Elliptic curves with bounded ranks in function field towers

TL;DR

This paper investigates the ranks of elliptic curves over function field towers, showing that most such families have Mordell-Weil groups of rank zero over certain extensions, using combinatorial and rank formula techniques.

Contribution

It introduces a new analysis of elliptic curve families over function fields, demonstrating that their Mordell-Weil groups typically have rank zero in tower extensions.

Findings

Most families have rank zero over $k(t^{1/d})$ for large $d$

Combines combinatorial analysis with Ulmer's rank formula

Results hold for all but finitely many families

Abstract

Let $k$ denote an algebraically closed field. We revisit a construction of the author of families of elliptic curves over the rational function field $k(t)$. Combining a combinatorial analysis with a rank formula of Ulmer we prove that, for all but finitely many families of these curves, the Mordell-Weil groups over $k(t^{1/d})$ have rank zero, as $d$ ranges over positive integers prime to the characteristic of $k$.