The average rank of elliptic $n$-folds

TL;DR

This paper investigates the distribution of elliptic curves over function fields of varieties, showing that the proportion with positive rank is zero in higher dimensions and bounded in surfaces, revealing new insights into their arithmetic properties.

Contribution

It provides a precise density estimate for elliptic curves with positive rank over function fields of varieties of different dimensions.

Findings

Density of positive rank elliptic curves is zero for varieties of dimension ≥ 3.

For surfaces, the density is at most 1 - ζ_V(3)^{-1}.

Advances understanding of rank distribution over function fields.

Abstract

Let $V/\mathbb{F}_q$ be a variety of dimension at least two. We show that the density of elliptic curves $E/\mathbb{F}_q(V)$ with positive rank is zero if $V$ has dimension at least 3 and is at most $1-\zeta_V(3)^{-1}$ if $V$ is a surface.