On ranks of Jacobian varieties in prime degree extensions
Volume 161 / 2013
Acta Arithmetica 161 (2013), 241-248 MSC: Primary 11G05. DOI: 10.4064/aa161-3-3
T. Dokchitser [Acta Arith. 126 (2007)] showed 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 $f(y) = g(x)$ where $f$ and $g$ are polynomials of coprime degree.