjecture (up to a constant factor in the exponent). On the other hand, for fixed g = 1 g=1 g = 1
κ ( E ) : = h F ( E ) log N \kappa(E) := \frac{h_{\mathrm{F}}(E)}{\log N} κ ( E ) := log N h F ( E ) as g → ∞ g \to\infty g → ∞
The question
Definition 1.1 (Tower). A curve tower over a number field K K K X n / K X_n/K X n / K n ≥ 1 n \ge1 n ≥ 1 g n : = g ( X n ) → ∞ g_n := g(X_n) \to\infty g n := g ( X n ) → ∞
(T2) there is a finite set S S S X n X_n X n S S S
(T3) writing A n : = Jac ( X n ) A_n := \operatorname{Jac}(X_n) A n := Jac ( X n ) L n : = log cond ( A n ) L_n := \log\operatorname{cond}(A_n) L n := log cond ( A n ) f ^ n : = L n / g n \hat{f}_n := L_n/g_n f ^ n := L n / g n f ^ n → ∞ \hat{f}_n \to\infty f ^ n → ∞
Such a tower is said to lie in the tower world .
Problem 1.2. For a tower { X n } \{X_n\} { X n }
κ n = h F ( A n ) L n , A n = Jac ( X n ) , L n = log cond ( A n ) ? \kappa_n = \frac{h_F(A_n)}{L_n}, \qquad A_n = \operatorname{Jac}(X_n), \qquad L_n = \log\operatorname{cond}(A_n)? κ n = L n h F ( A n ) , A n = Jac ( X n ) , L n = log cond ( A n )? Is there a tower-independent bound?
The point is that the question places Szpiro (g = 1 g=1 g = 1 g g g g = 1 g=1 g = 1
Overview of results
Table 1.
Notation and the master identity
Basic quantities
Basic quantities. Let X / K X/K X / K g ≥ 1 g \ge1 g ≥ 1 A = Jac ( X ) A = \operatorname{Jac}(X) A = Jac ( X ) [21 Cor. 1.2] :
12 h F ( A ) = ω ˉ 2 + Δ D e l + δ F a l . 12h_F(A) = \bar{\omega}^{2} + \Delta_{\mathrm{Del}} + \delta_{\mathrm{Fal}}. 12 h F ( A ) = ω ˉ 2 + Δ Del + δ Fal . where h F ( A ) h_F(A) h F ( A ) ω ˉ 2 = ω ˉ A / O K 2 \bar{\omega}^{2} = \bar{\omega}^{2}_{A/\mathcal{O}_K} ω ˉ 2 = ω ˉ A / O K 2 Δ D e l \Delta_{\mathrm{Del}} Δ Del δ F a l \delta_{\mathrm{Fal}} δ Fal
Remark 2.1 (Normalization). Another common convention is 12 h F = ω ˉ 2 + Δ D e l + δ F a l − 4 g log 2 π 12h_F = \bar{\omega}^{2}+\Delta_{\mathrm{Del}}+\delta_{\mathrm{Fal}}-4g\log2\pi 12 h F = ω ˉ 2 + Δ Del + δ Fal − 4 g log 2 π g g g
Definition 2.2 (Three dimensionless ratios).
κ : = h F ( A ) log cond ( A ) , σ : = Δ D e l log cond ( A ) , R : = ω ˉ 2 Δ D e l . \kappa:= \frac{h_F(A)}{\log\operatorname{cond}(A)}, \qquad\sigma:= \frac{\Delta_{\mathrm{Del}}}{\log\operatorname{cond}(A)}, \qquad R := \frac{\bar{\omega}^{2}}{\Delta_{\mathrm{Del}}}. κ := log cond ( A ) h F ( A ) , σ := log cond ( A ) Δ Del , R := Δ Del ω ˉ 2 . κ \kappa κ σ \sigma σ R ≤ 3 R \le3 R ≤ 3
The master identity
Proposition 2.3. For every A = Jac ( X ) A = \operatorname{Jac}(X) A = Jac ( X )
κ = σ ( 1 + R ) 12 + δ F a l 12 log cond ( A ) . (1) \kappa= \frac{\sigma(1+R)}{12} + \frac{\delta_{\mathrm{Fal}}}{12\log\operatorname{cond}(A)}. \tag*{(1)} κ = 12 σ ( 1 + R ) + 12 log cond ( A ) δ Fal . (1) Proof. Divide Wilms’ Noether formula by log cond \log\operatorname{cond} log cond ω ˉ 2 = R Δ D e l = R σ log cond \bar{\omega}^{2}=R\Delta_{\mathrm{Del}}=R\sigma\log\operatorname{cond} ω ˉ 2 = R Δ Del = R σ log cond
This identity is the hub of the paper: every assertion about κ \kappa κ σ \sigma σ R R R
Four a priori inequalities
Table 2.3.
Lemma 2.4 (Wild necessity). Let { X n } \{X_n\} { X n } ( T 1 ) (T1) ( T 1 ) ( T 3 ) (T3) ( T 3 ) S S S f p = dim ( V / V I p ) + Sw p f_p=\dim(V/V^{I_p})+\operatorname{Sw}_p f p = dim ( V / V I p ) + Sw p dim ( V / V I p ) ≤ 2 g n \dim(V/V^{I_p})\le2g_n dim ( V / V I p ) ≤ 2 g n
f ^ n ≤ 2 ∑ p ∈ S log p + ∑ p ∈ S Sw p log p g n . (2) \hat{f}_n \le2\sum_{p\in S}\log p+\frac{\sum_{p\in S}\operatorname{Sw}_p\log p}{g_n}. \tag*{(2)} f ^ n ≤ 2 p ∈ S ∑ log p + g n ∑ p ∈ S Sw p log p . (2) Consequently ( T 3 ) (T3) ( T 3 ) ∑ p ∈ S Sw p log p g n → ∞ \frac{\sum_{p\in S}\operatorname{Sw}_p\log p}{g_n}\to\infty g n ∑ p ∈ S Sw p l o g p → ∞ p ∈ S p\in S p ∈ S Sw p = 0 \operatorname{Sw}_p=0 Sw p = 0 ( T 3 ) (T3) ( T 3 ) f ^ ≤ ∑ p ∈ S log p = O ( 1 ) \hat{f}\le\sum_{p\in S}\log p=O(1) f ^ ≤ ∑ p ∈ S log p = O ( 1 )
Proof. f p = dim ( V / V I p ) + Sw p f_p=\dim(V/V^{I_p})+\operatorname{Sw}_p f p = dim ( V / V I p ) + Sw p dim V = 2 g n \dim V=2g_n dim V = 2 g n f ^ n \hat{f}_n f ^ n
Conversely, f p ≥ Sw p f_p\ge\operatorname{Sw}_p f p ≥ Sw p dim ( V / V I p ) ≥ 0 \dim(V/V^{I_p})\ge0 dim ( V / V I p ) ≥ 0 ∑ p ∈ S Sw p log p g n → ∞ \frac{\sum_{p\in S}\operatorname{Sw}_p\log p}{g_n}\to\infty g n ∑ p ∈ S Sw p l o g p → ∞ f ^ n → ∞ \hat{f}_n\to\infty f ^ n → ∞
an identity plus a trivial inequality, containing no conjecture. Scope: any genus, any residue characteristic, no semistability, no hyperellipticity required.
Citation note. Liu’s Prop. 1 appears in a genus-2 paper but is routinely cited as the standard result for arbitrary genus by later literature (Obus–Srinivasan [14 Prop. 1.3] , [15 Prop. 1.8] ). The attribution of Eq. (2) to [13 ] is withdrawn: that paper (L. Melnikas, on root numbers of genus 1 and 2 curves) does not contain the identity. We follow this common usage and flag it explicitly; an independent first-principles proof for arbitrary genus is listed as Problem 10.11.
Terminological warning. Throughout we use the Deligne–Saito discriminant, which is not the minimal Weierstrass discriminant of a hyperelliptic curve. The two agree for g = 1 g=1 g = 1 g ≥ 2 g \ge2 g ≥ 2 ν ( Δ min ) ≥ − Art = Δ D e l \nu(\Delta_{\min}) \ge-\operatorname{Art} = \Delta_{\mathrm{Del}} ν ( Δ m i n ) ≥ − Art = Δ Del g = 2 g=2 g = 2 [12 Théorème 1] gives ν ( Δ min ) = − Art + d d − 1 2 \nu(\Delta_{\min}) = -\operatorname{Art} + d\frac{d-1}{2} ν ( Δ m i n ) = − Art + d 2 d − 1 d d d [14 Thm. 1.1] (Mac Lane valuations) prove − Art ≤ ν ( Δ min ) -\operatorname{Art} \le\nu(\Delta_{\min}) − Art ≤ ν ( Δ m i n ) y n = f ( x ) y^n=f(x) y n = f ( x ) char ( k ) ∤ n \operatorname{char}(k) \nmid n char ( k ) ∤ n [15 ] . Consequently, estimates made via the valuation of the polynomial discriminant give only upper bounds for σ \sigma σ
Anchor calibration
Proposition 2.3 holds place by place on the following known towers (rational number comparisons, no floating point):
Table 2.4.
The anchors lie on the line κ = 1 + R 12 \kappa= \frac{1+R}{12} κ = 12 1 + R σ → 1 \sigma\to1 σ → 1 d → 0 d \to0 d → 0 R ≤ 5 3 R \le\frac{5}{3} R ≤ 3 5 σ ≤ 2 \sigma\le2 σ ≤ 2 X ( p n ) X(p^n) X ( p n ) κ = 5 12 > 1 3 \kappa= \frac{5}{12} > \frac{1}{3} κ = 12 5 > 3 1 σ → 1 \sigma\to1 σ → 1 κ = 5 24 \kappa= \frac{5}{24} κ = 24 5 R R R 1 2 \frac{1}{2} 2 1 § 5 and with κ F e r m a t = 1 8 \kappa_{\mathrm{Fermat}} = \frac{1}{8} κ Fermat = 8 1 σ = 1 \sigma= 1 σ = 1 δ F a l / log cond → 0 \delta_{\mathrm{Fal}}/\log\operatorname{cond} \to0 δ Fal / log cond → 0
The Fermat/CM tower: slope and second-order intercept
Throughout this section N = p n N=p^n N = p n F N : x N + y N = 1 F_N: x^N+y^N=1 F N : x N + y N = 1
g N = ( N − 1 ) ( N − 2 ) 2 , (3) g_N=\frac{(N-1)(N-2)}{2}, \tag*{(3)} g N = 2 ( N − 1 ) ( N − 2 ) , (3) and J ( F N ) J(F_N) J ( F N ) Q ( ζ N ) \mathbb{Q}(\zeta_N) Q ( ζ N )
The main closed form
Theorem 3.1 (Main chain). Let h ^ n : = h F ( J ( F p n ) ) / g p n \hat{h}_n := h_F(J(F_{p^n}))/g_{p^n} h ^ n := h F ( J ( F p n )) / g p n be the per-dimension stable height and A = 1.2824271291 … A=1.2824271291\ldots A = 1.2824271291 … the Glaisher–Kinkelin constant. Then
h ^ n = log p 4 n − 6 log A − p log p 4 ( p 2 − 1 ) + o ( 1 ) . (4) \hat{h}_n=\frac{\log p}{4}n-6\log A-\frac{p\log p}{4(p^2-1)}+o(1). \tag*{(4)} h ^ n = 4 log p n − 6 log A − 4 ( p 2 − 1 ) p log p + o ( 1 ) . (4) Derivation (five steps). 1. Colmez’ formula [4 ] : for a CM abelian variety,
h F ( A ) = ∑ Φ m ( Φ , χ ) ( − 1 2 L ′ L ( 0 , χ ) − 1 2 log 2 π ) . h_F(A)=\sum_{\Phi}m(\Phi,\chi)\left(-\frac{1}{2}\frac{L'}{L}(0,\chi)-\frac{1}{2}\log2\pi\right). h F ( A ) = Φ ∑ m ( Φ , χ ) ( − 2 1 L L ′ ( 0 , χ ) − 2 1 log 2 π ) . CM-type decomposition: J ( F p n ) J(F_{p^n}) J ( F p n ) Q ( ζ p j ) \mathbb{Q}(\zeta_{p^j}) Q ( ζ p j ) j = v p ( d 0 ) j=v_p(d_0) j = v p ( d 0 ) d 0 = N / gcd ( r , s , t , N ) d_0=N/\gcd(r,s,t,N) d 0 = N / g cd( r , s , t , N ) w χ = c v ∣ B 1 , χ ∣ 2 w_\chi=c_v|B_{1,\chi}|^2 w χ = c v ∣ B 1 , χ ∣ 2
c v = 3 p ( p v + 1 − 1 ) ( p − 1 ) 2 . c_v=\frac{3p(p^{v+1}-1)}{(p-1)^2}. c v = ( p − 1 ) 2 3 p ( p v + 1 − 1 ) . (verified numerically to 10 − 14 10^{-14} 1 0 − 14 L ′ / L ( 0 , χ ) L'/L(0,\chi) L ′ / L ( 0 , χ )
∏ k = 0 m − 1 Γ ( z + k m ) = ( 2 π ) m − 1 2 m 1 2 − m z Γ ( m z ) \prod_{k=0}^{m-1}\Gamma\left(z+\frac{k}{m}\right)=(2\pi)^{\frac{m-1}{2}}m^{\frac{1}{2}-mz}\Gamma(mz) k = 0 ∏ m − 1 Γ ( z + m k ) = ( 2 π ) 2 m − 1 m 2 1 − m z Γ ( m z ) collapses the Gamma products into closed form (residual < 4 × 10 − 12 <4\times10^{-12} < 4 × 1 0 − 12 Theorem 3.2 .
The exact order of the remainder
Let d p ( a ) d_p(a) d p ( a ) j j j p p p a a a
D p ( a ) = 1 2 d p ( a ) ( c 0 − c 0 d p ( a ) ) , D c ( a ) = c p c − 2 d p ( a ) 2 d p ( a ) − c − 1 c − 1 ( c ≥ 2 ) . D_p(a)=\frac{1}{2}d_p(a)\left(c_0-c_0^{d_p(a)}\right),\qquad D_c(a)=cp^{c-2d_p(a)}\frac{2d_p(a)-c-1}{c-1}\quad(c\ge2). D p ( a ) = 2 1 d p ( a ) ( c 0 − c 0 d p ( a ) ) , D c ( a ) = c p c − 2 d p ( a ) c − 1 2 d p ( a ) − c − 1 ( c ≥ 2 ) . These have mean zero over digits, so the ∑ t \sum_t ∑ t
Grouping by r ≡ a ( m o d p c ) r\equiv a\pmod{p^c} r ≡ a ( mod p c ) m = p n − c m=p^{n-c} m = p n − c
h ^ n g n = p − 1 2 ∑ c = 1 n c n − c [ c log p 2 p c + 1 S a ( c ) + R ( c ) p ] , c v = 3 p ( p v + 1 − 1 ) ( p − 1 ) 2 . (5) \widehat{h}_n g_n=\frac{p-1}{2}\sum_{c=1}^{n}c_{n-c}\left[\frac{c\log p}{2p^{c+1}}S_a(c)+\frac{R(c)}{p}\right],\qquad c_v=\frac{3p(p^{v+1}-1)}{(p-1)^2}. \tag*{(5)} h n g n = 2 p − 1 c = 1 ∑ n c n − c [ 2 p c + 1 c log p S a ( c ) + p R ( c ) ] , c v = ( p − 1 ) 2 3 p ( p v + 1 − 1 ) . (5) where S a ( c ) S_a(c) S a ( c )
R ( c ) = ∑ 1 ≤ r < p c p ∣ r D c ( r ) log Γ ( r p c ) = p c − 1 ( F ~ ( c ) − F ~ ( c − 1 ) ) R(c)=\sum_{\substack{1\le r<p^c\\p\mid r}}D_c(r)\log\Gamma\left(\frac{r}{p^c}\right)=p^{c-1}\left(\widetilde{F}(c)-\widetilde{F}(c-1)\right) R ( c ) = 1 ≤ r < p c p ∣ r ∑ D c ( r ) log Γ ( p c r ) = p c − 1 ( F ( c ) − F ( c − 1 ) ) is the Riemann-sum term. With w p ( x ) = 2 ⌊ p x ⌋ − p + 1 w_p(x)=2\lfloor px\rfloor-p+1 w p ( x ) = 2 ⌊ p x ⌋ − p + 1
F ~ ( c ) = ∑ r = 1 p c − 1 w p ( r p c ) log Γ ( r p c ) , F ~ ( c ) p c → I p : = ∫ 0 1 w p log Γ . \widetilde{F}(c)=\sum_{r=1}^{p^c-1}w_p\left(\frac{r}{p^c}\right)\log\Gamma\left(\frac{r}{p^c}\right),\qquad\frac{\widetilde{F}(c)}{p^c}\to I_p:=\int_0^1w_p\log\Gamma. F ( c ) = r = 1 ∑ p c − 1 w p ( p c r ) log Γ ( p c r ) , p c F ( c ) → I p := ∫ 0 1 w p log Γ. Theorem 3.2 (Exact order of the remainder). For p = 3 , 5 , 7 p=3,5,7 p = 3 , 5 , 7
h ^ n = log p 4 p n − 6 log A − p log p 4 ( p 2 − 1 ) + 9 4 ( log p ) n + b p p n + o ( p − n ) , \widehat{h}_n=\frac{\log p}{4}p^n-6\log A-\frac{p\log p}{4(p^2-1)}+\frac{\frac{9}{4}(\log p)n+b_p}{p^n}+o(p^{-n}), h n = 4 log p p n − 6 log A − 4 ( p 2 − 1 ) p log p + p n 4 9 ( log p ) n + b p + o ( p − n ) , where b p b_p b p p p p b 3 ≈ − 3.127 b_3\approx-3.127 b 3 ≈ − 3.127 b 5 ≈ − 3.176 b_5\approx-3.176 b 5 ≈ − 3.176 b 7 ≈ − 3.223 b_7\approx-3.223 b 7 ≈ − 3.223 O ( p − n ) O(p^{-n}) O ( p − n ) Θ ( n log p ⋅ p − n ) \Theta(n\log p\cdot p^{-n}) Θ ( n log p ⋅ p − n )
Proof sketch. Separate the singularity on [ 0 , 1 ] [0,1] [ 0 , 1 ] w p log Γ = p ( p − 1 ) log x + φ ( x ) w_p\log\Gamma=p(p-1)\log x+\varphi(x) w p log Γ = p ( p − 1 ) log x + φ ( x ) φ \varphi φ ∑ log r \sum\log r ∑ log r ∑ φ \sum\varphi ∑ φ ∑ log r \sum\log r ∑ log r
E c : = F ~ ( c ) − p c I p = p − 1 2 log ( 2 π p c ) + C 0 ( p ) + O ( p − c ) , E_c:=\widetilde{F}(c)-p^cI_p=\frac{p-1}{2}\log(2\pi p^c)+C_0(p)+O(p^{-c}), E c := F ( c ) − p c I p = 2 p − 1 log ( 2 π p c ) + C 0 ( p ) + O ( p − c ) , we get E c − E c − 1 = p − 1 2 log p + O ( p − c ) E_c-E_{c-1}=\frac{p-1}{2}\log p+O(p^{-c}) E c − E c − 1 = 2 p − 1 log p + O ( p − c )
Δ ( c ) : = R ( c ) − ( p − 1 ) p 2 c − 2 I p = p − 1 2 ( log p ) p c − 1 + O ( 1 ) . \Delta(c):=R(c)-(p-1)p^{2c-2}I_p=\frac{p-1}{2}(\log p)p^{c-1}+O(1). Δ ( c ) := R ( c ) − ( p − 1 ) p 2 c − 2 I p = 2 p − 1 ( log p ) p c − 1 + O ( 1 ) . Substitution into the master identity gives the Riemann part
∑ c c n − c p c − 1 ⋅ log p 2 = 3 log p 4 g n ( n p n − p n − 1 p − 1 ) ∼ 3 2 n log p p n ; \sum_c c_{n-c}p^{c-1}\cdot\frac{\log p}{2}=\frac{3\log p}{4g_n}\left(np^n-\frac{p^n-1}{p-1}\right)\sim\frac{3}{2}\frac{n\log p}{p^n}; c ∑ c n − c p c − 1 ⋅ 2 log p = 4 g n 3 log p ( n p n − p − 1 p n − 1 ) ∼ 2 3 p n n log p ; the remaining closed-form matching part is a pure algebraic truncation of the same order, 3 4 n log p p n \frac{3}{4}\frac{n\log p}{p^n} 4 3 p n n l o g p 3 2 + 3 4 = 9 4 \frac{3}{2}+\frac{3}{4}=\frac{9}{4} 2 3 + 4 3 = 4 9
Numerical check (slope of X ⋅ n X\cdot n X ⋅ n log p \log p log p
Table 3.
