Euler Systems and Selmer Bounds for GU(2,1)

TL;DR

This paper advances the understanding of Selmer groups and Iwasawa main conjectures for automorphic representations of GU(2,1) over imaginary quadratic fields, using Euler systems and p-adic L-functions.

Contribution

It adapts Euler system techniques to prove divisibility results for the Iwasawa main conjecture in the context of GU(2,1), including split and inert prime cases.

Findings

Proves one divisibility of the rank 1 Iwasawa main conjecture.

Establishes bounds on Selmer groups in terms of p-adic L-values when p splits in E.

Provides integral level bounds on Bloch--Kato Selmer groups.

Abstract

We investigate properties of the Euler system associated to certain automorphic representations of the unitary similitude group GU(2,1) with respect to an imaginary quadratic field $E$, constructed by Loeffler-Skinner-Zerbes. By adapting Mazur and Rubin's Euler system machinery we prove one divisibility of the ``rank 1" Iwasawa main conjecture under some mild hypotheses. When $p$ is split in $E$ we also prove a ``rank 0" statement of the main conjecture, bounding a particular Selmer group in terms of a $p$-adic distribution conjecturally interpolating complex $L$-values. We then prove descended versions of these results, at integral level, where we bound certain Bloch--Kato Selmer groups. We will also discuss the case where $p$ is inert, which is a work in progress.