On the typical rank of elliptic curves over ${\mathbb Q}(T)$

TL;DR

This paper establishes upper bounds on the number of rational elliptic surfaces with positive rank over certain families, confirming Cowan's conjecture for cases where parameters are at most 2, and shows these surfaces are rare.

Contribution

It provides new upper bounds for the density of positive rank elliptic surfaces over ${f Q}(T)$, confirming Cowan's conjecture in specific low-dimensional cases.

Findings

Positive rank elliptic surfaces form a density zero subset in certain families.

Confirmed Cowan's conjecture for cases with parameters m,n ≤ 2.

Established upper bounds for the count of such surfaces.

Abstract

We give upper bounds for the number of rational elliptic surfaces in some families having positive rank, obtaining in particular that these form a subset of density zero. This confirms Cowan's conjecture (arXiv:2009.08622v2) in the case $m,n\leq2$.