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

TL;DR

This paper constructs a $p$-adic Eigenvariety for Picard modular forms on $U(2,1)(E)$, enabling new insights into the Bloch-Kato conjecture for certain Galois characters, especially in cases with positive sign.

Contribution

It develops a novel method to interpolate automorphic sheaves for Picard modular forms, extending previous techniques to cases with no ordinary locus.

Findings

Reproves a case of the Bloch-Kato conjecture for specific Galois characters.

Introduces a new $p$-adic Eigenvariety for Picard modular forms on $U(2,1)(E)$.

Extends interpolation techniques to cases with positive sign in the 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.