Families of Picard modular forms and an application to the Bloch-Kato conjecture

Abstract

In this article we construct a \(p\)-adic three dimensional Eigenvariety for the group \(U(2,1)(E)\), where \(E\) is a quadratic imaginary field and \(p\) is inert in \(E\). The Eigenvariety parametrizes Hecke eigensystems on the space of overconvergent, locally analytic, cuspidal Picard modular forms of finite slope. The method generalized the one developed in Andreatta-Iovita-Pilloni by interpolating the coherent automorphic sheaves when the ordinary locus is empty. As an application of this construction, we reprove a particular case of the Bloch-Kato conjecture for some Galois characters of \(E\), extending the result of Bellaiche-Chenevier to the case of a positive sign.