(theoretical value 2.25).
Corollary 3.3 (Attribution of the slope). View the master identity as an affine function of n n n h ^ n = log p 4 n + V n \hat{h}_n=\frac{\log p}{4}n+V_n h ^ n = 4 l o g p n + V n Numerical fitting gives the slope of V n V_n V n 10 − 11 10^{-11} 1 0 − 11 S a S_a S a V n V_n V n
c p = u 0 ( p ) + 3 p p 2 − 1 I p , c_p=u_0(p)+\frac{3p}{p^2-1}\mathcal{I}_p, c p = u 0 ( p ) + p 2 − 1 3 p I p , matching the ledger constant − 6 log A − p log p 4 ( p 2 − 1 ) -6\log A-\frac{p\log p}{4(p^2-1)} − 6 log A − 4 ( p 2 − 1 ) p l o g p 6.5 × 10 − 9 6.5\times10^{-9} 6.5 × 1 0 − 9 9.1 × 10 − 12 9.1\times10^{-12} 9.1 × 1 0 − 12 9.4 × 10 − 12 9.4\times10^{-12} 9.4 × 1 0 − 12 p = 3 , 5 , 7 p=3,5,7 p = 3 , 5 , 7
The origin of the Glaisher–Kinkelin constant is thereby located: it is not a product of Colmez' formula or the CM decomposition, but enters the second-order intercept from the limiting integral of the Riemann sum via the coefficient p p − 1 ⋅ 3 p p 2 − 1 \frac{p}{p-1}\cdot\frac{3p}{p^2-1} p − 1 p ⋅ p 2 − 1 3 p
Numerical check (main-chain residual):
Table 4.
Conductor density
Theorem 3.4.
log cond ( J ( F p n ) ) g p n log ( p n ) ⟶ c = 2. \frac{\log\operatorname{cond}(J(F_{p^n}))}{g_{p^n}\log(p^n)}\longrightarrow c=2. g p n log ( p n ) log cond ( J ( F p n )) ⟶ c = 2. Proof. A factor with CM by Q ( ζ p j ) \mathbb{Q}(\zeta_{p^j}) Q ( ζ p j ) ∣ d K ∣ 1 / 2 |d_K|^{1/2} ∣ d K ∣ 1/2 log ∣ d Q ( ζ p j ) ∣ = ( j p j − 1 ( p − 1 ) − p j − 1 ) log p \log|d_{\mathbb{Q}(\zeta_{p^j})}|=(jp^{j-1}(p-1)-p^j-1)\log p log ∣ d Q ( ζ p j ) ∣ = ( j p j − 1 ( p − 1 ) − p j − 1 ) log p d ~ j = φ ( p j ) / 2 \tilde{d}_j=\varphi(p^j)/2 d ~ j = φ ( p j ) /2
log cond ( A O ) dim = ( 2 j + 2 p − 1 ) log p . \frac{\log\operatorname{cond}(A_{\mathcal{O}})}{\dim}=\left(2j+\frac{2}{p-1}\right)\log p. dim log cond ( A O ) = ( 2 j + p − 1 2 ) log p . The top layer j = n j=n j = n h ^ n = h F / g n ∼ 1 4 log ( p n ) \hat{h}_n=h_F/g_n\sim\frac{1}{4}\log(p^n) h ^ n = h F / g n ∼ 4 1 log ( p n ) c = 2 c=2 c = 2
Corollary 3.5. κ F e r m a t = h ^ n log cond / g n = 1 4 log N 2 log N = 1 8 \kappa_{\mathrm{Fermat}}=\frac{\hat{h}_n}{\log\operatorname{cond}/g_n}=\frac{\frac{1}{4}\log N}{2\log N}=\frac{1}{8} κ Fermat = l o g cond / g n h ^ n = 2 l o g N 4 1 l o g N = 8 1
Three independent cross-checks: (a) Colmez height + Liu–Saito conductor; (b) additivity triangle back-substitution of axiom (III)(a) (σ = 1 \sigma=1 σ = 1 R = 1 2 R=\frac{1}{2} R = 2 1
Orbit stratification
Theorem 3.6. The height–conductor ratio on orbit j j j
κ O ( j ) = H ( j ) C ( j ) = 1 4 j log p + c p ( 2 j + 2 p − 1 ) log p , c p = − 6 log A − p log p 4 ( p 2 − 1 ) . \kappa_O(j)=\frac{H(j)}{C(j)}=\frac{\frac{1}{4}j\log p+c_p}{\left(2j+\frac{2}{p-1}\right)\log p},\qquad c_p=-6\log A-\frac{p\log p}{4(p^2-1)}. κ O ( j ) = C ( j ) H ( j ) = ( 2 j + p − 1 2 ) log p 4 1 j log p + c p , c p = − 6 log A − 4 ( p 2 − 1 ) p log p . It has four properties: (1) it depends only on j j j n n n κ O ( j ) < 0 \kappa_O(j)<0 κ O ( j ) < 0 p = 3 p=3 p = 3 j = 1 j=1 j = 1 − 0.227 -0.227 − 0.227
lim j → ∞ κ O ( j ) = 1 / 4 2 = 1 8 \lim_{j\to\infty}\kappa_O(j)=\frac{1/4}{2}=\frac{1}{8} j → ∞ lim κ O ( j ) = 2 1/4 = 8 1 from below.
Numerical (p = 3 p=3 p = 3 j = 1 , … , 6 j=1,\ldots,6 j = 1 , … , 6 − 0.227 -0.227 − 0.227 − 0.127 -0.127 − 0.127 − 0.075 -0.075 − 0.075 − 0.039 -0.039 − 0.039 − 0.013 -0.013 − 0.013 + 0.007 +0.007 + 0.007 j = 5 j=5 j = 5 j = 6 j=6 j = 6
Remark 3.7 (Robustness). If h ^ / dim ^ \hat h/\widehat{\dim} h ^ / dim C C C κ O ( j ) = 1 4 j log p + c p − C ( 2 j + 2 p − 1 ) log p → 1 8 \kappa_O(j)=\frac{\frac{1}{4}j\log p+c_p-C}{\left(2j+\frac{2}{p-1}\right)\log p}\to\frac{1}{8} κ O ( j ) = ( 2 j + p − 1 2 ) l o g p 4 1 j l o g p + c p − C → 8 1 j j j κ F e r m a t = 1 8 \kappa_{\mathrm{Fermat}}=\frac{1}{8} κ Fermat = 8 1 κ F e r m a t \kappa_{\mathrm{Fermat}} κ Fermat j = n j=n j = n
Counting closed forms
Theorem 3.8.
M n , j = ( p + 1 ) p j − 1 − 3 , ∣ Ω n ∣ = ( p + 1 ) p n − 1 p − 1 − 3 n . M_{n,j}=(p+1)p^{j-1}-3,\qquad|\Omega_n|=\frac{(p+1)p^n-1}{p-1}-3n. M n , j = ( p + 1 ) p j − 1 − 3 , ∣ Ω n ∣ = p − 1 ( p + 1 ) p n − 1 − 3 n . Table 5.
Two additional checks: M n , j M_{n,j} M n , j d ~ j \tilde d_j d ~ j
Remark 3.9 (Label). The layer–condensation identity
∑ χ o d d w χ i n t r = g n / 2 \sum_{\chi\ \mathrm{odd}}w_\chi^{\mathrm{intr}}=g_n/2 χ odd ∑ w χ intr = g n /2 holds numerically to 10 − 14 10^{-14} 1 0 − 14 10 − 14 10^{-14} 1 0 − 14
The one-third barrier for GL2-type towers
In this section we prove that for GL2-type towers (X 1 X_1 X 1 X 0 X_0 X 0 κ ≤ 1 3 \kappa\leq\frac{1}{3} κ ≤ 3 1
Two geometric inputs
Theorem 4.1 (Systole of X 1 X_1 X 1 p n ≥ 5 p^n\geq5 p n ≥ 5
sys ( X 1 ( p n ) ) = 2 arccosh ( p n − 2 2 ) ∼ 2 n log p ⟶ ∞ . \operatorname{sys}\left(X_1(p^n)\right)=2\operatorname{arccosh}\left(\frac{p^n-2}{2}\right)\sim2n\log p\longrightarrow\infty. sys ( X 1 ( p n ) ) = 2 arccosh ( 2 p n − 2 ) ∼ 2 n log p ⟶ ∞. For the remaining levels the formula is evaluated directly: p n = 2 , 3 , 4 p^n=2,3,4 p n = 2 , 3 , 4 2 arccosh ( 0 ) = 2\operatorname{arccosh}(0)= 2 arccosh ( 0 ) =
Proof. Four steps. 1. ( 1 − p n 1 − p n 1 ) ∈ Γ 1 ( p n ) ‾ \begin{pmatrix} 1-p^n & 1 \\ -p^n & 1 \end{pmatrix}\in\overline{\Gamma_1(p^n)} ( 1 − p n − p n 1 1 ) ∈ Γ 1 ( p n ) det = 1 \det=1 det = 1 tr = 2 − p n \operatorname{tr}=2-p^n tr = 2 − p n γ ∈ Γ 1 ( p n ) \gamma\in\Gamma_1(p^n) γ ∈ Γ 1 ( p n ) a ≡ d ≡ 1 a\equiv d\equiv1 a ≡ d ≡ 1 c ≡ 0 ( m o d p n ) c\equiv0\pmod{p^n} c ≡ 0 ( mod p n ) tr γ = a + d ≡ 2 ( m o d p n ) \operatorname{tr}\gamma=a+d\equiv2\pmod{p^n} tr γ = a + d ≡ 2 ( mod p n ) ∣ tr ∣ ≥ 3 |\operatorname{tr}|\ge3 ∣ tr ∣ ≥ 3 tr ≡ 2 ( m o d p n ) \operatorname{tr}\equiv2\pmod{p^n} tr ≡ 2 ( mod p n ) ∣ tr ∣ ≥ p n − 2 |\operatorname{tr}|\ge p^n-2 ∣ tr ∣ ≥ p n − 2 □ \square □
Cusp region. Since c γ ≡ 0 ( m o d p n ) c_\gamma\equiv0\pmod{p^n} c γ ≡ 0 ( mod p n ) { y > 1 / p n } \{y>1/p^n\} { y > 1/ p n } R = e − 2 π / ( w p n ) → 1 R=e^{-2\pi/(wp^n)}\to1 R = e − 2 π / ( w p n ) → 1 Ψ ≤ 1 4 π \Psi\le\frac{1}{4\pi} Ψ ≤ 4 π 1
Theorem 4.2 (Short geodesic count on X 0 X_0 X 0
N s h o r t ( X 0 ( p n ) ) = p n − a − 1 ( p + 1 ) ( n ≥ a + 1 ) , N_{\mathrm{short}}(X_0(p^n))=p^{n-a-1}(p+1)\qquad(n\ge a+1), N short ( X 0 ( p n )) = p n − a − 1 ( p + 1 ) ( n ≥ a + 1 ) , where t 0 2 − 4 = p 2 a c t_0^2-4=p^{2a}c t 0 2 − 4 = p 2 a c p ∤ c p\nmid c p ∤ c c c c
Derivation. 1. Write γ = ( u v − k t 0 − u ) \gamma=\begin{pmatrix}u&v\\-k&t_0-u\end{pmatrix} γ = ( u − k v t 0 − u ) u v − N k = 1 uv-Nk=1 uv − N k = 1 tr γ = t 0 \operatorname{tr}\gamma=t_0 tr γ = t 0 u ( t 0 − u ) ≡ 1 ( m o d N ) u(t_0-u)\equiv1\pmod N u ( t 0 − u ) ≡ 1 ( mod N ) ( 2 u − t 0 ) 2 ≡ t 0 2 − 4 ( m o d 4 p n ) (2u-t_0)^2\equiv t_0^2-4\pmod{4p^n} ( 2 u − t 0 ) 2 ≡ t 0 2 − 4 ( mod 4 p n ) t 0 2 − 4 = p 2 a c t_0^2-4=p^{2a}c t 0 2 − 4 = p 2 a c c c c 2 u − t 0 ≡ ± p a r ( m o d p a + 1 ) 2u-t_0\equiv\pm p^ar\pmod{p^{a+1}} 2 u − t 0 ≡ ± p a r ( mod p a + 1 ) r 2 ≡ c r^2\equiv c r 2 ≡ c p n − a − 1 p^{n-a-1} p n − a − 1 p n + 1 p^{n+1} p n + 1 ⇒ \Rightarrow ⇒ u ↔ v u\leftrightarrow v u ↔ v N s h o r t = p n − a − 1 ( p + 1 ) N_{\mathrm{short}}=p^{n-a-1}(p+1) N short = p n − a − 1 ( p + 1 )
Threshold analysis. N s h o r t ≍ p n − a N_{\mathrm{short}}\asymp p^{n-a} N short ≍ p n − a g ≍ p 2 n g\asymp p^{2n} g ≍ p 2 n a ≥ 0 ⇒ N s h o r t / g → 0 a\ge0\Rightarrow N_{\mathrm{short}}/g\to0 a ≥ 0 ⇒ N short / g → 0 o ( g ) o(g) o ( g ) o ( g log N ) o(g\log N) o ( g log N ) ω ˉ 2 \bar{\omega}^2 ω ˉ 2 Γ 0 \Gamma_0 Γ 0
Note. One must use ( t 0 2 − 4 , p ) = 1 (t_0^2-4,p)=1 ( t 0 2 − 4 , p ) = 1 ( t 0 , p ) = 1 (t_0,p)=1 ( t 0 , p ) = 1 p = 3 p=3 p = 3 t 0 = 11 t_0=11 t 0 = 11 11 2 − 4 = 117 = 9 ⋅ 13 11^2-4=117=9\cdot13 1 1 2 − 4 = 117 = 9 ⋅ 13 a = 1 a=1 a = 1 a = 0 a=0 a = 0 p = 3 p=3 p = 3 t 0 = 11 t_0=11 t 0 = 11 N s h o r t / g 1 / 2 → p + 1 = 4 N_{\mathrm{short}}/g^{1/2}\to p+1=4 N short / g 1/2 → p + 1 = 4
The systole dichotomy
Table 6.
