Modeling families of L-functions

Summary

Abstract

We discuss the idea of a “family of L-functions” and describe various methods which have been used to make predictions about L-function families. The methods involve a mixture of random matrix theory and heuristics from number theory. Particular attention is paid to families of elliptic curve L-functions. We describe two random matrix models for elliptic curve families: the Independent Model and the Interaction Model.

Introduction

Using ensembles of random matrices to model the statistical properties of a family of L-functions has led to a wealth of interesting conjectures and results in number theory. In this paper we survey recent results in the hopes of conveying our best current answers to these questions:

1. What is a family of L-functions?

2. How do we model a family of L-functions?

3. What properties of the family can the model predict?

In the remainder of this section we briefly review some commonly studied families and describe some of the properties which have been modeled using ideas from random matrix theory. In Section 2 we provide a definition of “family of L-functions” which has been successful in permitting precise conjectures, and we briefly describe how to model such a family. In Section 3 we discuss families of elliptic curve L-functions and show that there is an additional subtlety which requires us to slightly broaden the class of random matrix models we use.