On the arithmetic of a family of twisted constant elliptic curves

Richard Griffon and Douglas Ulmer

Vol. 305 (2020), No. 2, 597–640

DOI: 10.2140/pjm.2020.305.597

Abstract

Let ${\mathbb{𝔽}}_r$ be a finite field of characteristic $p>3$. For any power $q$ of $p$, consider the elliptic curve $E=E_{q,r}$ defined by $y^2=x^3+t^q-t$ over $K={\mathbb{𝔽}}_r\left(t\right)$. We describe several arithmetic invariants of $E$ such as the rank of its Mordell–Weil group $E\left(K\right)$, the size of its Néron–Tate regulator Reg$\left(E\right)$, and the order of its Tate–Shafarevich group Ш$\left(E\right)$ (which we prove is finite). These invariants have radically different behaviors depending on the congruence class of $p$ modulo 6. For instance Ш$\left(E\right)$ either has trivial $p$-part or is a $p$-group. On the other hand, we show that the product $|$Ш$\left(E\right)|$ Reg$\left(E\right)$ has size comparable to $r^{q∕6}$ as $q\to \infty$, regardless of $p\left(mod6\right)$. Our approach relies on the BSD conjecture, an explicit expression for the $L$-function of $E$, and a geometric analysis of the Néron model of $E$.

Keywords

elliptic curves over function fields, Mordell–Weil rank, Néron–Tate regulator, Tate–Shafarevich group, L-function and BSD conjecture

Mathematical Subject Classification 2010

Primary: 11G05, 14J27

Secondary: 11G40, 11G99, 14G10, 14G99

Milestones

Received: 29 May 2019

Revised: 11 November 2019

Accepted: 13 November 2019

Published: 29 April 2020