Saturation values (from the Z p \mathbb{Z}_p Z p t 2 − 4 = p a c t^2-4=p^ac t 2 − 4 = p a c a a a c c c
Table 8.
The unique difference is the condition a ≡ d ≡ 1 a\equiv d\equiv1 a ≡ d ≡ 1
Two analytic identities
Theorem 4.3. Let { ω j } \{\omega_j\} { ω j }
Ψ ( x ) : = sup j ∣ ω ( x ) ∣ 2 ∑ j ∣ ω j ∣ h y p 2 , mean Ψ = ∑ j ∣ ω j ∣ h y p 2 vol ( X ) . \Psi(x):=\sup_j\frac{|\omega(x)|^2}{\sum_j|\omega_j|_{\mathrm{hyp}}^2},\qquad\operatorname{mean}\Psi=\frac{\sum_j|\omega_j|_{\mathrm{hyp}}^2}{\operatorname{vol}(X)}. Ψ ( x ) := j sup ∑ j ∣ ω j ∣ hyp 2 ∣ ω ( x ) ∣ 2 , mean Ψ = vol ( X ) ∑ j ∣ ω j ∣ hyp 2 . Then the disc-comparison bound
Ψ ( x ) ≤ 1 4 π 1 ( 1 − tanh 2 ( inj ( x ) / 2 ) ) 2 = 1 4 π cosh 4 ( inj ( x ) / 2 ) . (6) \Psi(x)\le\frac{1}{4\pi}\frac{1}{(1-\tanh^2(\operatorname{inj}(x)/2))^2}=\frac{1}{4\pi}\cosh^4(\operatorname{inj}(x)/2). \tag*{(6)} Ψ ( x ) ≤ 4 π 1 ( 1 − tanh 2 ( inj ( x ) /2 ) ) 2 1 = 4 π 1 cosh 4 ( inj ( x ) /2 ) . (6) Derivation. The hyperbolic disc Bergman kernel is K H ( z , w ) = − 1 π ( z − w ˉ ) 2 K_{\mathbb{H}}(z,w)=-\frac{1}{\pi(z-\bar{w})^2} K H ( z , w ) = − π ( z − w ˉ ) 2 1 R = tanh ( r / 2 ) R=\tanh(r/2) R = tanh ( r /2 ) K H ( i , i ) = 1 4 π = 0.0795775 … K_{\mathbb{H}}(i,i)=\frac{1}{4\pi}=0.0795775\ldots K H ( i , i ) = 4 π 1 = 0.0795775 … 10 − 5 10^{-5} 1 0 − 5
Corollary 4.4. In a cusp neighborhood the injectivity radius tends to 0 and the right-hand side of (6) blows up like inj ( x ) − 4 \operatorname{inj}(x)^{-4} inj ( x ) − 4 X 1 X_1 X 1 o ( vol ) o(\operatorname{vol}) o ( vol ) Ψ \Psi Ψ
The barrier
Theorem 4.5 (One-third barrier). Let { X n } \{X_n\} { X n } B = 1 B=1 B = 1 σ → 1 \sigma\to1 σ → 1 ω ˉ 2 ≤ 3 Δ D e l + O ( g ) \bar{\omega}^{2} \le3\Delta_{\mathrm{Del}} + O(g) ω ˉ 2 ≤ 3 Δ Del + O ( g ) R ≤ 3 + o ( 1 ) R \le3 + o(1) R ≤ 3 + o ( 1 )
lim sup n κ n ≤ 1 3 . \limsup_{n} \kappa_n \le\frac{1}{3}. n lim sup κ n ≤ 3 1 . On the modular towers X 0 ( N ) X_0(N) X 0 ( N ) X 1 ( N ) X_1(N) X 1 ( N ) ω ˉ 2 \bar{\omega}^{2} ω ˉ 2 [5 ] , Mayer [26 ] ), so the barrier is unconditional there. On a general GL2-type tower it remains Problem 10.1; we do not claim it unconditionally.
Proof sketch, three routes. Route 1 (X 1 X_1 X 1 inj → ∞ \operatorname{inj} \to\infty inj → ∞ (6) to an upper bound on ω ˉ 2 \bar{\omega}^{2} ω ˉ 2 X 0 X_0 X 0 N s h o r t = o ( g ) N_{\mathrm{short}} = o(g) N short = o ( g ) σ → 1 \sigma\to1 σ → 1 d ≤ C σ + o ( 1 ) d \le C\sigma+ o(1) d ≤ C σ + o ( 1 ) κ ≤ 1 12 ( 1 ) ( 1 + 3 ) + o ( 1 ) = 1 3 \kappa\le\frac{1}{12}(1)(1+3) + o(1) = \frac{1}{3} κ ≤ 12 1 ( 1 ) ( 1 + 3 ) + o ( 1 ) = 3 1
Remark 4.6 (Proof status of the barrier). Route 1 reduces the barrier to [Arith-BMY]; the step from the Bergman-kernel bound of Theorem 4.3 to an upper bound on ω ˉ 2 \bar{\omega}^{2} ω ˉ 2 [25 Thm. 5.5] ) is unconditional and is what makes the barrier unconditional on modular towers.
Function-field photograph
Construction: K = F q ( t ) K = \mathbb{F}_q(t) K = F q ( t ) a = 1 a=1 a = 1 b = t M b=t^M b = t M c = 1 + t M c=1+t^M c = 1 + t M q ∤ M q \nmid M q ∤ M E M : y 2 = x ( x − 1 ) ( x + t M ) E_M : y^2 = x(x-1)(x+t^M) E M : y 2 = x ( x − 1 ) ( x + t M ) c ′ = M t M − 1 ≠ 0 c' = Mt^{M-1} \ne0 c ′ = M t M − 1 = 0 gcd ( c , c ′ ) = 1 \gcd(c,c')=1 g cd( c , c ′ ) = 1 c c c max deg = M = deg rad ( a b c ) − 1 \max\deg= M = \deg\operatorname{rad}(abc)-1 max deg = M = deg rad ( ab c ) − 1 rad ( a b c ) = t ( 1 + t M ) \operatorname{rad}(abc)=t(1+t^M) rad ( ab c ) = t ( 1 + t M ) deg = M + 1 \deg=M+1 deg = M + 1 deg Δ min = 4 M \deg\Delta_{\min}=4M deg Δ m i n = 4 M deg j ˉ = 3 ( 2 M ) − 4 M = 2 M \deg\bar{j}=3(2M)-4M=2M deg j ˉ = 3 ( 2 M ) − 4 M = 2 M n ( E M ) = M + 1 n(E_M)=M+1 n ( E M ) = M + 1
ρ f f ( M ) = 2 M M + 1 → 2 , κ f f = ρ f f 12 → 1 6 . \rho_{\mathrm{ff}}(M)=\frac{2M}{M+1}\to2,\qquad\kappa_{\mathrm{ff}}=\frac{\rho_{\mathrm{ff}}}{12}\to\frac{1}{6}. ρ ff ( M ) = M + 1 2 M → 2 , κ ff = 12 ρ ff → 6 1 . Numerics: ( q , M ) = ( 5 , 12 ) → 1.8462 (q,M)=(5,12)\to1.8462 ( q , M ) = ( 5 , 12 ) → 1.8462 ( 7 , 30 ) , ( 11 , 30 ) → 1.9355 (7,30),(11,30)\to1.9355 ( 7 , 30 ) , ( 11 , 30 ) → 1.9355
Verdict. The exact function-field extreme values are ρ = 2 \rho=2 ρ = 2 κ = 1 6 \kappa=\frac{1}{6} κ = 6 1 1 3 \frac{1}{3} 3 1 1 6 \frac{1}{6} 6 1 1 3 \frac{1}{3} 3 1 W = f g ′ − f ′ g W=fg'-f'g W = f g ′ − f ′ g W W W ≥ m α − 1 \ge m_\alpha-1 ≥ m α − 1 Δ D e l , p = f p + m p − 1 ≥ f p \Delta_{\mathrm{Del},p}=f_p+m_p-1\ge f_p Δ Del , p = f p + m p − 1 ≥ f p deg W ≤ deg f + deg g − 1 \deg W\le\deg f+\deg g-1 deg W ≤ deg f + deg g − 1 a b c abc ab c d / d t d/dt d / d t Z \mathbb{Z} Z D ( 1 ) = 2 D ( 1 ) ⇒ D = 0 D(1)=2D(1)\Rightarrow D=0 D ( 1 ) = 2 D ( 1 ) ⇒ D = 0 κ ≥ 1 12 \kappa\ge\frac{1}{12} κ ≥ 12 1 κ ≤ 1 3 \kappa\le\frac{1}{3} κ ≤ 3 1
Theorem 4.7. G L 2 \mathrm{GL}_2 GL 2 B = 1 B = 1 B = 1 B = 2 B = 2 B = 2 p 2 j p 2 − 1 \frac{p^{2j}}{p^2-1} p 2 − 1 p 2 j B ≥ 1 B \ge1 B ≥ 1
κ > 1 3 ⟺ ω ˉ 2 log N > 3 ( B = 1 ) . \kappa> \frac{1}{3} \Longleftrightarrow\frac{\bar{\omega}^{2}}{\log N} > 3 \qquad(B = 1). κ > 3 1 ⟺ log N ω ˉ 2 > 3 ( B = 1 ) . Identification and sharpness of the barrier
The barrier is Parshin’s arithmetic BMY constant
Proposition 5.1. Write R = ω ˉ 2 / Δ D e l R = \bar{\omega}^{2}/\Delta_{\mathrm{Del}} R = ω ˉ 2 / Δ Del σ → 1 \sigma\to1 σ → 1
κ ≤ 1 3 ⟺ R ≤ 3. \kappa\le\frac{1}{3} \Longleftrightarrow R \le3. κ ≤ 3 1 ⟺ R ≤ 3. Literature identification. Parshin [16 ] (1988, Sémin. Théor. Nombres Paris; same form in Moret–Bailly) states the arithmetic Bogomolov–Miyaoka–Yau inequality
ω ˉ 2 ≤ 3 Δ D e l + 3 δ a r c h + ( 2 g − 2 ) log ∣ D K ∣ ; \bar{\omega}^{2} \le3\Delta_{\mathrm{Del}} + 3\delta_{\mathrm{arch}} + (2g-2)\log|D_K|; ω ˉ 2 ≤ 3 Δ Del + 3 δ arch + ( 2 g − 2 ) log ∣ D K ∣ ; the constant 3 is the arithmetic analogue of c 1 2 ≤ 3 c 2 c_1^2 \le3c_2 c 1 2 ≤ 3 c 2
Literature status (must-read). The lower-bound direction (Zhang [23 ] , Moriwaki, Yuan–Zhang [22 ] , Kühn–Müller) is known and effective; the upper bound remains a conjecture. Yuan–Zhang state explicitly: “The first inequality is in the opposite direction of the conjectural arithmetic BMY. . . ” — one may not cite arithmetic BMY as a theorem. Theorem 4.5 proves R ≤ 3 R \le3 R ≤ 3
Theorem 5.2 (Sharpness). Let N → ∞ N \to\infty N → ∞ X 0 ( p n ) X_0(p^n) X 0 ( p n ) p ≥ 11 p \ge11 p ≥ 11 g > 0 g > 0 g > 0 ( N , 6 ) = 1 (N,6) = 1 ( N , 6 ) = 1 [5 ] ). From Dolce–Mercuri [5 ] , ω ˉ 2 ∼ 3 g log N \bar{\omega}^{2} \sim3g \log N ω ˉ 2 ∼ 3 g log N Δ D e l ∼ g log N \Delta_{\mathrm{Del}} \sim g \log N Δ Del ∼ g log N
R ( X 0 ( N ) ) = ω ˉ 2 Δ D e l ⟶ 3. R(X_0(N)) = \frac{\bar{\omega}^{2}}{\Delta_{\mathrm{Del}}} \longrightarrow3. R ( X 0 ( N )) = Δ Del ω ˉ 2 ⟶ 3. The convergence is slow and one-sided; the finite-level value is R ≈ 3 g g + 1 R \approx\frac{3g}{g+1} R ≈ g + 1 3 g R = 3 R = 3 R = 3 N = 10 N = 10 N = 10 10 4 10^4 1 0 4 10 6 10^6 1 0 6 g ( X 0 ( 10 ) ) = 0 g(X_0(10)) = 0 g ( X 0 ( 10 )) = 0 κ \kappa κ 10 4 10^4 1 0 4 10 6 10^6 1 0 6 ( N , 6 ) = 1 (N,6) = 1 ( N , 6 ) = 1
Proof that κ → 1 3 \kappa\to\frac{1}{3} κ → 3 1 For X 0 ( p ) X_0(p) X 0 ( p ) p p p P 1 \mathbb{P}^1 P 1 g + 1 g+1 g + 1 b 1 = g b_1 = g b 1 = g f p = g f_p = g f p = g m p = 2 m_p = 2 m p = 2 Δ D e l = ( g + 1 ) log p \Delta_{\mathrm{Del}} = (g+1)\log p Δ Del = ( g + 1 ) log p log cond = g log p \log\operatorname{cond} = g\log p log cond = g log p σ = 1 + 1 g \sigma= 1 + \frac{1}{g} σ = 1 + g 1 ω ˉ 2 ∼ 3 g log p \bar{\omega}^{2} \sim3g\log p ω ˉ 2 ∼ 3 g log p
σ ( 1 + R ) = ( 1 + 1 g ) ( 1 + 3 g g + 1 ) = 4 g + 1 g = 4 + 1 g , \sigma(1+R) = \left(1+\frac{1}{g}\right)\left(1+\frac{3g}{g+1}\right) = \frac{4g+1}{g} = 4+\frac{1}{g}, σ ( 1 + R ) = ( 1 + g 1 ) ( 1 + g + 1 3 g ) = g 4 g + 1 = 4 + g 1 , hence
κ = 1 3 + 1 12 g + d 12 , \kappa= \frac{1}{3}+\frac{1}{12g}+\frac{d}{12}, κ = 3 1 + 12 g 1 + 12 d , and κ → 1 3 \kappa\to\frac{1}{3} κ → 3 1 d → 0 d \to0 d → 0 f ^ = log cond / g = log p \hat{f} = \log\operatorname{cond}/g = \log p f ^ = log cond / g = log p d = δ F a l / ( g log p ) d = \delta_{\mathrm{Fal}}/(g\log p) d = δ Fal / ( g log p ) d > − 10.544 / log p → 0 d > -10.544/\log p \to0 d > − 10.544/ log p → 0 [25 Thm. 5.6 and Rem. 5.8] : δ F a l ( X 0 ( N ) ) = O ( g ) \delta_{\mathrm{Fal}}(X_0(N)) = O(g) δ Fal ( X 0 ( N )) = O ( g ) X 1 ( N ) X_1(N) X 1 ( N ) X ( N ) X(N) X ( N ) d = O ( 1 / log p ) → 0 d = O(1/\log p) \to0 d = O ( 1/ log p ) → 0 X 0 ( N ) X_0(N) X 0 ( N ) δ F a l ( X 0 ( N ) ) ≤ 2 g log N + o ( g log N ) \delta_{\mathrm{Fal}}(X_0(N)) \le2g\log N + o(g\log N) δ Fal ( X 0 ( N )) ≤ 2 g log N + o ( g log N )
At prime level; for the tower X 0 ( p n ) X_0(p^n) X 0 ( p n ) p p p σ → 1 \sigma\to1 σ → 1 σ ( 1 + R ) → 4 \sigma(1+R) \to4 σ ( 1 + R ) → 4 4 + 1 / g 4+1/g 4 + 1/ g κ → 1 3 \kappa\to\frac{1}{3} κ → 3 1 δ F a l \delta_{\mathrm{Fal}} δ Fal
The barrier is strictly stronger than BMY
Theorem 5.3. For every tower satisfying (L1) (σ ≥ 1 \sigma\ge1 σ ≥ 1
κ ≤ 1 3 ⟹ R ≤ 4 σ − 1 ≤ 3 , \kappa\le\frac{1}{3} \Longrightarrow R \le\frac{4}{\sigma}-1 \le3, κ ≤ 3 1 ⟹ R ≤ σ 4 − 1 ≤ 3 , i.e. the barrier implies arithmetic BMY, and strictly so when σ > 1 \sigma> 1 σ > 1
Proof. From Proposition 2.3 , dropping the last term (which is o ( 1 ) o(1) o ( 1 )
κ = σ ( 1 + R ) 12 ≤ 1 3 ⟹ σ ( 1 + R ) ≤ 4 ⟹ R ≤ 4 σ − 1. \kappa= \frac{\sigma(1+R)}{12} \le\frac{1}{3} \Longrightarrow\sigma(1+R) \le4 \Longrightarrow R \le\frac{4}{\sigma}-1. κ = 12 σ ( 1 + R ) ≤ 3 1 ⟹ σ ( 1 + R ) ≤ 4 ⟹ R ≤ σ 4 − 1. By (L1), σ ≥ 1 \sigma\ge1 σ ≥ 1 4 σ − 1 ≤ 3 \frac{4}{\sigma}-1 \le3 σ 4 − 1 ≤ 3
Corollary 5.4. Read jointly with Proposition 5.1 : given [Arith-BMY], arithmetic BMY holds on GL2 _2 2 [5 ] , Mayer [26 ] ), so both hold unconditionally there; in general both are open (Problem 10.1).
