Iwasawa theory of automorphic representations of $\mathrm{GL}_{2n}$ at non-ordinary primes

TL;DR

This paper constructs bounded $p$-adic $L$-functions for automorphic representations of $ ext{GL}_{2n}$ at non-ordinary primes and formulates related Iwasawa main conjectures using signed Selmer groups.

Contribution

It extends previous work by constructing bounded $p$-adic $L$-functions under relaxed conditions and defines signed Selmer groups for non-ordinary automorphic representations.

Findings

Constructed two bounded $p$-adic $L$-functions for $ ext{GL}_{2n}$ automorphic representations.

Formulated Iwasawa main conjectures using signed Selmer groups.

Extended earlier work by relaxing the Pollack condition.

Abstract

Let $\Pi$ be a cuspidal automorphic representation of $\mathrm{GL}_{2n}(\mathbb{A_Q})$ and let $p$ be an odd prime at which $\Pi$ is unramified. In a recent work, Barrera, Dimitrov and Williams constructed possibly unbounded $p$-adic $L$-functions interpolating complex $L$-values of $\Pi$ in the non-ordinary case. Under certain assumptions, we construct two \textit{bounded} $p$-adic $L$-functions for $\Pi$, thus extending an earlier work of Rockwood by relaxing the Pollack condition. Using Langlands local-global compatibility, we define signed Selmer groups over the $p$-adic cyclotomic extension of $\mathbb{Q}$ attached to the $p$-adic Galois representation of $\Pi$ and formulate Iwasawa main conjectures in the spirit of Kobayashi's plus and minus main conjectures for $p$-supersingular elliptic curves.