Elliptic curves over function fields

Abstract

Background on curves and function fields This "Lecture 0" covers definitions and notations that are probably familiar to many readers and that were reviewed very quickly during the PCMI lectures. Readers are invited to skip it and refer back as necessary.

Explicit points on the Legendre curve

“…We refer to [Ulm11] and the references there for basic results on elliptic curves over function fields. Throughout the paper, p will be an odd prime number and E will be the elliptic curve E : y 2 = x(x + 1)(x + t) (2.1) over F p (t) or one of its extensions.…”

“…There are now several constructions of elliptic curves (and higher-dimensional Jacobians) of large rank over F p (t) or F p (t). The first results in this direction are due to Shafarevich and Tate [TS67], and their arguments as well as more recent results on large ranks are discussed in [Ulm11]. Most of these constructions rely on relatively sophisticated mathematics, such as the theory of algebraic surfaces, cohomology, and L-functions.…”

Curves and Jacobians over Function Fields

“…One checks that the degree of the conductor of E ′ is 4 and so (by [69], Lecture 1, Theorem 9.3 and Theorem 12.1(1)) the rank of E ′ (K) is zero. Also, E ′ (K) has no 2-torsion.…”

“…Isomorphism in (8.1.1) in turn implies T 1 (X , ℓ) for products of curves and abelian varieties (and more generally for products of varieties for which T 1 is known). This is explained, for example, in [60] or [69].…”

“…We also discuss more recent results such as those of Kato and Trihan. Constructions leading to high ranks are only briefly reviewed, because they are discussed in detail in other recent and forthcoming publications.These notes may be viewed as a continuation of my Park City notes [69]. In those notes, the focus was on elliptic curves and finite constant fields, whereas here we discuss curves of high genera and results over more general base fields.It is a pleasure to thank the organizers of the CRM Research Program, especially Francesc Bars, for their efficient organization of an exciting meeting, the audience for their interest and questions, the National Science Foundation for funding to support the travel of junior participants (grant DMS 0968709), and the editors for their patience in the face of many delays.…”