L-Functions of Elliptic Curves in Positive Characteristic (Part II : Studying L-Functions of Elliptic Curves over Function Fields via their Reduction Modulo Integers)

Abstract: Elliptic curves are a central object of study in number theory. In this talk, we focus on those defined over function fields and with nonconstant j-invariant. The L-function of such an elliptic curve E/K is polynomial with integer coefficients.

Inspired by Schoof's algorithm, we study the reduction modulo integers of the L-function. More precisely, when E(K) has nontrivial N-torsion, we give formulas for the reductions modulo 2 and N for any quadratic twist of E/K. This generalizes a formula obtained by Chris Hall for E/K. We give examples where we can compute the global root number of the quadratic twists, the order of vanishing of the L-function at a special value and even the whole L-function from these reductions. However, the group E(K) is finitely generated and in particular has finite torsion. Time permiting, we discuss some of our work in progress in this situation. More precisely, given a prime ell different from char(K), we provide, in absence of nontrivial ell-torsion and in a quite general context, expressions for the reduction modulo ell of the L-function.

algebraic geometrynumber theory

Audience: researchers in the topic

Comments: The talk will be given in English

Algebra and Number Theory Seminars at Université Laval