Rank gain of Jacobians over finite Galois extensions

TL;DR

This paper proves that Jacobians of certain algebraic curves, including the Klein quartic, can gain rank over infinitely many finite Galois extensions, with applications to elliptic curves over cyclic cubic extensions.

Contribution

It demonstrates the existence of infinitely many degree extensions over which Jacobians and elliptic curves increase rank, including specific cases with automorphisms and branch point configurations.

Findings

Jacobians gain rank over infinitely many degree d-extensions.

Explicit construction for Klein quartic and rank gain over degree 7 extensions.

Existence of elliptic curves gaining rank over infinitely many cyclic cubic extensions.

Abstract

Let $\mathcal{X}$ be a Riemann surface of genus $g>0$ defined over a number field $K$ which is a degree $d$-covering of $\mathbb{P}^1_K$. In this paper we show the existence of infinitely many linearly disjoint degree $d$-extensions $L/K$ over which the Jacobian of $\mathcal{X}$ gains rank. In the case where 0, 1 and $\infty$ are the only branch points, and there is an automorphism $\sigma$ of $\mathcal{X}$ which cyclically permutes these branch points, we obtain the same result for the Jacobian of $\mathcal{X}/\sigma$. In particular if $\mathcal{X}$ is the Klein quartic, then the construction provides an elliptic curve which gains rank over infinitely many degree $7$-extensions of $\mathbb{Q}$. As an application, we show the existence of infinitely many elliptic curves that gain rank over infinitely many cyclic cubic extensions of $\mathbb{Q}$.