Elliptic curves over Galois number fields

Abstract

This thesis is concerned with the statistical behaviour of elliptic curves over extension fields. That is, if K/Q is a finite extension, we study the arithmetic of E/K as E ranges in natural families of elliptic curves defined over Q. We study the statistical properties of the action of the group Aut(K) on E(K) and on the p-Selmer groups Selp(E/K) where p is a prime number. We construct special generalised Selmer groups, and show that these are related to certain representation-theoretic invariants of Selp(E/K). The sizes of these groups are related to the cokernels of the norm maps over the completions of K, which we go on to compute in several cases. In the statistical component of this thesis, we study quadratic twist families of elliptic curves and the family of ‘all elliptic curves’. For quadratic twist families we consider the behaviour over quadratic extensions. Using methods similar to those of Heath-Brown [HB93, HB94] and of Fouvry–Klüners [FK07], we determine the complete distribution of the 2-Selmer groups as Galois modules. This also allows us to determine representation-theoretic properties for the Mordell–Weil groups of 100% of twists. For the family of all elliptic curves over Q, we consider the behaviour with respect to a general finite Galois extension K/F. Writing G = Gal(K/F), our first main result is that the difference in dimension between Selp(E/K) G and Selp(E/F) has bounded average in this family. Using this we are able, with additional assumptions on K/F and p, to bound the average dimension of Selp(E/K) and so the average rank of the Mordell–Weil group E(K). Our methods also allow us to bound how often certain Z[G]-lattices occur as summands of E(K), with additional assumptions on F. We refine our results in the setting where K/Q is multiquadratic and p = 2, and prove strong upper and lower bounds for the average dimension of the 2-Selmer group.