Analytic rank-one elliptic curves over function fields and their rank over certain ring class fields

TL;DR

This paper proves that elliptic curves over function fields with analytic rank 1 have infinite rank over certain ring class fields, extending known rank results in the context of function fields.

Contribution

It establishes the infinite rank over fixed subfields for elliptic curves with analytic rank 1, under specific conditions involving ring class extensions and Heegner hypotheses.

Findings

Elliptic curves with analytic rank 1 have infinite rank over certain ring class fields.

For rank 0 curves, the same holds if an associated quadratic extension has rank 1 and satisfies the Heegner hypothesis.

The results connect analytic rank with algebraic rank growth over specialized field extensions.

Abstract

Let $E/k$ be a non-isotrivial elliptic curve over a global function field $k$ of characteristic $p>3$, and $G\subset \mathrm{Gal}(k^{\mathrm{sep}}/k)$ be a topologically finitely generated subgroup. We prove that if $E/k$ has analytic rank $1$, then its rank over the fixed subfield $L^G$ is infinite, where $L$ is the infinite ring class extension of some finite separable extension $K/k$. If $E/k$ has analytic rank $0$, then we prove that the same holds provided there exists an imaginary quadratic extension $K/k$ such that $E/K$ has analytic rank $1$ and satisfies the Heegner hypothesis.