Rank statistics for a family of elliptic curves over a function field

TL;DR

This paper demonstrates that in a specific family of elliptic curves over a function field, both average and typical ranks grow without bound as the parameter increases, contrasting with the bounded average rank over all such curves.

Contribution

It provides the first example of a family of elliptic curves over a function field with unbounded average and typical ranks as the parameter tends to infinity.

Findings

Average rank tends to infinity as parameter increases

Typical rank also tends to infinity in the family

Contrasts with known bounded average rank over all elliptic curves

Abstract

We show that the average and typical ranks in a certain parametric family of elliptic curves described by D. Ulmer tend to infinity as the parameter $d \to\infty$. This is perhaps unexpected since by a result of A. Brumer, the average rank for all elliptic curves over a function field of positive characteristic is asymptotically bounded above by 2.3.