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

Let L = Q(i) or Q(√−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.