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Richard Griffon
Explicit $L$-functions and a Brauer–Siegel theorem for Hessian elliptic curves
Journal de théorie des nombres de Bordeaux, 30 no. 3 (2018), p. 1059-1084, doi: 10.5802/jtnb.1065
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Class. Math.: 11G05, 11G40, 14G10, 11F67, 11M38
Keywords: Elliptic curves over function fields, Explicit computation of $L$-functions, Special values of $L$-functions and BSD conjecture, Estimates of special values, Analogue of the Brauer–Siegel theorem.

Résumé - Abstract

For a finite field $\mathbb{F}_q$ of characteristic $p\ge 5$ and $K=\mathbb{F}_q(t)$, we consider the family of elliptic curves $E_d$ over $K$ given by $y^2+xy - t^dy=x^3$ for all integers $d$ coprime to $q$.

We provide an explicit expression for the $L$-functions of these curves. Moreover, we deduce from this calculation that the curves $E_d$ satisfy an analogue of the Brauer–Siegel theorem. Precisely, we show that, for $d\rightarrow \infty $ ranging over the integers coprime with $q$, one has

$$ \log \left(|\Sha(E_d/K)|\cdot \mathrm{Reg}(E_d/K)\right) \sim \log H(E_d/K), $$

where $H(E_d/K)$ denotes the exponential differential height of $E_d$, $\Sha(E_d/K)$ its Tate–Shafarevich group and $\mathrm{Reg}(E_d/K)$ its Néron–Tate regulator.

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