On Ranks of Jacobian Varieties in Prime Degree Extensions

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.