Analogue of the BrauerSiegel theorem for Legendre elliptic curves
Abstract
We prove an analogue of the BrauerSiegel 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 TateShafarevich group of $E_d$ (which is known to be finite) with its NéronTate regulator, in terms of the exponential differential height of $E_d$, as $d\to\infty$.
 Publication:

arXiv eprints
 Pub Date:
 June 2017
 arXiv:
 arXiv:1706.07728
 Bibcode:
 2017arXiv170607728G
 Keywords:

 Mathematics  Number Theory
 EPrint:
 15 pages, comments welcome