Three regions in the (σ \sigma σ R R R
Table 1 (PDF p. 11).
The second region is nonempty exactly when σ > 1 \sigma> 1 σ > 1 4 σ 4\sigma 4 σ R ≤ 3 R \le3 R ≤ 3 σ = 1 \sigma= 1 σ = 1 R = 3 R = 3 R = 3 X 0 ( N ) X_0(N) X 0 ( N )
Remark 5.5 (Hazard: BMY true and κ → ∞ \kappa\to\infty κ → ∞ σ → ∞ \sigma\to\infty σ → ∞ R ≤ 3 R \le3 R ≤ 3
κ ≥ σ 12 − W 12 f ^ ⟶ ∞ , κ ≤ σ ( 1 + R ) 12 ≤ σ 3 , \kappa\ge\frac{\sigma}{12} - \frac{W}{12\hat{f}} \longrightarrow\infty,\qquad\kappa\le\frac{\sigma(1+R)}{12} \le\frac{\sigma}{3}, κ ≥ 12 σ − 12 f ^ W ⟶ ∞ , κ ≤ 12 σ ( 1 + R ) ≤ 3 σ , where the lower bound uses (L2) and (L3) with f ^ → ∞ \hat{f} \to\infty f ^ → ∞ R ≤ 3 R \le3 R ≤ 3 κ → ∞ \kappa\to\infty κ → ∞ ⇒ \Rightarrow ⇒ [16 ] , whose argument we have not independently verified, so we flag every use of it as conditional.
R R R Table 2.
Scope: covers only towers with explicit known values. Warning: the three families are distinct and their coefficients differ: X 0 X_0 X 0 X 1 X_1 X 1 X ( N ) X(N) X ( N ) X ( N ) → 3 X(N) \to3 X ( N ) → 3 X ( N ) X(N) X ( N ) X 0 ( N ) X_0(N) X 0 ( N ) X ( N ) X(N) X ( N ) 4 g log N 4g\log N 4 g log N G L 2 \mathrm{GL}_2 GL 2
Hierarchy of geometric analogies
From K S = ω S / C ⊗ f ∗ K C K_S = \omega_{S/C} \otimes f^*K_C K S = ω S / C ⊗ f ∗ K C e ( S ) = e ( C ) e ( F ) + δ e(S) = e(C)e(F) + \delta e ( S ) = e ( C ) e ( F ) + δ
c 1 2 − 3 c 2 = ω 2 − 4 ( g − 1 ) ( g 0 − 1 ) − 3 δ . c_1^2 - 3c_2 = \omega^2 - 4(g-1)(g_0-1) - 3\delta. c 1 2 − 3 c 2 = ω 2 − 4 ( g − 1 ) ( g 0 − 1 ) − 3 δ . Table 5.5.
Geometric BMY ⇏ \not\Rightarrow ⇒ δ ∼ g n \delta\sim g_n δ ∼ g n
Arakelov semipositivity ⊊ \subsetneq ⊊ ⊊ \subsetneq ⊊ ⊊ \subsetneq ⊊
The Heisenberg tower and the floor
Structure
The Heisenberg group: H N = { ( a , b , c ) ∈ ( Z / N ) 3 } H_N = \{(a,b,c) \in(\mathbb{Z}/N)^3\} H N = {( a , b , c ) ∈ ( Z / N ) 3 }
( a , b , c ) ( a ′ , b ′ , c ′ ) = ( a + a ′ , b + b ′ , c + c ′ + a b ′ ) , ∣ H N ∣ = N 3 , (a,b,c)(a',b',c') = (a+a',b+b',c+c'+ab'), \qquad|H_N| = N^3, ( a , b , c ) ( a ′ , b ′ , c ′ ) = ( a + a ′ , b + b ′ , c + c ′ + a b ′ ) , ∣ H N ∣ = N 3 , nilpotency class 2. With x = ( 1 , 0 , 0 ) x=(1,0,0) x = ( 1 , 0 , 0 ) y = ( 0 , 1 , 0 ) y=(0,1,0) y = ( 0 , 1 , 0 ) N N N [ x , y ] = ( 0 , 0 , 1 ) [x,y]=(0,0,1) [ x , y ] = ( 0 , 0 , 1 ) N N N z = ( x y ) − 1 z=(xy)^{-1} z = ( x y ) − 1 N N N H N H_N H N Δ ( N , N , N ) \Delta(N,N,N) Δ ( N , N , N )
Theorem 6.1. The cover H N → F N H_N \to F_N H N → F N N N N
Chevalley–Weil decomposition:
Table 6.1.
Non-CM criterion: End ( Prym ) ⊇ ∏ M N ( Q ( ζ N ) ) \operatorname{End}(\operatorname{Prym}) \supseteq\prod M_N(\mathbb{Q}(\zeta_N)) End ( Prym ) ⊇ ∏ M N ( Q ( ζ N )) N < φ ( N ) N < \varphi(N) N < φ ( N ) dim Prym = 40 , 3808 , 989800 \dim\operatorname{Prym}=40, 3808, 989800 dim Prym = 40 , 3808 , 989800
Determination
Formula correction (Appendix B, E4): one formerly wrote κ X = 1 4 c \kappa_X=\frac{1}{4c} κ X = 4 c 1
κ X = c + 1 12 c , c : = F X n g X . \kappa_X=\frac{c+1}{12c}, \qquad c:=\frac{F_X}{n g_X}. κ X = 12 c c + 1 , c := n g X F X . Theorem 6.2. The Heisenberg tower satisfies
κ X ∈ ( 1 12 , 1 8 ) , κ X → 1 12 . \kappa_X \in\left(\frac{1}{12},\frac{1}{8}\right), \qquad\kappa_X \to\frac{1}{12}. κ X ∈ ( 12 1 , 8 1 ) , κ X → 12 1 . Two pillars. Pillar A (no good reduction), two independent proofs: (i) supersingular — the Fermat factors are CM by Q ( ζ p j ) \mathbb{Q}(\zeta_{p^j}) Q ( ζ p j ) p p p ⇒ A [ p ∞ ] = A [ π ∞ ] \Rightarrow A[p^\infty]=A[\pi^\infty] ⇒ A [ p ∞ ] = A [ π ∞ ] 2 g 2g 2 g g g g ⇒ \Rightarrow ⇒ 1 2 \frac{1}{2} 2 1 ⇒ \Rightarrow ⇒ Z / p n \mathbb{Z}/p^n Z / p n − 2 ∣ G ∣ + 3 ∣ G ∣ N ( N − 1 ) -2|G|+3\frac{|G|}{N}(N-1) − 2∣ G ∣ + 3 N ∣ G ∣ ( N − 1 ) 2 g X − 2 2g_X-2 2 g X − 2 N = 5 N=5 N = 5 50 = 50 50=50 50 = 50 N = 17 N=17 N = 17 4046 = 4046 4046 = 4046 4046 = 4046 Z / p n \mathbb{Z}/p^n Z / p n p p p Sw ≥ p n − 1 ≫ 2 n \operatorname{Sw} \ge p^{n-1} \gg2n Sw ≥ p n − 1 ≫ 2 n a ( V i ) = dim V i ⋅ a ( χ i ) a(V_i) = \dim V_i \cdot a(\chi^i) a ( V i ) = dim V i ⋅ a ( χ i )
c = F X n g X ≥ 2 p n − 1 n ≥ 2. c = \frac{F_X}{n g_X} \ge\frac{2p^{n-1}}{n} \ge2. c = n g X F X ≥ n 2 p n − 1 ≥ 2. From κ X = c + 1 12 c \kappa_X = \frac{c+1}{12c} κ X = 12 c c + 1 κ X ≤ 1 8 \kappa_X \le\frac{1}{8} κ X ≤ 8 1 c → ∞ c \to\infty c → ∞ κ X → 1 12 \kappa_X \to\frac{1}{12} κ X → 12 1 p = 7 p = 7 p = 7 n = 6 n = 6 n = 6 c = 5602 c = 5602 c = 5602 κ X = 0.0833 \kappa_X = 0.0833 κ X = 0.0833
Remark 6.3 (Proof status). Four sub-gaps, each presently one or two lines (E30). (i) Supersingularity: A [ p ∞ ] = A [ π ∞ ] A[p^\infty] = A[\pi^\infty] A [ p ∞ ] = A [ π ∞ ] 2 g 2g 2 g g ⇒ g \Rightarrow g ⇒ 1 2 ⇒ \frac{1}{2} \Rightarrow 2 1 ⇒ Z / p n \mathbb{Z}/p^n Z / p n p p p p p p ∼ 2 \sim2 ∼ 2 − 2 ∣ G ∣ + 3 ∣ G ∣ N ( N − 1 ) = 2 g X − 2 -2|G| + 3\frac{|G|}{N}(N-1) = 2g_X - 2 − 2∣ G ∣ + 3 N ∣ G ∣ ( N − 1 ) = 2 g X − 2 ∼ 1 \sim1 ∼ 1 Sw ≥ p n − 1 \operatorname{Sw} \ge p^{n-1} Sw ≥ p n − 1 a ( V i ) = dim V i ⋅ a ( χ i ) a(V_i) = \dim V_i \cdot a(\chi^i) a ( V i ) = dim V i ⋅ a ( χ i ) Sw ( χ ) = a n − 1 ∼ n p n \operatorname{Sw}(\chi) = a_n - 1 \sim np^n Sw ( χ ) = a n − 1 ∼ n p n ∼ 3 \sim3 ∼ 3 π : X → Y \pi: \mathcal{X} \to\mathcal{Y} π : X → Y ω X ≅ π ∗ ω Y \omega_{\mathcal{X}} \cong\pi^*\omega_{\mathcal{Y}} ω X ≅ π ∗ ω Y ω ˉ X 2 = N ω ˉ Y 2 \bar{\omega}_{\mathcal{X}}^2 = N\bar{\omega}_{\mathcal{Y}}^2 ω ˉ X 2 = N ω ˉ Y 2 Ω X = O ( log N ) \Omega_X = O(\log N) Ω X = O ( log N )
Local Kummer–ASW closed form. Let F = Q p ( ζ p n ) F = \mathbb{Q}_p(\zeta_{p^n}) F = Q p ( ζ p n ) π = 1 − ζ p n \pi= 1 - \zeta_{p^n} π = 1 − ζ p n L n = F ( w ) L_n = F(w) L n = F ( w ) w p n = π a w^{p^n} = \pi^a w p n = π a p ∤ a p \nmid a p ∤ a
d ( L n / F ) = n ( p − 1 ) p 2 n − 1 + a ( p n − 1 ) , a n ( a ) = n p n − ( n − 1 ) p n − 1 + a , d(L_n/F) = n(p-1)p^{2n-1} + a(p^n-1), \qquad a_n(a) = np^n - (n-1)p^{n-1} + a, d ( L n / F ) = n ( p − 1 ) p 2 n − 1 + a ( p n − 1 ) , a n ( a ) = n p n − ( n − 1 ) p n − 1 + a , Sw ( χ ) = a n ( a ) − 1 ∼ n p n . \operatorname{Sw}(\chi) = a_n(a) - 1 \sim np^n. Sw ( χ ) = a n ( a ) − 1 ∼ n p n . Table 9.
Dependency labels. Étale scaling ω ˉ X 2 ≈ N ω ˉ F 2 \bar{\omega}_{\mathcal{X}}^2 \approx N\bar{\omega}_{\mathcal{F}}^2 ω ˉ X 2 ≈ N ω ˉ F 2 Ω X = O ( log N ) \Omega_X = O(\log N) Ω X = O ( log N ) Ω X \Omega_X Ω X c c c κ X → 1 12 \kappa_X \to\frac{1}{12} κ X → 12 1
Floor theorem
Theorem 6.4 (Tower-world floor). Let { X n } \{X_n\} { X n } f ^ n = L n / g n → ∞ \hat{f}_n = L_n/g_n \to\infty f ^ n = L n / g n → ∞
lim inf n κ n ≥ 1 12 . \liminf_{n} \kappa_n \ge\frac{1}{12}. n lim inf κ n ≥ 12 1 . Proof. By Proposition 2.3:
κ n = σ n ( 1 + R n ) 12 + δ F a l , n 12 L n . \kappa_n = \frac{\sigma_n(1+R_n)}{12} + \frac{\delta_{\mathrm{Fal},n}}{12L_n}. κ n = 12 σ n ( 1 + R n ) + 12 L n δ Fal , n . With (L2) R n ≥ 0 R_n \ge0 R n ≥ 0 σ n ≥ 1 \sigma_n \ge1 σ n ≥ 1 ≥ 1 ⋅ ( 1 + 0 ) 12 = 1 12 \ge\frac{1\cdot(1+0)}{12} = \frac{1}{12} ≥ 12 1 ⋅ ( 1 + 0 ) = 12 1 δ F a l , n > − 2 g n log ( 2 π 4 ) \delta_{\mathrm{Fal},n} > -2g_n\log(2\pi^4) δ Fal , n > − 2 g n log ( 2 π 4 )
− 2 g n log ( 2 π 4 ) 12 L n = − log ( 2 π 4 ) 6 f ^ n → 0. \frac{-2g_n\log(2\pi^4)}{12L_n} = -\frac{\log(2\pi^4)}{6\hat{f}_n} \to0. 12 L n − 2 g n log ( 2 π 4 ) = − 6 f ^ n log ( 2 π 4 ) → 0. Hence κ n ≥ 1 12 + o ( 1 ) \kappa_n \ge\frac{1}{12} + o(1) κ n ≥ 12 1 + o ( 1 )
Remark 6.5. Note that the second term only needs the Wilms lower bound (δ F a l \delta_{\mathrm{Fal}} δ Fal δ F a l = O ( g ) \delta_{\mathrm{Fal}} = O(g) δ Fal = O ( g ) δ F a l \delta_{\mathrm{Fal}} δ Fal L n / g n = f ^ n → ∞ L_n/g_n = \hat{f}_n \to\infty L n / g n = f ^ n → ∞ Remark 2.5 .
Remark 6.6 (This is not a “universal floor”). For a single abelian variety (no tower limit), κ \kappa κ j = 0 j = 0 j = 0 N = 2 2 ⋅ 3 3 N = 2^2 \cdot3^3 N = 2 2 ⋅ 3 3 h F = − 1.667691 h_F = -1.667691 h F = − 1.667691 κ = − 0.506 \kappa= -0.506 κ = − 0.506 y 2 = x 5 + 1 y^2 = x^5 + 1 y 2 = x 5 + 1 κ = − 0.282 \kappa= -0.282 κ = − 0.282 g = 1 g = 1 g = 1 12 h F = − log ∣ Δ min ∣ − 6 log ( Area ) − 6 log 2 12h_F = -\log|\Delta_{\min}| - 6\log(\operatorname{Area}) - 6\log2 12 h F = − log ∣ Δ m i n ∣ − 6 log ( Area ) − 6 log 2 1 12 \frac{1}{12} 12 1
The attainable set in the tower world
In this section we prove: the set of limit values of κ \kappa κ [ 1 12 , 1 3 ] \left[\frac{1}{12}, \frac{1}{3}\right] [ 12 1 , 3 1 ]
Convexity lemma
Lemma 7.1 (Convexity). For abelian varieties A A A B B B
κ ( A × B ) = L A κ ( A ) + L B κ ( B ) L A + L B , L ∙ : = log cond ( ∙ ) , \kappa(A \times B) = \frac{L_A\kappa(A) + L_B\kappa(B)}{L_A + L_B}, \qquad L_{\bullet} := \log\operatorname{cond}(\bullet), κ ( A × B ) = L A + L B L A κ ( A ) + L B κ ( B ) , L ∙ := log cond ( ∙ ) , i.e. κ \kappa κ
Proof. Three steps, additivity only. 1. The Hodge bundle splits: ω A × B = ω A ⊕ ω B \omega_{A \times B} = \omega_A \oplus\omega_B ω A × B = ω A ⊕ ω B h F ( A × B ) = h F ( A ) + h F ( B ) h_F(A \times B) = h_F(A) + h_F(B) h F ( A × B ) = h F ( A ) + h F ( B ) L L L ⇒ \Rightarrow ⇒ ⇒ \Rightarrow ⇒ h F ( A ) = κ ( A ) L A h_F(A) = \kappa(A)L_A h F ( A ) = κ ( A ) L A h F ( B ) = κ ( B ) L B h_F(B) = \kappa(B)L_B h F ( B ) = κ ( B ) L B
Corollary 7.2 (Convexity safety).
min ( κ A , κ B ) ≤ κ ( A × B ) ≤ max ( κ A , κ B ) , \min(\kappa_A,\kappa_B) \le\kappa(A \times B) \le\max(\kappa_A,\kappa_B), min ( κ A , κ B ) ≤ κ ( A × B ) ≤ max ( κ A , κ B ) , i.e. a weighted average never escapes the barrier or breaks the floor.
