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

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.