#### Vol. 305, No. 2, 2020

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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 $p\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}6\right)$. 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
##### Authors
 Richard Griffon Departement Mathematik und Informatik Universität Basel Spiegelgasse Basel Switzerland Douglas Ulmer Department of Mathematics University of Arizona Tucson, AZ 85721-0089 United States