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 𝔽r be a finite field of characteristic p > 3. For any power q of p, consider the elliptic curve E = Eq,r defined by y2 = x3 + tq t over K = 𝔽r(t). We describe several arithmetic invariants of E such as the rank of its Mordell–Weil group E(K), the size of its Néron–Tate regulator Reg(E), and the order of its Tate–Shafarevich group Ш(E) (which we prove is finite). These invariants have radically different behaviors depending on the congruence class of p modulo 6. For instance Ш(E) either has trivial p-part or is a p-group. On the other hand, we show that the product |Ш(E)| Reg(E) has size comparable to rq6 as q , regardless of p(mod6). 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