# Analogue of the Brauer-Siegel theorem for Legendre elliptic curves

## TL;DR

This paper establishes an analogue of the Brauer-Siegel theorem for Legendre elliptic curves over function fields, providing asymptotic estimates for key arithmetic invariants as the parameter grows.

## Contribution

It introduces a new asymptotic relationship for the product of the Tate-Shafarevich group order and the Néron-Tate regulator for Legendre elliptic curves over $_q(t)$.

## Key findings

- Asymptotic estimate of the product of Tate-Shafarevich group order and regulator
- Explicit relation involving the exponential differential height of the curves
- Extension of Brauer-Siegel type results to a new class of 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 N\'eron-Tate regulator, in terms of the exponential differential height of $E_d$, as $d\to\infty$.