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

Abstract

We compute the stable Faltings height of the hyperelliptic curve $X_n:y^2=x^n-1$ for every odd integer $n\ge 3$ in terms of special values of Euler's gamma function. In particular, we prove the bounds $$-0.975n<h_{\mathrm{Fal}}(X_n)-\frac{n}{8}\log n<\frac{9}{64}n\log \log n-0.263n.$$ 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 $\frac{n}{8}\log n+\frac{9}{64}n\log \log n-0.136n$.

20 pages, 1 figure. Comments are welcome!