On Ranks of Jacobian Varieties in Prime Degree Extensions

TL;DR

This paper generalizes a known result about elliptic curve ranks over degree 3 extensions to all prime degree extensions, using properties of Jacobian varieties of certain algebraic curves.

Contribution

It proves that for any prime degree extension, the rank of the Jacobian variety can be increased, extending previous results limited to degree 3.

Findings

Rank increases in prime degree extensions for Jacobian varieties.

Generalization from elliptic curves to broader classes of algebraic curves.

Establishment of a property for Jacobian varieties associated with specific polynomial equations.

Abstract

In Dokchitser (2007) it is shown that given an elliptic curve $E$ defined over a number field $K$ then there are infinitely many degree 3 extensions $L/K$ for which the rank of $E(L)$ is larger than $E(K)$. In the present paper we show that the same is true if we replace 3 by any prime number. This result follows from a more general result establishing a similar property for the Jacobian varieties associated with curves defined by an equation of the shape $g(y) = f(x)$ where $f$ and $g$ are polynomials of coprime degree.