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Families of Picard modular forms and an application to the Bloch–Kato conjecture
- Part of
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- Journal:
- Compositio Mathematica / Volume 155 / Issue 7 / July 2019
- Published online by Cambridge University Press:
- 25 June 2019, pp. 1327-1401
- Print publication:
- July 2019
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- Article
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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 and Stevens [
$p$-adic families of Siegel modular cuspforms Ann. of Math. (2) 181, (2015), 623–697] 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 results of Bellaiche and Chenevier to the case of a positive sign.
