Iwasawa Theory for GU(2,1) at inert primes

TL;DR

This paper extends Iwasawa theory to the unitary group GU(2,1) at inert primes, linking automorphic L-values to Selmer groups and verifying cases of the Bloch--Kato conjecture in this context.

Contribution

It introduces a novel approach using Schneider--Venjakob regulators and locally analytic distributions to adapt Iwasawa theory for GU(2,1) at inert primes.

Findings

Vanishing of certain Bloch--Kato Selmer groups when specific p-adic distributions are non-zero

Verification of cases of the Bloch--Kato conjecture for GU(2,1) at inert primes in rank 0

Development of new algebraic methods linking automorphic L-values to Selmer groups in this setting

Abstract

Many problems of arithmetic nature rely on the computation or analysis of values of $L$-functions attached to objects from geometry. Whilst basic analytic properties of the $L$-functions can be difficult to understand, recent research programs have shown that automorphic $L$-values are susceptible to study via algebraic methods linking them to Selmer groups. Iwasawa theory, pioneered first by Iwasawa in the 1960s and later Mazur and Wiles provides an algebraic recipe to obtain a $p$-adic analogue of the $L$-function. In this work we aim to adapt Iwasawa theory to a new context of representations of the unitary group GU(2,1) at primes inert in the respective imaginary quadratic field. This requires a novel approach using the Schneider--Venjakob regulator map, working over locally analytic distribution algebras. Subsequently, we show vanishing of some Bloch--Kato Selmer groups when a certain $p$-adic distribution is non-vanishing. These results verify cases of the Bloch--Kato conjecture for GU(2,1) at inert primes in rank 0.