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arXiv · 2601.15271

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

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

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Robert Wilms. 2026-01-21. On the Faltings height of the curve $y^2=x^n-1$. https://arxiv.org/abs/2601.15271

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