Analogue of the Brauer-Siegel theorem for Legendre elliptic curves

@article{Griffon2017AnalogueOT,
  title={Analogue of the Brauer-Siegel theorem for Legendre elliptic curves},
  author={Richard Griffon},
  journal={arXiv: Number Theory},
  year={2017}
}
  • Richard Griffon
  • Published 2017
  • Mathematics
  • arXiv: Number Theory
  • 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 $d\to\infty$. 
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