Points Defined over Cyclic Quartic Extensions on an Elliptic Curve and Generalized Kummer Surfaces

Abstract

Let E be an elliptic curve over a number field k. By the Mordell-Weil theorem the group E(K) of K-rational points on E, where K/k is a finite extension of k,is a finitely generated abelian group. We fix E/k once and for all, and we study the behavior of the rank of the group E(K) as K varies through a certain family. We are particularly interested in the family F _k (G) of all Galois extensions K/k whose Galois group Gal(K/k) is isomorphic to a prescribed finite group G. In this article we focus on the case $G = \mathbb{Z}/4\mathbb{Z}$