the Iwasawa Theory for Unitary groups

Abstract

In this thesis we generalize earlier work of Skinner and Urban to construct ($p$-adic families of) nearly ordinary Klingen Eisensten series for the unitary groups $U(r,s)\hookrightarrow U(r+1,s+1)$ and do some preliminary computations of their Fourier Jacobi coefficients. As an application, using the case of the embedding $U(1,1)\hookrightarrow U(2,2)$ over totally real fields in which the odd prime $p$ splits completely, we prove that for a Hilbert modular form $f$ of parallel weight $2$, trivial character, and good ordinary reduction at all places dividing $p$, if the central critical $L$-value of $f$ is $0$ then the associated Bloch Kato Selmer group has infinite order. We also state a consequence for the Tate module of elliptic curves over totally real fields that are known to be modular.