First explicit reciprocity law for unitary Friedberg—Jacquet periods

Abstract

In the early 2000's, Bertolini and Darmon introduced a new technique to bound Selmer groups of elliptic curves via level raising congruences. This was the first example of what is now termed a "bipartite Euler system", and over the last decade we have seen many breakthroughs on constructing such systems for other Galois representations, including settings such as twisted and cubic triple product, symmetric cube, and Rankin--Selberg, with applications to the Bloch--Kato conjecture and to Iwasawa theory.

This thesis studies the case of Galois representations attached to automorphic representations on a totally definite unitary group U(2r) over a CM field which are distinguished by the subgroup U(r) x U(r). We prove a new ``first explicit reciprocity law'' in this setting, which has applications to the rank 0 case of the corresponding Bloch--Kato conjecture.