Iwasawa theory for $\mathrm{U}(r,s)$, Bloch-Kato conjecture and Functional Equation

TL;DR

This paper introduces a new approach to Iwasawa theory for unitary groups, proving nonvanishing of central L-values under certain conditions and establishing p-adic functional equations for related L-functions.

Contribution

It develops a novel method that avoids Fourier-Jacobi coefficient computations, extending results to general signatures of unitary groups and broadening the scope of Iwasawa theory applications.

Findings

Proves nonvanishing of central L-values when Selmer groups have rank 0.

Establishes p-adic functional equations for L-functions.

Constructs p-adic families of Klingen Eisenstein series.

Abstract

In this paper we develop a new method to study Iwasawa theory and Eisenstein families for unitary groups $\mathrm{U}(r,s)$ of general signature over a totally real field $F$. As a consequence we prove that for a motive corresponding to a regular algebraic cuspidal automorphic representation $\pi$ on $\mathrm{U}(r,s)_{/F}$ which is ordinary at $p$, twisted by a Hecke character, if its Selmer group has rank $0$, then the corresponding central $L$-value is nonzero. This generalizes a result of Skinner-Urban in their ICM 2006 report in the special case when $F=\mathbb{Q}$ and the motive is conjugate self-dual. Along the way we also obtain $p$-adic functional equations for the corresponding $p$-adic $L$-functions and $p$-adic families of Klingen Eisenstein series. Our method does not involve computing Fourier-Jacobi coefficients (as opposed to previous work which only work in low rank cases, e.g. $\mathrm{U}(1,1)$, $\mathrm{U}(2,0)$ and $\mathrm{U}(1,0)$) whose automorphic interpretation is unclear in general.