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On the Faltings height of the curve $y^2=x^n-1$

Robert Wilms

TL;DR

This work computes the stable Faltings height of the hyperelliptic family $X_n: y^2=x^n-1$ for odd $n\ge 3$ by decomposing contributions from finite and infinite places, yielding an explicit gamma-function formula and sharp bounds that reveal the leading term $\frac{n}{8}\log n$. The method combines cluster-picture analysis for finite places, hyperelliptic period matrices for archimedean places, and determinant evaluations to express $h_{\text{Fal}}(X_n)$ in terms of gamma values. The authors then apply Colmez's CM theory to cyclotomic CM-types, showing that Jacobians of $X_p$ (prime) have CM and obtaining an upper bound for the CM-type height $h_{\text{Fal}}(A_n)$. The results illuminate precise height behavior in a CM setting and provide tools to bound Faltings heights of CM abelian varieties arising from cyclotomic fields.

Abstract

We compute the stable Faltings height of the hyperelliptic curve $X_n\colon y^2=x^{n}-1$ for every odd integer $n\ge 3$ in terms of special values of Euler's gamma function. In particular, we prove the bounds $$-0.975n< h_{\mathrm{Fal}}(X_n)-\tfrac{n}{8}\log n<\tfrac{9}{64}n\log\log n-0.263n.$$ As an application, we bound the Faltings height of any abelian variety with complex multiplication by the canonical CM-type of the $n$-th cyclotomic field by $\frac{n}{8}\log n+\frac{9}{64}n\log\log n-0.136n$.

On the Faltings height of the curve $y^2=x^n-1$

TL;DR

This work computes the stable Faltings height of the hyperelliptic family for odd by decomposing contributions from finite and infinite places, yielding an explicit gamma-function formula and sharp bounds that reveal the leading term . The method combines cluster-picture analysis for finite places, hyperelliptic period matrices for archimedean places, and determinant evaluations to express in terms of gamma values. The authors then apply Colmez's CM theory to cyclotomic CM-types, showing that Jacobians of (prime) have CM and obtaining an upper bound for the CM-type height . The results illuminate precise height behavior in a CM setting and provide tools to bound Faltings heights of CM abelian varieties arising from cyclotomic fields.

Abstract

We compute the stable Faltings height of the hyperelliptic curve for every odd integer in terms of special values of Euler's gamma function. In particular, we prove the bounds As an application, we bound the Faltings height of any abelian variety with complex multiplication by the canonical CM-type of the -th cyclotomic field by .
Paper Structure (10 sections, 11 theorems, 112 equations, 1 figure)

This paper contains 10 sections, 11 theorems, 112 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The stable Faltings height
  3. Cluster pictures
  4. Contributions of finite places
  5. Hyperelliptic period matrices
  6. Contributions of infinite places
  7. Proof of Theorem \ref{['thm_faltings-height-hyperelliptic']}
  8. Bounds for products of gamma values
  9. Proof of Corollary \ref{['cor_bounds']}
  10. Complex multiplication by cyclotomic fields

Key Result

Theorem 1.1

Let $X_n$ be the hyperelliptic curve defined by the equation $y^2=x^n-1$ for any odd integer $n\ge 3$. We write $g=\frac{n-1}{2}$ for its genus. The stable Faltings height of $X_n$ is given by

Figures (1)

  • Figure 1: Plot of the points $(n,h_{\mathrm{Fal}}(X_n))$ for all odd $3\le n\le 6001$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Remark 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • proof : Proof of Lemma \ref{['lem_cluster']}
  • Lemma 4.1
  • proof
  • ...and 13 more