Remark 7.3. In the decomposition of Proposition 2.3 , σ \sigma σ R R R Δ D e l \Delta_{\mathrm{Del}} Δ Del δ F a l \delta_{\mathrm{Fal}} δ Fal
Trap: same-index pairing collapses convexity
Pair two towers at the same index, A n ⊕ a ⊕ B n ⊕ b A_n^{\oplus a} \oplus B_n^{\oplus b} A n ⊕ a ⊕ B n ⊕ b
w n = a L A ( n ) a L A ( n ) + b L B ( n ) = a ρ n a ρ n + b , ρ n : = L A ( n ) L B ( n ) . w_n = \frac{aL_A(n)}{aL_A(n) + bL_B(n)} = \frac{a\rho_n}{a\rho_n + b}, \qquad\rho_n := \frac{L_A(n)}{L_B(n)}. w n = a L A ( n ) + b L B ( n ) a L A ( n ) = a ρ n + b a ρ n , ρ n := L B ( n ) L A ( n ) . A limit for κ \kappa κ ρ n \rho_n ρ n ρ n → ∞ \rho_n \to\infty ρ n → ∞ w → 1 w \to1 w → 1 κ → κ A \kappa\to\kappa_A κ → κ A n = k n = k n = k g ∼ p n − 1 ( p + 1 ) / 12 … g \sim p^{n-1}(p + 1)/12\ldots g ∼ p n − 1 ( p + 1 ) /12 … g F ∼ p 2 k / 2 g_F \sim p^{2k/2} g F ∼ p 2 k /2 X 0 X_0 X 0 g ∼ p 2 k / 2 g \sim p^{2k/2} g ∼ p 2 k /2 L X ∼ p n n log p L_X \sim p^{n}n\log p L X ∼ p n n log p ρ k ∼ p k → ∞ \rho_k \sim p^k \to\infty ρ k ∼ p k → ∞
Table 1 (PDF p. 14).
Whatever a , b a,b a , b κ → κ F e r m a t = 1 8 \kappa\to\kappa_{\mathrm{Fermat}} = \frac{1}{8} κ → κ Fermat = 8 1
Conclusion. The naive weight a a + b \frac{a}{a+b} a + b a a ρ a ρ + b \frac{a\rho}{a\rho+b} a ρ + b a ρ
Correction: reindexing restores convexity
Construction: take the same prime p p p
T k ( a , b ) : = J ( F p k ) ⊕ a × J 0 ( p 2 k ) ⊕ b T_k^{(a,b)} := J(F_{p^k})^{\oplus a} \times J_0(p^{2k})^{\oplus b} T k ( a , b ) := J ( F p k ) ⊕ a × J 0 ( p 2 k ) ⊕ b (Fermat at level k k k 2 k 2k 2 k { p } \{p\} { p }
Theorem 7.4 (Reindexing). For every prime p p p T k ( a , b ) T_k^{(a,b)} T k ( a , b )
κ ( T ( a , b ) ) ⟶ a ρ κ F + b κ X a ρ + b , ρ = 6 p p + 1 , κ F = 1 8 , κ X = 1 3 . \kappa(T^{(a,b)}) \longrightarrow\frac{a\rho\kappa_F+b\kappa_X}{a\rho+b}, \qquad\rho=\frac{6p}{p+1},\quad\kappa_F=\frac{1}{8},\quad\kappa_X=\frac{1}{3}. κ ( T ( a , b ) ) ⟶ a ρ + b a ρ κ F + b κ X , ρ = p + 1 6 p , κ F = 8 1 , κ X = 3 1 . Derivation. g F ( p k ) ∼ p 2 k / 2 g_F(p^k)\sim p^{2k}/2 g F ( p k ) ∼ p 2 k /2 g X ( p n ) ∼ p n − 1 ( p + 1 ) 12 g_X(p^n)\sim\frac{p^{n-1}(p+1)}{12} g X ( p n ) ∼ 12 p n − 1 ( p + 1 ) L F = 2 g F log ( p k ) L_F=2g_F\log(p^k) L F = 2 g F log ( p k ) L X = g X log ( p n ) L_X=g_X\log(p^n) L X = g X log ( p n ) n = 2 k n=2k n = 2 k
ρ k = p 2 k k log p p 2 k − 1 ( p + 1 ) ( 2 k ) log p / 12 = 6 p p + 1 + O ( p − k ) . \rho_k=\frac{p^{2k}k\log p}{p^{2k-1}(p+1)(2k)\log p/12}=\frac{6p}{p+1}+O(p^{-k}). ρ k = p 2 k − 1 ( p + 1 ) ( 2 k ) log p /12 p 2 k k log p = p + 1 6 p + O ( p − k ) . Convention 7.5 (E27). Here ρ : = lim k L ( F p k ) / L ( X 0 ( p 2 k ) ) \rho:=\lim_k L(F_{p^k})/L(X_0(p^{2k})) ρ := lim k L ( F p k ) / L ( X 0 ( p 2 k )) ρ = 6 p p + 1 \rho=\frac{6p}{p+1} ρ = p + 1 6 p κ = 1 4 \kappa=\frac{1}{4} κ = 4 1 b / ( a ρ ) = 3 2 b/(a\rho)=\frac{3}{2} b / ( a ρ ) = 2 3 b / a = 9 p p + 1 b/a=\frac{9p}{p+1} b / a = p + 1 9 p ρ ′ = 1 / ρ \rho'=1/\rho ρ ′ = 1/ ρ κ \kappa κ b / a = 2 p 3 b/a=\frac{2p}{3} b / a = 3 2 p
Density
Theorem 7.6 (Density). The weight set W = { a ρ a ρ + b : a , b ∈ Z > 0 } W=\left\{\frac{a\rho}{a\rho+b}:a,b\in\mathbb{Z}_{>0}\right\} W = { a ρ + b a ρ : a , b ∈ Z > 0 } is dense in ( 0 , 1 ) (0,1) ( 0 , 1 ) [ 1 8 , 1 3 ] \left[\frac{1}{8},\frac{1}{3}\right] [ 8 1 , 3 1 ] ( → 1 12 ) (\to\frac{1}{12}) ( → 12 1 ) ( → 1 3 ) (\to\frac{1}{3}) ( → 3 1 ) [ 1 12 , 1 3 ] \left[\frac{1}{12},\frac{1}{3}\right] [ 12 1 , 3 1 ]
Proof. Run over t : = a / b ∈ Q > 0 t:=a/b\in\mathbb{Q}_{>0} t := a / b ∈ Q > 0 s ↦ s ρ / ( s ρ + 1 ) s\mapsto s\rho/(s\rho+1) s ↦ s ρ / ( s ρ + 1 ) ( 0 , ∞ ) → ( 0 , 1 ) (0,\infty)\to(0,1) ( 0 , ∞ ) → ( 0 , 1 ) ρ \rho ρ x = 6 p a x=6pa x = 6 p a y = ( p + 1 ) b y=(p+1)b y = ( p + 1 ) b
Measured maximal gaps (p = 3 p=3 p = 3 ρ = 4.5 \rho=4.5 ρ = 4.5 a , b ≤ 5 a,b\leq5 a , b ≤ 5 ≤ 20 \leq20 ≤ 20 ≤ 100 \leq100 ≤ 100 M M M
Robustness of the lower end
Theorem 7.7. If the Heisenberg tower is defined over Q ( ζ p n ) \mathbb{Q}(\zeta_{p^n}) Q ( ζ p n )
κ ( Res ) ⟶ 1 12 . \kappa(\operatorname{Res})\longrightarrow\frac{1}{12}. κ ( Res ) ⟶ 12 1 . The main term grows exponentially, the discriminant term polynomially, so the ratio → 0 \to0 → 0
Balancing Heisenberg (ρ → 3 p + 1 ∈ ( 0 , ∞ ) \rho\to\frac{3}{p+1}\in(0,\infty) ρ → p + 1 3 ∈ ( 0 , ∞ ) X 0 X_0 X 0 m ≈ 4 n − log p n m\approx4n-\log_p n m ≈ 4 n − log p n [ 1 12 , 1 8 ] \left[\frac{1}{12},\frac{1}{8}\right] [ 12 1 , 8 1 ]
Two residual rulings.
Jacobian realization — negative, but unneeded.
Theorem 7.8. Let A = J ( C 1 ) × J ( C 2 ) A = J(C_1) \times J(C_2) A = J ( C 1 ) × J ( C 2 ) g i ≥ 1 g_i \ge1 g i ≥ 1 A A A
Proof. The theta divisor of the product polarization is Θ A = ( Θ 1 × J 2 ) ∪ ( J 1 × Θ 2 ) \Theta_A = (\Theta_1 \times J_2) \cup(J_1 \times\Theta_2) Θ A = ( Θ 1 × J 2 ) ∪ ( J 1 × Θ 2 ) g ≥ 2 g \ge2 g ≥ 2 Θ = W g − 1 = image ( Sym g − 1 C ) \Theta= W_{g-1} = \operatorname{image}(\operatorname{Sym}^{g-1} C) Θ = W g − 1 = image ( Sym g − 1 C ) g = 1 g = 1 g = 1
But the convexity tower does not need it: Lemma 7.1 applies to the product abelian variety itself; both inputs are precise theorems and involve no isogeny. Thus the attainable-set statement is in the abelian-variety category.
Category warning. The dense filling of the attainable set is an abelian-variety statement; the barrier/BMY/κ → ∞ \sqrt{\kappa} \to\infty κ → ∞ κ → ∞ \kappa\to\infty κ → ∞
ρ = C A C ^ A d B C B C ^ B d A \rho= \frac{C_A \widehat{C}_A d_B}{C_B \widehat{C}_B d_A} ρ = C B C B d A C A C A d B (reindexing condition α d A = β d B \alpha d_A = \beta d_B α d A = β d B
For instance ρ → 3 p + 1 ∈ ( 0 , ∞ ) \rho\to\frac{3}{p+1} \in(0,\infty) ρ → p + 1 3 ∈ ( 0 , ∞ ) X 0 X_0 X 0 [ 1 12 , 1 8 ] [\frac{1}{12}, \frac{1}{8}] [ 12 1 , 8 1 ]
Main theorem
Theorem 7.9 (Attainable set). In the abelian-variety tower world (products of Jacobians allowed; cf. the category warning in § 7 ), the set of limit values of κ \kappa κ [ 1 12 , 1 3 ] [\frac{1}{12}, \frac{1}{3}] [ 12 1 , 3 1 ] [ 1 12 , 1 3 ] [\frac{1}{12}, \frac{1}{3}] [ 12 1 , 3 1 ] 1 12 \frac{1}{12} 12 1 1 3 \frac{1}{3} 3 1 v v v
v ∈ { a ρ κ F + b κ X a ρ + b : a , b ∈ Z > 0 } ; v \in\left\{ \frac{a\rho\kappa_F + b\kappa_X}{a\rho+b} : a,b \in\mathbb{Z}_{>0} \right\}; v ∈ { a ρ + b a ρ κ F + b κ X : a , b ∈ Z > 0 } ; in particular the attainable set is not closed. Whether every interior value is attained by a curve tower (single Jacobian) is open (Problem 10.12).
Table 7.
Double-column accounting (do not mix). Column A (single abelian variety / constant tower): κ \kappa κ [ 1 12 , 1 3 ] [\frac{1}{12}, \frac{1}{3}] [ 12 1 , 3 1 ] Theorem 6.4 and Remark 6.6: finite level κ \kappa κ − 0.282 -0.282 − 0.282 y 2 = x 5 + 1 y^2 = x^5 + 1 y 2 = x 5 + 1 J ( F 5 ) , J ( F 7 ) , J ( F 9 ) J(\mathbb{F}_5), J(\mathbb{F}_7), J(\mathbb{F}_9) J ( F 5 ) , J ( F 7 ) , J ( F 9 ) κ < 1 8 \kappa< \frac{1}{8} κ < 8 1 κ N ≈ 1 8 − 0.8 log N \kappa_N \approx\frac{1}{8} - \frac{0.8}{\log N} κ N ≈ 8 1 − l o g N 0.8 N ≲ 550 N \lesssim550 N ≲ 550 lim κ = 1 8 \lim\kappa= \frac{1}{8} lim κ = 8 1
The discriminant channel: reduction, closure, and cancellation
By Proposition 2.3 , κ → ∞ \kappa\to\infty κ → ∞ σ → ∞ \sigma\to\infty σ → ∞ σ \sigma σ R → ∞ R \to\infty R → ∞
The local identity and the g = 1 g = 1 g = 1
Proposition 8.1 (Ogg–Saito/Liu; the place-by-place form of (L1)). For a regular arithmetic surface of any genus,
Δ D e l , p = f p + m p − 1 , \Delta_{\mathrm{Del},p} = f_p + m_p - 1, Δ Del , p = f p + m p − 1 , where f p f_p f p m p m_p m p g = 1 g = 1 g = 1 ν p ( Δ min ) = f p + m p − 1 \nu_p(\Delta_{\min}) = f_p + m_p - 1 ν p ( Δ m i n ) = f p + m p − 1
Check (full Kodaira table):
Table 10.
