Modeling families of L-functions

TL;DR

This paper explores the concept of families of L-functions, especially elliptic curve L-functions, using random matrix theory and heuristics to predict their properties.

Contribution

It introduces two new random matrix models, the Independent Model and the Interaction Model, for analyzing elliptic curve L-function families.

Findings

Two random matrix models for elliptic curve families are proposed.

The models provide insights into the statistical behavior of L-functions.

Heuristics from number theory complement the models.

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.