Ranks of elliptic curves over function fields

TL;DR

This paper provides experimental evidence supporting the belief that half of all elliptic curves over function fields have infinitely many rational points, using a refined algorithm based on advanced cohomology techniques.

Contribution

It introduces a refined algorithm leveraging rigid and crystalline cohomology to gather evidence for the distribution of rational points on elliptic curves over function fields.

Findings

Supports the conjecture that half of all elliptic curves have infinitely many rational points

Uses a novel computational approach based on advanced cohomology theories

Provides empirical data consistent with theoretical predictions

Abstract

We present experimental evidence to support the widely held belief that one half of all elliptic curves have infinitely many rational points. The method used to gather this evidence is a refinement of an algorithm due to the author which is based upon rigid and crystalline cohomology.