Corollary 8.2. For g = 1 g = 1 g = 1 σ = 1 + m − 1 f \sigma= 1 + \frac{m-1}{f} σ = 1 + f m − 1 σ ≤ 5 \sigma\le5 σ ≤ 5 I 0 ∗ … I I ∗ I_0^* \ldots II^* I 0 ∗ … I I ∗ I n I_n I n σ = n \sigma= n σ = n
Szpiro ( g = 1 ) ⟺ n ≤ 6 + ε ⟺ boundedness of long ( − 2 ) -chains . \operatorname{Szpiro}(g = 1) \Longleftrightarrow n \le6 + \varepsilon\Longleftrightarrow\text{boundedness of long }(-2)\text{-chains}. Szpiro ( g = 1 ) ⟺ n ≤ 6 + ε ⟺ boundedness of long ( − 2 ) -chains . Remark 8.3 (Outside semistability the conductor strictly exceeds the “effective conductor”). Liu and Saito define the effective conductor
Effcond ( X K / K ) : = length O K Coker ( H 0 ( X , Ω ) → H 0 ( X , ω ) ) \operatorname{Effcond}(X_K/K) := \operatorname{length}_{\mathcal{O}_K}\operatorname{Coker}\left(H^0(X,\Omega) \to H^0(X,\omega)\right) Effcond ( X K / K ) := length O K Coker ( H 0 ( X , Ω ) → H 0 ( X , ω ) ) (the conducteur efficace of Pesenti–Szpiro) and prove Effcond ≤ Art \operatorname{Effcond} \le\operatorname{Art} Effcond ≤ Art − Art = m ^ − 1 + f ^ -\operatorname{Art} = \hat{m} - 1 + \hat{f} − Art = m ^ − 1 + f ^ ( − 2 ) (-2) ( − 2 ) m m m Δ D e l \Delta_{\mathrm{Del}} Δ Del f f f
Combinatorial dictionary in the semistable world. Semistable: (dual graph) # E = m \#E = m # E = m # V = b 1 + 1 \#V = b_1 + 1 # V = b 1 + 1 ν ( Δ ) = # { nodes } \nu(\Delta) = \#\{\text{nodes}\} ν ( Δ ) = # { nodes } = b 1 ( Γ d u a l ) = b_1(\Gamma_{\mathrm{dual}}) = b 1 ( Γ dual )
Szpiro ⟺ # nodes ≤ ( 6 + ε ) ⋅ b 1 . \operatorname{Szpiro} \Longleftrightarrow\#\text{nodes} \le(6+\varepsilon)\cdot b_1. Szpiro ⟺ # nodes ≤ ( 6 + ε ) ⋅ b 1 . Mestre forcing law and bounds for stable fibers
Theorem 8.4. Let the bad set of a tower be fixed S S S q max : = max S q_{\max} := \max S q m a x := max S [24 ] (requiring analyticity of L ( A , s ) L(A,s) L ( A , s ) m ≤ 2 g − 2 m \le2g - 2 m ≤ 2 g − 2
σ ≤ 1 + 2 g − 3 g log q max log B M = 1 + c S ⋅ log q max log B M , c S = 2 g − 3 g ≤ 2. \sigma\le1 + \frac{2g-3}{g}\frac{\log q_{\max}}{\log B_M}
= 1 + c_S\cdot\frac{\log q_{\max}}{\log B_M},
\qquad c_S = \frac{2g-3}{g} \le2. σ ≤ 1 + g 2 g − 3 log B M log q m a x = 1 + c S ⋅ log B M log q m a x , c S = g 2 g − 3 ≤ 2. Proof sketch. By (L1), σ = 1 + ∑ p ( m p − 1 ) log p ∑ p f p log p \sigma= 1 + \frac{\sum_p(m_p-1)\log p}{\sum_p f_p\log p} σ = 1 + ∑ p f p l o g p ∑ p ( m p − 1 ) l o g p m p − 1 ≤ 2 g − 3 m_p - 1 \le2g - 3 m p − 1 ≤ 2 g − 3 m ≤ 2 g − 2 m \le2g - 2 m ≤ 2 g − 2 log p ≤ log q max \log p \le\log q_{\max} log p ≤ log q m a x ( 2 g − 3 ) log q max ⋅ # S (2g-3)\log q_{\max}\cdot\#S ( 2 g − 3 ) log q m a x ⋅ # S f ≥ log B M f \ge\log B_M f ≥ log B M L ( A , s ) L(A,s) L ( A , s ) σ ≤ 1 + c S log q max log B M \sigma\le1 + c_S\frac{\log q_{\max}}{\log B_M} σ ≤ 1 + c S l o g B M l o g q m a x c S = 2 g − 3 g c_S = \frac{2g-3}{g} c S = g 2 g − 3 ( − 2 ) (-2) ( − 2 ) m m m
Remark 8.5 (Convention unification, E43). All constants in this table use log B M = 2.334374 \log B_{\mathrm{M}}=2.334374 log B M = 2.334374 B M ≈ 10.323 B_{\mathrm{M}}\approx10.323 B M ≈ 10.323 log 10 \log10 log 10 log B M \log B_{\mathrm{M}} log B M B M ≈ 10 B_{\mathrm{M}}\approx10 B M ≈ 10 c S = 2 g − 3 g c_{\mathrm{S}}=\frac{2g-3}{g} c S = g 2 g − 3
Two corollaries: f ^ → ∞ ⇒ σ → ∞ \hat{f}\to\infty\Rightarrow\sigma\to\infty f ^ → ∞ ⇒ σ → ∞ f ≤ g f\le g f ≤ g f ^ ≥ c S log q max log B M > 1 \hat{f}\ge c_{\mathrm{S}}\frac{\log q_{\max}}{\log B_{\mathrm{M}}}>1 f ^ ≥ c S l o g B M l o g q m a x > 1 c S ≤ 1 c_{\mathrm{S}}\le1 c S ≤ 1 f ^ < 1 \hat{f}<1 f ^ < 1 q max ∈ { 2 , 3 , 5 , 7 } q_{\max}\in\{2,3,5,7\} q m a x ∈ { 2 , 3 , 5 , 7 }
For q max = 11 q_{\max}=11 q m a x = 11 rad N S ≤ 313 ⇒ σ < 6 \operatorname{rad}N_{\mathrm{S}}\le313\Rightarrow\sigma<6 rad N S ≤ 313 ⇒ σ < 6
Remark 8.6 (Limited range). Theorem 8.4 covers only “semistable stable fibers” (minimal regular stable, no ( − 2 ) (-2) ( − 2 ) ( − 2 ) (-2) ( − 2 ) m m m ( − 2 ) (-2) ( − 2 ) n n n σ \sigma σ = e 2 =e_{2} = e 2 m ≤ 2 g − 2 m\le2g-2 m ≤ 2 g − 2
Cancellation of the archimedean channel
The third route once proposed was d : = δ F a l / log cond → ∞ d:=\delta_{\mathrm{Fal}}/\log\operatorname{cond}\to\infty d := δ Fal / log cond → ∞
Theorem 8.7. δ F a l \delta_{\mathrm{Fal}} δ Fal
δ F a l = − c g log ∣ Δ d i s c ∣ ∞ + O ( g ) , Δ D e l = − ∑ p log ∣ Δ d i s c ∣ p = − log ∣ Δ d i s c ∣ ∞ + O ( 1 ) . \delta_{\mathrm{Fal}=-c_g\log\lvert\Delta_{\mathrm{disc}}\rvert_{\infty}+O(g),\qquad
\Delta_{\mathrm{Del}}=-\sum_p\log\lvert\Delta_{\mathrm{disc}}\rvert_p=-\log\lvert\Delta_{\mathrm{disc}}\rvert_{\infty}+O(1).} δ Fal = − c g l o g ∣ Δ disc ∣ ∞ + O ( g ) , Δ Del = − ∑ p l o g ∣ Δ disc ∣ p = − l o g ∣ Δ disc ∣ ∞ + O ( 1 ) . Hence
δ F a l ≤ C Δ D e l + O ( g ) ⟹ d ≤ C σ + o ( 1 ) , \delta_{\mathrm{Fal}}\le C\Delta_{\mathrm{Del}}+O(g)\Longrightarrow d\le C\sigma+o(1), δ Fal ≤ C Δ Del + O ( g ) ⟹ d ≤ C σ + o ( 1 ) , i.e. the third channel is fully absorbed into the discriminant channel: d → ∞ d\to\infty d → ∞ σ → ∞ \sigma\to\infty σ → ∞
Singularity order. The logarithmic singularity of δ F a l \delta_{\mathrm{Fal}} δ Fal [21 ] : near a degeneration with vanishing cycle of length δ \delta δ δ F a l ∼ − ( δ + ε ) log ∣ t ∣ − 6 log det Im Ω ( t ) \delta_{\mathrm{Fal}}\sim-(\delta+\varepsilon)\log\lvert t\rvert-6\log\det\operatorname{Im}\Omega(t) δ Fal ∼ − ( δ + ε ) log ∣ t ∣ − 6 log det Im Ω ( t )
Sharp calibration for g = 1 g=1 g = 1 From η ( i y ) ∼ e − π y / 12 \eta(iy)\sim e^{-\pi y/12} η ( i y ) ∼ e − π y /12 − 24 log ∣ η ∣ ∼ 2 π y -24\log\lvert\eta\rvert\sim2\pi y − 24 log ∣ η ∣ ∼ 2 π y δ F a l ∼ 2 π Im τ = log ∣ j ∣ \delta_{\mathrm{Fal}}\sim2\pi\operatorname{Im}\tau=\log\lvert j\rvert δ Fal ∼ 2 π Im τ = log ∣ j ∣ N ⇒ λ = 1 / N N\Rightarrow\lambda=1/N N ⇒ λ = 1/ N δ F a l ∼ 2 log N \delta_{\mathrm{Fal}}\sim2\log N δ Fal ∼ 2 log N Δ D e l ∼ 4 log N \Delta_{\mathrm{Del}}\sim4\log N Δ Del ∼ 4 log N log cond ∼ log N \log\operatorname{cond}\sim\log N log cond ∼ log N
δ F a l Δ D e l → 1 2 , d → 2 , σ → 4. \frac{\delta_{\mathrm{Fal}}}{\Delta_{\mathrm{Del}}}\to\frac{1}{2},\qquad d\to2,\qquad\sigma\to4. Δ Del δ Fal → 2 1 , d → 2 , σ → 4. The third channel not only fails to diverge but converges to a constant, and it is half of the discriminant channel.
Correction (Appendix B, E3): one formerly wrote δ F a l ∼ 1 3 log ∣ j ∣ \delta_{\mathrm{Fal}}\sim\frac{1}{3}\log\lvert j\rvert δ Fal ∼ 3 1 log ∣ j ∣ 2 π Im τ = log ∣ j ∣ 2\pi\operatorname{Im}\tau=\log\lvert j\rvert 2 π Im τ = log ∣ j ∣ S S S σ \sigma σ S S S
Two closure theorems (negative results)
Theorem 8.8 (Evertse closure). Let y 2 = f ( x ) y^2=f(x) y 2 = f ( x ) f ∈ Q [ x ] f\in\mathbb{Q}[x] f ∈ Q [ x ] deg f = 2 g + 1 \deg f=2g+1 deg f = 2 g + 1 S S S disc ( f ) = ∏ i < j ( r i − r j ) 2 \operatorname{disc}(f)=\prod_{i<j}(r_i-r_j)^2 disc ( f ) = ∏ i < j ( r i − r j ) 2 S S S r 1 = 0 r_1=0 r 1 = 0 r 2 = 1 r_2=1 r 2 = 1 r 3 = c r_3=c r 3 = c k ≥ 4 k\geq4 k ≥ 4 r k r_k r k x = r k c x=\frac{r_k}{c} x = c r k y = − r k − c c y=-\frac{r_k-c}{c} y = − c r k − c x + y = 1 x+y=1 x + y = 1 x , y x,y x , y S S S [6 ] (x + y = 1 x+y=1 x + y = 1 S S S N S N_S N S ∣ S ∣ |S| ∣ S ∣
n ≤ 3 + N S , g ≤ N S + 2 2 . n\leq3+N_S,\qquad g\leq\frac{N_S+2}{2}. n ≤ 3 + N S , g ≤ 2 N S + 2 . Totally rational branch-point hyperelliptic curves cannot form a tower with g → ∞ g\to\infty g → ∞
Theorem 8.9 (Belyi/PU route closed). More generally, if the branch points α i \alpha_i α i S S S S S S ⇒ \Rightarrow ⇒ S S S ⇒ \Rightarrow ⇒ S S S ⇒ \Rightarrow ⇒ ⇒ g ↛ ∞ \Rightarrow g\not\to\infty ⇒ g → ∞
The numerical side is likewise negative. After correcting the root-collision count (⌊ n / 3 ⌋ \lfloor n/3\rfloor ⌊ n /3 ⌋ p = 3 p=3 p = 3 n = 5 , 7 , 9 , 11 n=5,7,9,11 n = 5 , 7 , 9 , 11 3.5 × 10 7 3.5\times10^7 3.5 × 1 0 7 n = 11 n=11 n = 11 S S S Θ ( w ) / w ≈ 0.28 \Theta(w)/w\approx0.28 Θ ( w ) / w ≈ 0.28
Corollary 8.10 (Sharpening of the search space). A candidate tower for the discriminant channel must satisfy at least one of:
Table 8.10.
All known towers (Fermat, X 0 ( p n ) X_0(p^n) X 0 ( p n ) σ → 1 \sigma\to1 σ → 1
Cost (honest label). Once one leaves the hyperelliptic world, there is no integral-point/branch-point formula to compute, and the decomposition ω ˉ 2 Δ D e l ∼ ∏ ( α i − α j ) 2 \bar{\omega}^2\Delta_{\mathrm{Del}}\sim\prod(\alpha_i-\alpha_j)^2 ω ˉ 2 Δ Del ∼ ∏ ( α i − α j ) 2
:::
Unconditional exclusion of on Fermat/wild towers
Theorem 8.11. On Fermat/wild CM towers, κ → ∞ \kappa\to\infty κ → ∞
Proof. By Proposition 2.3 , κ = σ ( 1 + R ) 12 + d 12 \kappa=\frac{\sigma(1+R)}{12}+\frac{d}{12} κ = 12 σ ( 1 + R ) + 12 d R ≥ 0 R\geq0 R ≥ 0 d d d δ F a l > − W g \delta_{\mathrm{Fal}}>-Wg δ Fal > − W g W = 2 log ( 2 π 4 ) W=2\log(2\pi^4) W = 2 log ( 2 π 4 ) d = δ F a l / ( g f ^ ) > − W / f ^ d=\delta_{\mathrm{Fal}}/(g\hat{f})>-W/\hat{f} d = δ Fal / ( g f ^ ) > − W / f ^
κ ≥ σ 12 − W 12 f ^ . \kappa\geq\frac{\sigma}{12}-\frac{W}{12\hat{f}}. κ ≥ 12 σ − 12 f ^ W . By (T3) f ^ → ∞ \hat{f}\to\infty f ^ → ∞ κ ≥ σ 12 − o ( 1 ) \kappa\geq\frac{\sigma}{12}-o(1) κ ≥ 12 σ − o ( 1 ) σ → ∞ \sigma\to\infty σ → ∞ κ → ∞ \kappa\to\infty κ → ∞ d ≤ C σ + o ( 1 ) d\leq C\sigma+o(1) d ≤ C σ + o ( 1 ) κ F e r m a t = 1 8 \kappa_{\mathrm{Fermat}}=\frac{1}{8} κ Fermat = 8 1 σ ≤ 12 κ + o ( 1 ) = 3 2 + o ( 1 ) \sigma\leq12\kappa+o(1)=\frac{3}{2}+o(1) σ ≤ 12 κ + o ( 1 ) = 2 3 + o ( 1 ) σ \sigma σ
Remark 8.12 (Scope). This covers Fermat/CM towers; an upper bound on σ \sigma σ § 10 ).
The curvature channel
By Proposition 2.3 and Theorem 8.7 ,
no κ → ∞ ⟺ σ = O ( 1 ) & R = O ( 1 ) , \text{no }\kappa\to\infty\Longleftrightarrow\sigma= O(1)\ \&\ R = O(1), no κ → ∞ ⟺ σ = O ( 1 ) & R = O ( 1 ) , i.e. Szpiro and arithmetic Bogomolov control the two factors separately. This section treats the second factor, R = O ( 1 ) R = O(1) R = O ( 1 )
An unconditional reduction theorem
Theorem 9.1.
ω ˉ 2 ≤ 12 h F + 2 log ( 2 π 4 ) g = 12 h F + 10.544133 g . \bar{\omega}^{2} \le12h_{F} + 2\log(2\pi^{4})g = 12h_{F} + 10.544133g. ω ˉ 2 ≤ 12 h F + 2 log ( 2 π 4 ) g = 12 h F + 10.544133 g . Three inputs (all theorems). 1. Noether (Wilms): 12 h F = ω ˉ 2 + Δ D e l + δ F a l 12h_{F} = \bar{\omega}^{2} + \Delta_{\mathrm{Del}} + \delta_{\mathrm{Fal}} 12 h F = ω ˉ 2 + Δ Del + δ Fal Δ D e l ≥ 0 \Delta_{\mathrm{Del}} \ge0 Δ Del ≥ 0 ≥ 0 \ge0 ≥ 0 [21 Cor. 1.2] : for every compact connected Riemann surface X X X g ≥ 1 g \ge1 g ≥ 1
δ F a l ( X ) > − 2 g log ( 2 π 4 ) , δ F a l = − 24 H + 2 φ − 8 g log 2 π . \delta_{\mathrm{Fal}}(X) > -2g\log(2\pi^{4}), \qquad\delta_{\mathrm{Fal}} = -24H + 2\varphi- 8g\log2\pi. δ Fal ( X ) > − 2 g log ( 2 π 4 ) , δ Fal = − 24 H + 2 φ − 8 g log 2 π . Derivation: solve Noether for ω ˉ 2 = 12 h F − Δ D e l − δ F a l \bar{\omega}^{2} = 12h_{F} - \Delta_{\mathrm{Del}} - \delta_{\mathrm{Fal}} ω ˉ 2 = 12 h F − Δ Del − δ Fal Δ D e l ≥ 0 \Delta_{\mathrm{Del}} \ge0 Δ Del ≥ 0 − Δ D e l ≤ 0 -\Delta_{\mathrm{Del}} \le0 − Δ Del ≤ 0 − δ F a l < 2 g log ( 2 π 4 ) -\delta_{\mathrm{Fal}} < 2g\log(2\pi^{4}) − δ Fal < 2 g log ( 2 π 4 )
Corollary 9.2 (Reduction).
F a l t i n g s – S h a f a r e v i c h ( h F = O S ( g ) ) ⟹ a r i t h m e t i c B o g o m o l o v ( ω ˉ 2 = O ( g ) ) . \mathit{Faltings\text{–}Shafarevich}\ (h_{F} = O_{S}(g)) \Longrightarrow\mathit{arithmetic\ Bogomolov}\ (\bar{\omega}^{2} = O(g)). Faltings – Shafarevich ( h F = O S ( g )) ⟹ arithmetic Bogomolov ( ω ˉ 2 = O ( g )) . Meaning: the task “bound ω ˉ 2 \bar{\omega}^{2} ω ˉ 2 h F h_{F} h F
A necessary correction
A statement made in three rounds — “fixed bad set S S S ω ˉ 2 ≤ C ( S ) g \bar{\omega}^{2} \le C(S)g ω ˉ 2 ≤ C ( S ) g S = { p } S = \{p\} S = { p } [5 ] give ω ˉ 2 ∼ 3 g log N \bar{\omega}^{2} \sim3g\log N ω ˉ 2 ∼ 3 g log N X 0 ( p n ) X_{0}(p^{n}) X 0 ( p n ) ω ˉ 2 → ∞ \bar{\omega}^{2} \to\infty ω ˉ 2 → ∞ O ( 1 ) O(1) O ( 1 )
S fixed & f ^ = log cond g = O ( 1 ) ⟹ ω ˉ 2 = O S ( g ) . S\ \text{fixed}\ \&\ \hat{f} = \frac{\log\operatorname{cond}}{g} = O(1) \Longrightarrow\bar{\omega}^{2} = O_{S}(g). S fixed & f ^ = g log cond = O ( 1 ) ⟹ ω ˉ 2 = O S ( g ) . Since f ^ = O ( 1 ) \hat{f} = O(1) f ^ = O ( 1 ) κ \kappa κ
A non-circular cross-check
Solving Proposition 2.3 against (L3) (write f ^ = log cond / g \hat{f} = \log\operatorname{cond}/g f ^ = log cond / g
R < 12 κ σ − 1 + 2 log ( 2 π 4 ) σ f ^ . R < \frac{12\kappa}{\sigma} - 1 + \frac{2\log(2\pi^{4})}{\sigma\hat{f}}. R < σ 12 κ − 1 + σ f ^ 2 log ( 2 π 4 ) . Substituting the Fermat values κ = 1 8 \kappa= \frac{1}{8} κ = 8 1 σ = 1 \sigma= 1 σ = 1 f ^ → ∞ \hat{f} \to\infty f ^ → ∞
R ⟶ 12 ⋅ 1 8 1 − 1 − 1 2 , R \longrightarrow\frac{12\cdot\frac{1}{8}}{1} - 1 - \frac{1}{2}, R ⟶ 1 12 ⋅ 8 1 − 1 − 2 1 , matching the independently obtained R ≤ 1 2 R \le\frac{1}{2} R ≤ 2 1 L L L R R R X 0 X_{0} X 0
Difference from a withdrawn “check”: an earlier version solved the identity for κ = 1 8 \kappa= \frac{1}{8} κ = 8 1 R R R R R R
The reverse direction is unreachable
From 12 h F = ω ˉ 2 + Δ D e l + δ F a l 12h_F=\bar{\omega}^{2}+\Delta_{\mathrm{Del}}+\delta_{\mathrm{Fal}} 12 h F = ω ˉ 2 + Δ Del + δ Fal ω ˉ 2 = O S ( g ) \bar{\omega}^{2}=O_S(g) ω ˉ 2 = O S ( g ) h F h_F h F [16 ] . We merely named the intermediate step. In the f ^ = O ( 1 ) \hat{f}=O(1) f ^ = O ( 1 ) κ → ∞ \kappa\to\infty κ → ∞ ω ˉ 2 = O ( g ) ⟺ R σ = O ( 1 ) \bar{\omega}^{2}=O(g)\Longleftrightarrow R_{\sigma}=O(1) ω ˉ 2 = O ( g ) ⟺ R σ = O ( 1 )
What survives
Arithmetic Bogomolov is strictly weaker than BMY: if σ = O ( 1 ) \sigma=O(1) σ = O ( 1 ) ω ˉ 2 ≤ 3 Δ D e l \bar{\omega}^{2}\leq3\Delta_{\mathrm{Del}} ω ˉ 2 ≤ 3 Δ Del 3 σ log cond 3\sigma\log\operatorname{cond} 3 σ log cond κ → ∞ \kappa\to\infty κ → ∞ R = O ( 1 ) R=O(1) R = O ( 1 )
Kept judgment: arithmetic Bogomolov R = O ( 1 ) R=O(1) R = O ( 1 )
Open problems and the function-field comparison
Function-field comparison: a missing pair of inputs
Table 1 (PDF p. 21).
