p-Sylow subgroup growth of elliptic curves over Galois extension of prime degree $q$ upon base change from quadratic cyclotomic number fields

Abstract

Let $ L = \mathbb{Q}(i) $ or  $\mathbb{Q}(\sqrt{-3}) $ be a quadratic cyclotomic number field, and let $K$ be a Galois extension of $ L $ of prime degree $ q $. This paper examines the behavior of $p$-Sylow subgroups of elliptic curves defined over $K$ $, focusing on their growth under base change from $L$. The study uncovers distinctive patterns in subgroup growth, shaped by both the arithmetic nature of the base field and the intrinsic properties of the elliptic curves.