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

TL;DR

This paper calculates the stable Faltings height of a family of hyperelliptic curves explicitly using special values of the gamma function, providing bounds and applications to CM abelian varieties.

Contribution

It provides an explicit formula for the Faltings height of $X_n$ in terms of gamma function values and establishes bounds with applications to CM abelian varieties.

Findings

Explicit formula for $h_{Fal}(X_n)$ in terms of gamma functions

Bounds on the Faltings height with logarithmic terms

Application to bounding heights of CM abelian varieties

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 $\frac{n}{8}\log n+\frac{9}{64}n\log\log n-0.136n$.