Relating Galois representations via L-series and dynamical systems

Abstract

We find invariants of number fields and of Galois representations of number fields that characterise them, i.e. if two Galois representations (or number fields) share this invariant, then they are necessarily isomorphic. These invariants take the form of the (Dirichlet) character groups (with some additional data) or a dynamical system. The thesis is split into four parts. In the first part, we show that number fields are characterized by their character groups along with their L-series. Moreover, we show that for a prime l, the isomorphisms of the l-torsion of character groups that respect L-series correspond bijectively with isomorphisms of number fields. If l is a large enough prime, the same holds for homomorphisms instead of isomorphisms. In the second part we show that number fields (along with their isomorphisms) are characterized by a dynamical system containing the integral adeles and the abelianized absolute Galois group. We characterise abelian varieties defined over number fields in the third part, showing that two abelian varieties are isogenous (up to an automorphism of the ground field) if their L-series twisted by Dirichlet characters of a high enough prime order agree, respecting the group structure on the Dirichlet characters. In particular, two elliptic curves are isogenous if their quadratic twists have matching L-series. The final part shows that under more restrictive conditions, only a single L-series twisted with a character suffices to characterise Galois representations over number fields, that is, if two Galois representations share this L-series, then they are isomorphic up to a twist with a character of small order and an automorphism of the underlying number field.