Explicit L-functions and a Brauer-Siegel theorem for Hessian elliptic curves

TL;DR

This paper explicitly computes the L-functions of a family of Hessian elliptic curves over function fields using Jacobi sums and establishes an analogue of the Brauer-Siegel theorem describing their asymptotic behavior.

Contribution

It provides an explicit formula for the L-functions of Hessian elliptic curves over function fields and proves a Brauer-Siegel type result for their Tate-Shafarevich groups and regulators.

Findings

Explicit L-function expressions in terms of Jacobi sums.

Asymptotic growth estimates matching Brauer-Siegel predictions.

Finite Tate-Shafarevich groups for the studied curves.

Abstract

For a finite field $\mathbb{F}_q$ of characteristic $p\geq 5$ 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 in terms of Jacobi sums. Moreover, we deduce from this calculation that the curves $E_d$ satisfy an analogue of the Brauer-Siegel theorem. More precisely, we estimate the asymptotic growth of the product of the order of the Tate-Shafarevich group of $E_d$ (which is known to be finite) by its N\'eron-Tate regulator, in terms of the exponential differential height of $E_d$, as $d\to\infty$.