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

Abstract

Let $\Pi$ be a cuspidal automorphic representation of ${\mathrm{GL}}_{2n}({\mathbb{A}}_{\mathbb{Q}})$ 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.

To appear in Res. Math. Sci

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