The function field seals both channels with two independent theorems; the number field has neither. The precise form of “the function field has Mason, the number field only the abc conjecture” is: Mason is only one of the two.
On the spectral row: the interpretation of the Néron local height pairing via the Laplacian of the dual graph on semistable curves is established in the published literature (Werner [20 ] , Wei) for Mumford curves; a recent preprint extends it to arbitrary semistable reduction and attempts an identification with boundary two-point functions in p p p σ = # nodes / b 1 \sigma=\#\text{nodes}/b_1 σ = # nodes / b 1
Open problems
Problem 10.1 (General arithmetic BMY). Does ω ˉ 2 ≤ 3 Δ D e l + O ( g ) \bar{\omega}^{2}\leq3\Delta_{\mathrm{Del}}+O(g) ω ˉ 2 ≤ 3 Δ Del + O ( g ) σ → 1 \sigma\to1 σ → 1
Table 2 (PDF p. 21).
In every known class σ → 1 \sigma\to1 σ → 1 R → c R\to c R → c c c c
Problem 10.2. Can κ n → ∞ \kappa_n \to\infty κ n → ∞ σ → ∞ \sigma\to\infty σ → ∞ σ → 1 \sigma\to1 σ → 1 R ≤ 3 R \le3 R ≤ 3
Problem 10.3. Search space: the tower must either have non-totally-rational branch points or be non-hyperelliptic (Corollary 8.10); this condition is mutually exclusive with computability.
Problem 10.4 (g = 1 g = 1 g = 1 ∣ c 4 ∣ ≥ 10 47 |c_4| \ge10^{47} ∣ c 4 ∣ ≥ 1 0 47 ∼ 10 \sim10 ∼ 10 m = 4 m = 4 m = 4 deg f = 8 \deg f = 8 deg f = 8 deg g = 12 \deg g = 12 deg g = 12 deg h = 5 = m + 1 \deg h = 5 = m + 1 deg h = 5 = m + 1 C ( 4 ) = 5 m − 1 m + 1 = 19 5 = 3.8 C(4) = \frac{5m-1}{m+1} = \frac{19}{5} = 3.8 C ( 4 ) = m + 1 5 m − 1 = 5 19 = 3.8 δ F a l / Δ D e l \delta_{\mathrm{Fal}}/\Delta_{\mathrm{Del}} δ Fal / Δ Del − 6 log log -6\log\log − 6 log log t 0 ∼ 10 16.4 t_0 \sim10^{16.4} t 0 ∼ 1 0 16.4 log ∣ Δ min ∣ ≈ 218 \log|\Delta_{\min}| \approx218 log ∣ Δ m i n ∣ ≈ 218 t 0 ∼ 10 50 t_0 \sim10^{50} t 0 ∼ 1 0 50
Problem 10.5 (Siegel/L-value channel). In all g > 1 g > 1 g > 1
Problem 10.6 (Mestre’s hypothesis). Mestre’s conductor bound f ≥ log B M f \ge\log B_M f ≥ log B M L ( A , s ) L(A,s) L ( A , s )
Problem 10.7 (Normalization tension). Tension between the two normalizations of h F h_F h F g = 1 g = 1 g = 1
Problem 10.8 (Curve realization of convexity towers). Single Jacobian realization of the interior filling: the abelian-variety version is complete (Theorem 7.8 ).
Problem 10.9. Towers neither non-hyperelliptic nor Belyi: no attainable construction currently.
Problem 10.10 (E19: resolved, negatively). Partially ramified orbits with B ˉ > 0 \bar{B} > 0 B ˉ > 0 B ≪ g B \ll g B ≪ g B ˉ > 0 \bar{B} > 0 B ˉ > 0 B ˉ / g → 0 \bar{B}/g \to0 B ˉ / g → 0 ν p / B ˉ → ∞ \nu_p/\bar{B} \to\infty ν p / B ˉ → ∞ B ˉ = 1 \bar{B} = 1 B ˉ = 1 ν p = 1 + 4 t \nu_p = 1 + 4t ν p = 1 + 4 t g = 4 , 5 , 7 , 10 , 13 g = 4,5,7,10,13 g = 4 , 5 , 7 , 10 , 13 h m o d p h \bmod p h mod p ν p = d − 1 = B ˉ \nu_p = d - 1 = \bar{B} ν p = d − 1 = B ˉ p p p ∣ S ∣ |S| ∣ S ∣ → \to → ν p / B ˉ \nu_p/\bar{B} ν p / B ˉ σ \sigma σ
Problem 10.11 (Independent proof of (L1)). An independent first-principles proof of (L1) for arbitrary genus: Liu 1994 Prop. 1 is commonly cited for arbitrary genus, but no explicit independent proof has been found (Remark 2.5 ).
Problem 10.12 (Single-curve realization of the interior filling). Does there exist a curve tower (single Jacobians, as in Definition 1.1) whose κ \kappa κ ( 1 12 , 1 3 ) \left(\frac{1}{12},\frac{1}{3}\right) ( 12 1 , 3 1 ) Theorem 7.8 ); the curve category is open.
The final form of the discriminant channel: thickness language and an explicit bound
Thickness formula. Let X s t X^{\mathrm{st}} X st i i i x y = π N i xy = \pi^{N_i} x y = π N i N i N_i N i N i − 1 N_i - 1 N i − 1 A 1 \mathbb{A}^1 A 1
( D ) ⟺ ν f → ∞ and ∑ p S w p g → ∞ (7) (\mathrm{D}) \Longleftrightarrow\frac{\nu}{f} \to\infty\ \text{and}\ \frac{\sum_p \mathrm{Sw}_p}{g} \to\infty \tag*{(7)} ( D ) ⟺ f ν → ∞ and g ∑ p Sw p → ∞ (7) (the second condition is exactly (T3), by the wild-necessity lemma). Thickness alone is insufficient: the P 1 \mathrm{P}^{1} P 1 ∑ i N i \sum_i N_i ∑ i N i f ^ = log p = O ( 1 ) \hat{f}=\log p=O(1) f ^ = log p = O ( 1 ) b 1 = # n o d e s b_1=\#\mathrm{nodes} b 1 = # nodes m = m s t + ∑ i ( N i − 1 ) m=m^{\mathrm{st}}+\sum_i(N_i-1) m = m st + ∑ i ( N i − 1 ) ν p = ∑ i N i \nu_p=\sum_i N_i ν p = ∑ i N i f p = b 1 ( Γ p ) + S w p f_p=b_1(\Gamma_p)+\mathrm{Sw}_p f p = b 1 ( Γ p ) + Sw p n n n 1 1 1 ν = n \nu=n ν = n n n n ν = n \nu=n ν = n # V ≤ 2 g − 2 \#V\leq2g-2 # V ≤ 2 g − 2 # V ≫ g \#V\gg g # V ≫ g
An extremal candidate passing Noether. Take the stable fiber to be a P 1 \mathrm{P}^{1} P 1 g g g N 1 , … , N g N_1,\ldots,N_g N 1 , … , N g f = b 1 = g f=b_1=g f = b 1 = g S w p = 0 \mathrm{Sw}_p=0 Sw p = 0 ν = ∑ i N i \nu=\sum_i N_i ν = ∑ i N i σ = ∑ i N i g \sigma=\frac{\sum_i N_i}{g} σ = g ∑ i N i N i N_i N i # V = 1 \#V=1 # V = 1 B = g B=g B = g L = 2 B − 2 = 2 g − 2 L=2B-2=2g-2 L = 2 B − 2 = 2 g − 2 F ⋅ b 1 = 1 F\cdot b_1=1 F ⋅ b 1 = 1 ν = B ( L + 1 ) \nu=B(L+1) ν = B ( L + 1 ) σ = L + 1 \sigma=L+1 σ = L + 1 N i N_i N i N i = O ( 1 ) N_i=O(1) N i = O ( 1 ) σ → 1 \sigma\to1 σ → 1 § 2.4 .
Semistability hypothesis. The thickness identity ν p = ∑ i N i \nu_p=\sum_i N_i ν p = ∑ i N i f p = b 1 + S w p f_p=b_1+\mathrm{Sw}_p f p = b 1 + Sw p dim ( V / V I p ) = t \dim(V/V^{I_p})=t dim ( V / V I p ) = t Δ D e l , p = f p + m p − 1 \Delta_{\mathrm{Del},p}=f_p+m_p-1 Δ Del , p = f p + m p − 1 X 1 ( p ) X_1(p) X 1 ( p ) p > 7 p>7 p > 7 [28 ] , Thm. 1.1.2), so m p = 1 m_p=1 m p = 1 Δ D e l , p = f p > 0 \Delta_{\mathrm{Del},p}=f_p>0 Δ Del , p = f p > 0 ν p = 0 \nu_p=0 ν p = 0
A new mechanism specific to g ≥ 2 g\geq2 g ≥ 2 For g = 1 g=1 g = 1 b 1 = 1 b_1=1 b 1 = 1 n n n b 1 = n b_1=n b 1 = n f p ≥ 1 f_p\geq1 f p ≥ 1 g ≥ 2 g\geq2 g ≥ 2 b 1 = 0 b_1=0 b 1 = 0 f p = 0 f_p=0 f p = 0 p p p ν p = m p − 1 > 0 \nu_p=m_p-1>0 ν p = m p − 1 > 0
Theorem 10.13. Write f ^ = log cond / g \hat{f}=\log\operatorname{cond}/g f ^ = log cond / g 12 κ = Ω + σ + d 12\kappa=\Omega+\sigma+d 12 κ = Ω + σ + d Ω ≥ 0 \Omega\geq0 Ω ≥ 0 δ F a l > − W g \delta_{\mathrm{Fal}}>-Wg δ Fal > − W g W = 2 log ( 2 π 4 ) = 10.544133 W=2\log(2\pi^{4})=10.544133 W = 2 log ( 2 π 4 ) = 10.544133 f ^ ≥ log B M \hat{f}\geq\log B_M f ^ ≥ log B M
d = δ F a l g f ^ ≥ − W log B M ⇒ σ ≤ 12 κ + W log B M . d=\frac{\delta_{\mathrm{Fal}}}{g\hat{f}}\geq-\frac{W}{\log B_M}\quad\Rightarrow\quad\sigma\leq12\kappa+\frac{W}{\log B_M}. d = g f ^ δ Fal ≥ − log B M W ⇒ σ ≤ 12 κ + log B M W . With Mestre’s bound f ^ ≥ log B M \hat{f}\geq\log B_M f ^ ≥ log B M log B M = 2.334374 \log B_M=2.334374 log B M = 2.334374 B M ≈ 10.323 B_M\approx10.323 B M ≈ 10.323 σ ≤ 12 κ + W log B M = 12 κ + 4.5169 \sigma\leq12\kappa+\frac{W}{\log B_M}=12\kappa+4.5169 σ ≤ 12 κ + l o g B M W = 12 κ + 4.5169 G L 2 \mathrm{GL}_2 GL 2 σ ≤ 8.5169 \sigma\leq8.5169 σ ≤ 8.5169 σ ≤ 4 \sigma\leq4 σ ≤ 4 f ^ → ∞ \hat{f}\to\infty f ^ → ∞ Notation warning: 2.334374 is log B M \log B_M log B M B M B_M B M
Channel (D) is closed for all G L 2 \mathrm{GL}_2 GL 2 § 2.4 .
The third input is conditional. Mestre’s lower bound requires, per four-source verification, that L ( J , s ) L(J,s) L ( J , s ) L L L [18 ] (assumption added in v2/v3), Mestre [24 ] (zbMATH abstract), Kowalski (MO #10860), and the Persiflage blog. Version-level evidence: the paper’s v1 abstract read “Any abelian variety over Q \mathbb{Q} Q g g g 10 0.323 g 10^{0.323g} 1 0 0.323 g
The structural mismatch between tools and loopholes
This is the last and methodologically most important observation of the paper.
What Mestre covers (automorphic towers) is exactly the class of the five anchors of § 2.4 , where Theorem 10.13 already gives σ < 8.5169 \sigma<8.5169 σ < 8.5169 f ^ ≥ O ( 1 ) \hat{f} \ge O(1) f ^ ≥ O ( 1 ) f ^ < 1 \hat{f} < 1 f ^ < 1 f ^ → 0 \hat{f} \to0 f ^ → 0 d = δ F a l / ( g f ^ ) → − ∞ d = \delta_{\mathrm{Fal}}/(g\hat{f}) \to-\infty d = δ Fal / ( g f ^ ) → − ∞
The inability to rule out κ → ∞ \kappa\to\infty κ → ∞
A theorem-level closure: cyclotomic towers
Let p p p { 2 , p } \{2,p\} { 2 , p }
Cluster picture and two exact quantities. C m : y 2 = x p m + 1 C_m : y^2 = x^{p^m} + 1 C m : y 2 = x p m + 1 g m = p m − 1 2 g_m = \frac{p^m-1}{2} g m = 2 p m − 1 { − ζ p m k } \{-\zeta_{p^m}^k\} { − ζ p m k } v ( 1 − ζ p m ) = 1 p m − 1 ( p − 1 ) v(1-\zeta_{p^m}) = \frac{1}{p^{m-1}(p-1)} v ( 1 − ζ p m ) = p m − 1 ( p − 1 ) 1 v v v p s p^s p s p m − s p^{m-s} p m − s s = 1 , … , m s = 1,\ldots,m s = 1 , … , m 1 p s − 1 ( p − 1 ) \frac{1}{p^{s-1}(p-1)} p s − 1 ( p − 1 ) 1 p p p ⇒ \Rightarrow ⇒ ⇒ \Rightarrow ⇒ ⇒ \Rightarrow ⇒ t = 0 t = 0 t = 0
f p = 2 g + Sw p , v p ( Δ ) = m p m , f_p = 2g + \operatorname{Sw}_p,\qquad v_p(\Delta) = mp^m, f p = 2 g + Sw p , v p ( Δ ) = m p m , i.e. the tower is potentially good, and Sw p \operatorname{Sw}_p Sw p
Conductor lower bound, verified. By the explicit form of the conductor formula for induced representations, a ( Ind ψ ) = val ( d L / k ) dim ψ + f L / k a ( ψ ) a(\operatorname{Ind}\psi) = \operatorname{val}(d_{L/k})\dim\psi+ f_{L/k}a(\psi) a ( Ind ψ ) = val ( d L / k ) dim ψ + f L / k a ( ψ ) ∣ disc Q ( μ n ) ∣ ⋅ N K / Q ( m ) |\operatorname{disc}\mathbb{Q}(\mu_n)|\cdot N_{K/\mathbb{Q}}(\mathfrak{m}) ∣ disc Q ( μ n ) ∣ ⋅ N K / Q ( m ) [3 ] gives the formula, and Rohrlich [31 ] proves f ∣ p 2 f \mid p^2 f ∣ p 2 c j ≤ 2 c_j \le2 c j ≤ 2 v p ( d Q ( ζ p j ) ) = p j − 1 ( j ( p − 1 ) − 1 ) v_p(d_{\mathbb{Q}(\zeta_{p^j})}) = p^{j-1}(j(p-1)-1) v p ( d Q ( ζ p j ) ) = p j − 1 ( j ( p − 1 ) − 1 ) B ‾ \overline{B} B Q ( ζ p j ) \mathbb{Q}(\zeta_{p^j}) Q ( ζ p j ) [9 ] prove each simple factor nondegenerate; Cain’s formula uses exactly the discriminant of Q ( ζ p j ) \mathbb{Q}(\zeta_{p^j}) Q ( ζ p j )
f p ≥ S m : = ∑ j = 1 m v p ( d Q ( ζ p j ) ) = p m ( m ( p − 1 ) − 2 ) + 2 p − 1 , f_p \ge S_m := \sum_{j=1}^{m} v_p(d_{\mathbb{Q}(\zeta_{p^j})}) = \frac{p^m\bigl(m(p-1)-2\bigr)+2}{p-1}, f p ≥ S m := j = 1 ∑ m v p ( d Q ( ζ p j ) ) = p − 1 p m ( m ( p − 1 ) − 2 ) + 2 , S m ≤ f p ≤ S m + 2 m ⟹ m p m S m + 2 m ≤ σ p ≤ m p m S m , S_m \le f_p \le S_m + 2m \Longrightarrow\frac{mp^m}{S_m+2m} \le\sigma_p \le\frac{mp^m}{S_m}, S m ≤ f p ≤ S m + 2 m ⟹ S m + 2 m m p m ≤ σ p ≤ S m m p m , both ends tending to 1. Quantitatively (E42):
σ p ≤ m p m ( p − 1 ) p m ( m ( p − 1 ) − 2 ) + 2 ⟶ 1 + 2 m ( p − 1 ) − 2 ( ⟶ m m − 1 at p = 3 ) , \sigma_p \le\frac{mp^m(p-1)}{p^m\bigl(m(p-1)-2\bigr)+2} \longrightarrow1+\frac{2}{m(p-1)-2}\left(\longrightarrow\frac{m}{m-1}\text{ at }p=3\right), σ p ≤ p m ( m ( p − 1 ) − 2 ) + 2 m p m ( p − 1 ) ⟶ 1 + m ( p − 1 ) − 2 2 ( ⟶ m − 1 m at p = 3 ) , and S m / ( 2 g m ) → m − 2 p − 1 S_m/(2g_m) \to m-\frac{2}{p-1} S m / ( 2 g m ) → m − p − 1 2 g m = p m − 1 2 g_m = \frac{p^m-1}{2} g m = 2 p m − 1
Numerics: for p = 3 p=3 p = 3 σ < 1.4727 , 1.3279 , 1.1997 \sigma< 1.4727, 1.3279, 1.1997 σ < 1.4727 , 1.3279 , 1.1997 m = 3 , 4 , 6 m=3,4,6 m = 3 , 4 , 6 m m − 1 = 1.5 , 1.333 , 1.200 \frac{m}{m-1}=1.5, 1.333, 1.200 m − 1 m = 1.5 , 1.333 , 1.200 q = 2 q=2 q = 2 v 2 = 4 g ∼ 4 log 2 v_2 = 4g \sim4\log2 v 2 = 4 g ∼ 4 log 2 o ( m p m ) o(mp^m) o ( m p m ) σ ≥ 1 \sigma\ge1 σ ≥ 1
σ → 1 (theorem level, no estimate needed) \sigma\to1\qquad\text{(theorem level, no estimate needed)} σ → 1 (theorem level, no estimate needed) Meaning: Theorem 10.13 gives only a constant upper bound on σ \sigma σ § 10.4 : there is more than one rescue mechanism.
