Analogue of the Brauer-Siegel theorem for Legendre elliptic curves

Abstract

We prove an analogue of the Brauer-Siegel theorem for the Legendre elliptic curves over ${\mathbb{F}}_q(t)$. More precisely, if $d$ is an integer coprime to $q$, we denote by $E_d$ the elliptic curve with model $y^2=x(x+1)(x+t^d)$ over $K={\mathbb{F}}_q(t)$. We give an asymptotic estimate of the product of the order of the Tate-Shafarevich group of $E_d$ (which is known to be finite) with its Néron-Tate regulator, in terms of the exponential differential height of $E_d$, as $d\to \infty$.

15 pages, comments welcome

References in corpus (3)

• Character formula for Weil representations in terms of Frobenius traces
• Character formula for conjugacy classes in a coset
• A Brauer-Siegel theorem for Fermat surfaces over finite fields

Cited by in corpus (3)

• On the Northcott property for special values of L-functions
• Bounds on special values of L-functions of elliptic curves in an Artin-Schreier family
• Explicit L-functions and a Brauer-Siegel theorem for Hessian elliptic curves