Maximal Galois group of L-functions of elliptic curves

TL;DR

This paper provides a quantitative analysis of the Galois groups of L-functions of elliptic curves over function fields, showing that most have maximal Galois groups as the field size increases.

Contribution

It offers a quantitative refinement of Katz's theorem by estimating the proportion of elliptic curves with L-functions having maximal Galois groups using large sieve methods.

Findings

Most elliptic curves have L-functions with maximal Galois groups as field size grows.

The paper quantifies the proportion of elliptic curves with maximal Galois groups.

Utilizes large sieve techniques and results on orthogonal monodromy.

Abstract

We give a quantitative version of a result due to N. Katz about L-functions of elliptic curves over function fields over finite fields. Roughly speaking, Katz's Theorem states that, on average over a suitably chosen algebraic family, the L-function of an elliptic curve over a function field becomes "as irreducible as possible" when seen as a polynomial with rational coefficients, as the cardinality of the field of constants grows. A quantitative refinement is obtained as a corollary of our main result which gives an estimate for the proportion of elliptic curves studied whose L-functions have "maximal" Galois group . To do so we make use of E. Kowalski's idea to apply large sieve methods in algebro-geometric contexts. Besides large sieve techniques, we use results of C. Hall on finite orthogonal monodromy and previous work of the author on orthogonal groups over finite fields.