Honest labels. Cain’s formula is for a single Fermat factor; the decomposition of f f f c j ≤ 2 c_j \le2 c j ≤ 2 n = 2 n=2 n = 2 c j ≤ 2 c_j \le2 c j ≤ 2
An unconditional conductor bound, and why it still does not suffice
The Brumer–Kramer bound. Let K K K p p p p p p h ( x ) ∈ K [ x ] h(x) \in K[x] h ( x ) ∈ K [ x ] deg h = 2 g + 1 \deg h = 2g+1 deg h = 2 g + 1 B = Jac ( y 2 = h ) B = \operatorname{Jac}(y^2=h) B = Jac ( y 2 = h ) h = h 1 ⋯ h r h=h_1\cdots h_r h = h 1 ⋯ h r M i = K ( θ i ) M_i=K(\theta_i) M i = K ( θ i ) [1 Lemma 6.3]
f p ( B / K ) ≥ B ˉ : = ∑ i = 1 r v K ( d M i / K ) f M i / K . f_p(B/K) \ge\bar{B} := \sum_{i=1}^{r} v_K(\mathfrak{d}_{M_i/K}) f_{M_i/K}. f p ( B / K ) ≥ B ˉ := i = 1 ∑ r v K ( d M i / K ) f M i / K . with equality when h h h B [ 2 ] B[2] B [ 2 ] L L L h h h K K K p p p y 2 = h y^2=h y 2 = h deg h \deg h deg h
Comparison with § 10.5 . B ˉ = S m \bar{B}=S_m B ˉ = S m y 2 = x p m + 1 y^2=x^{p^m}+1 y 2 = x p m + 1
x p m + 1 = ( x + 1 ) ∏ j = 1 m Φ 2 p j ( x ) , M j = Q p ( ζ p j ) , x^{p^m}+1=(x+1)\prod_{j=1}^{m}\Phi_{2p^j}(x), \qquad M_j=\mathbb{Q}_p(\zeta_{p^j}), x p m + 1 = ( x + 1 ) j = 1 ∏ m Φ 2 p j ( x ) , M j = Q p ( ζ p j ) , totally ramified, v p ( d M j / Q p ) = p j − 1 ( j ( p − 1 ) − 1 ) v_p(\mathfrak{d}_{M_j/\mathbb{Q}_p})=p^{j-1}(j(p-1)-1) v p ( d M j / Q p ) = p j − 1 ( j ( p − 1 ) − 1 )
B ˉ = ∑ j p j − 1 ( j ( p − 1 ) − 1 ) = S m . \bar{B}=\sum_j p^{j-1}(j(p-1)-1)=S_m. B ˉ = j ∑ p j − 1 ( j ( p − 1 ) − 1 ) = S m . Identical to § 10.5 place by place. But this route needs neither Cain, Rohrlich, nor Weil, nor reflex-field arguments — the risk drops from “a four-section corollary chain” to “one standard 1994 lemma”. Thus σ → 1 \sigma\to1 σ → 1
Counterexample: B ˉ \bar{B} B ˉ u = 2 u=2 u = 2 F 3 \mathbb{F}_3 F 3 p = 3 p=3 p = 3
h = ( x 2 − u ) ( x 2 − 2 p m x + ( p 2 m − u ) ) ( x − 1 ) . h=(x^2-u)(x^2-2p^m x+(p^{2m}-u))(x-1). h = ( x 2 − u ) ( x 2 − 2 p m x + ( p 2 m − u )) ( x − 1 ) . The two quadratic orbits have discriminants 4 u = 8 4u=8 4 u = 8 4 u = 8 4u=8 4 u = 8 B ˉ = 0 \bar{B}=0 B ˉ = 0 3 3 3 h 1 ≡ h 2 ≡ x 2 + 1 h_1\equiv h_2\equiv x^2+1 h 1 ≡ h 2 ≡ x 2 + 1 p = 3 p=3 p = 3
The criterion is clean: every M i / K M_i/K M i / K ⇒ B ˉ = 0 \Rightarrow\bar{B}=0 ⇒ B ˉ = 0
h = ∏ i = 1 s ( x 2 − u i ) ∏ i = 1 s ( x 2 − 2 p m x + p 2 m − u i ) ( x − c ) , g = 2 s , ν p = 2 g m , B ˉ = 0 , h=\prod_{i=1}^{s}(x^2-u_i)\prod_{i=1}^{s}(x^2-2p^m x+p^{2m}-u_i)(x-c),\qquad g=2s,\quad\nu_p=2gm,\quad\bar{B}=0, h = i = 1 ∏ s ( x 2 − u i ) i = 1 ∏ s ( x 2 − 2 p m x + p 2 m − u i ) ( x − c ) , g = 2 s , ν p = 2 g m , B ˉ = 0 , n ≍ g ⇒ ν p ≍ g 2 n\asymp g\Rightarrow\nu_p\asymp g^2 n ≍ g ⇒ ν p ≍ g 2 p = 5 p=5 p = 5 s = 2 s=2 s = 2 g = 4 g=4 g = 4 m = 1 , 2 , 3 m=1,2,3 m = 1 , 2 , 3 ν 5 = 8 , 16 , 24 \nu_5=8,16,24 ν 5 = 8 , 16 , 24
Neighboring–bad-set tension. Why does this not save (D)? Because Res = p 2 m ( p 2 m − 4 u i ) \operatorname{Res}=p^{2m}(p^{2m}-4u_i) Res = p 2 m ( p 2 m − 4 u i ) u i u_i u i p p p p 2 m − 4 u i ≡ − 4 u i ≢ 0 ( m o d p ) p^{2m}-4u_i\equiv-4u_i\not\equiv0\pmod p p 2 m − 4 u i ≡ − 4 u i ≡ 0 ( mod p ) p p p p = 7 p=7 p = 7 m = 3 m=3 m = 3 p 2 m − 4 u = 117641 = 47 × 2503 p^{2m}-4u=117641=47\times2503 p 2 m − 4 u = 117641 = 47 × 2503 p , m p,m p , m u u u n → ∞ n\to\infty n → ∞ p 2 n − 4 u p^{2n}-4u p 2 n − 4 u u u u m m m u u u
Table 11.
Cases (i) and (ii) are completely different mechanisms (S-unit equations vs. conductor-bound degeneration), yet are captured by the same criterion B ˉ = 0 \bar{B}=0 B ˉ = 0
Three comparison families: BK is region-optimal, not universal:
Table 12 (PDF p. 26).
Verdict: the gap is moved, not closed. BK does not close the gap of § 10.4 ; it only moves it from (D) into a weak version: for hyperelliptic Jacobians, case (iii) now has an unconditional bound independent of Mestre; but case (ii) shows the bound can vanish at genuine bad primes, and then the bad set must grow, so the conclusion retreats to “a weak version with growing bad set” rather than the original problem. The remaining exposure is Problem 10.10.
Honest labels. The three hard restrictions of [1 Lemma 6.3] (p p p y 2 = h y^2=h y 2 = h deg h \deg h deg h ν p = 2 g m \nu_p=2gm ν p = 2 g m s = 2 s=2 s = 2 p p p F 3 \mathbb{F}_3 F 3 u i u_i u i s ≥ 3 s\geq3 s ≥ 3 λ \lambda λ p = 3 p=3 p = 3 g = 40 g=40 g = 40
A meta-observation
Three “geometrically dangerous” instances are all rescued by arithmetic quantities, and each rescue covers only automorphic/CM towers:
Table 13 (PDF p. 26, second).
The geometric side (Bergman/systole) has zero effect on all three instances; the arithmetic side rescues three times, but all unconditionally only on automorphic towers. κ → ∞ \kappa\to\infty κ → ∞
A tension concerning BMY effective Mordell
BMY implies effective Mordell. If GRH ⇒ \Rightarrow ⇒ ⇒ \Rightarrow ⇒ ⇒ \Rightarrow ⇒ L L L
Appendix A. Falsification of an external framework: the rigidity-spectrum test
This appendix is independent of the main line, but records a complete falsifiability test, as a methodological specimen. It may be read separately or skipped.
Background
A recent phenomenological framework (self-described as “not a theorem, analogous to critical phenomena and renormalization group in physics”) postulates a universal set of dimensionless constants
L O W = 0.035 , B A N D = [ 0.62 , 0.71 ] , S A T ≈ 0.97 ∼ 1 , \mathrm{LOW}=0.035,\qquad\mathrm{BAND}=[0.62,0.71],\qquad\mathrm{SAT}\approx0.97\sim1, LOW = 0.035 , BAND = [ 0.62 , 0.71 ] , SAT ≈ 0.97 ∼ 1 , called the rigidity spectrum, and asserts that “most arithmetic problems fall in this band”, with the screening criterion “nonvanishing leading term of a functional equation at the critical point”. All five anchor towers of § 2.4 satisfy this screening criterion (all are standard automorphic/CM objects). Hence, if the framework is self-consistent, they should land in the band. Taking our κ \kappa κ
Pre-registration (written before computing)
Table 14.
Results
T 0 T_0 T 0 κ ∈ { 0.083 , 0.125 , 0.167 , 0.208 , 0.333 } \kappa\in\{0.083, 0.125, 0.167, 0.208, 0.333\} κ ∈ { 0.083 , 0.125 , 0.167 , 0.208 , 0.333 }
S 1 S_1 S 1
κ max κ min = 1 / 3 1 / 12 = 4 , 0.71 0.62 = 1.1452 , 4 > 1.145. \frac{\kappa_{\max}}{\kappa_{\min}}=\frac{1/3}{1/12}=4,\qquad\frac{0.71}{0.62}=1.1452,\qquad4>1.145. κ m i n κ m a x = 1/12 1/3 = 4 , 0.62 0.71 = 1.1452 , 4 > 1.145. Hence no c c c κ \kappa κ
S 2 S2 S 2 a + b κ a+b\kappa a + bκ ( a , b ) = ( 0.59 , 0.36 ) (a,b)=(0.59,0.36) ( a , b ) = ( 0.59 , 0.36 ) 0.59 + 0.36 ⋅ 1 12 = 0.62 0.59+0.36\cdot\frac{1}{12}=0.62 0.59 + 0.36 ⋅ 12 1 = 0.62 0.59 + 0.36 ⋅ 1 3 = 0.71 0.59+0.36\cdot\frac{1}{3}=0.71 0.59 + 0.36 ⋅ 3 1 = 0.71
T 1 T_1 T 1 T 4 T_4 T 4 3 κ ∈ { 0.25 , 0.375 , 0.5 , 0.625 , 1.0 } 3\kappa\in\{0.25,0.375,0.5,0.625,1.0\} 3 κ ∈ { 0.25 , 0.375 , 0.5 , 0.625 , 1.0 } [ 0.25 , 1 ] [0.25,1] [ 0.25 , 1 ] σ → 1 \sigma\to1 σ → 1 R ∈ { 0 , 1 2 , 1 , 3 2 , 3 } R\in\{0,\frac{1}{2},1,\frac{3}{2},3\} R ∈ { 0 , 2 1 , 1 , 2 3 , 3 }
Two side findings
Constant density. [ 0.62 , 0.71 ] [0.62,0.71] [ 0.62 , 0.71 ] ln 2 = 0.6931 \ln2=0.6931 ln 2 = 0.6931 1 − 1 / e = 0.6321 1-1/e=0.6321 1 − 1/ e = 0.6321 1 / 2 = 0.7071 1/\sqrt{2}=0.7071 1/ 2 = 0.7071 2 / 3 = 0.6667 2/3=0.6667 2/3 = 0.6667 1 / φ = 0.6180 1/\varphi=0.6180 1/ φ = 0.6180
The spectrum may not be independent. The framework states Φ T a t e ≈ 0.7 ∈ B A N D \Phi_{\mathrm{Tate}}\approx0.7\in\mathrm{BAND} Φ Tate ≈ 0.7 ∈ BAND Φ c u s p ≈ 0.07 \Phi_{\mathrm{cusp}}\approx0.07 Φ cusp ≈ 0.07 0.07 / 2 = 0.035 = L O W 0.07/2=0.035=\mathrm{LOW} 0.07/2 = 0.035 = LOW { Φ c u s p / 2 , Φ T a t e , 1 } \{\Phi_{\mathrm{cusp}}/2,\Phi_{\mathrm{Tate}},1\} { Φ cusp /2 , Φ Tate , 1 }
Multiple comparisons. 3 × 4 + 10 = 22 3\times4+10=22 3 × 4 + 10 = 22
Verdict and the disposition of this paper
Table 1 (PDF p. 28).
We do not merge the framework: the main line is theorem-driven, the framework is data-driven, their correctness standards differ, and merging would dilute the proved part. But two independent convergence points are retained as cross-checks: the framework’s “dangerous direction = bands with multiplicative parameter (−2)-chains” ↔ our independent identification of the (D) loophole (additive reduction + long (−2)-chains, § 10.3 ); and the framework’s “Ogg rewrite ⇒ Szpiro form” ↔ our independent discovery of the Ogg even-graph Euler characteristic \nu=#E , f = b 1 f=b_1 f = b 1 m=#V-1 (§ 8 ). Two programs, completely different methods, pointing to the same conclusion — such independent convergences are worth more than the framework’s own fits.
Final word (to the framework’s side): what it lacks is not more fitting but one a priori quantitative prediction about a new object — written down before measurement. Do it once, and the nature of the framework changes.
Honest labels. The band edges are taken from a paraphrase, not verified against the original verbatim; this test assumes κ \kappa κ
Appendix B. Revision history
This is the fourth version (2026-10-29). The third version (final, 2026-10-09) added §10.6 and E9, E10; the present revision incorporates a certified external revision list of 33 items (F1–F33), recorded as E16–E47 below and flagged inline, and upgrades the Jorgenson–Kramer input to [25 Thm. 5.6 and Rem. 5.8] . The second version (2026-10-08) added: § 3.2 (exact order of the remainder of the main chain; the first version’s Θ ( n log p ⋅ p − n ) \Theta(n\log p\cdot p^{-n}) Θ ( n log p ⋅ p − n ) § 10.6 (the Brumer–Kramer unconditional bound) and E9, E10 below. All corrections:
Table 15.
Appendix C. Verification and compilation note
Every conclusion in this paper has been reproduced place by place on at least one known anchor and cross-checked on a second, independent anchor; the anchors are the Fermat tower (κ = 1 8 \kappa= \frac{1}{8} κ = 8 1 X 0 ( N ) X_0(N) X 0 ( N ) κ = 1 3 \kappa= \frac{1}{3} κ = 3 1 j = 0 j = 0 j = 0 h F = − 1.667691 h_F = -1.667691 h F = − 1.667691 d = 163 d = 163 d = 163 h F = 0.231116 h_F = 0.231116 h F = 0.231116
