Adjoint Selmer groups of automorphic Galois representations of unitary type

James Newton,Jack A Thorne

doi:10.4171/jems/1228

Abstract

Let $\\rho$ be the $p$-adic Galois representation attached to a cuspidal, regular algebraic automorphic representation of $\\operatorname{GL}\_n$ of unitary type. Under very mild hypotheses on $\\rho$, we prove the vanishing of the (Bloch–Kato) adjoint Selmer group of $\\rho$. We obtain definitive results for the adjoint Selmer groups associated to non-CM Hilbert modular forms and elliptic curves over totally real fields.

Highlights

Let ρ be the p-adic Galois representation attached to a cuspidal, regular algebraic automorphic representation of GLn of unitary type
Kisin [Kis04] proved the analogue of Theorem A for the Galois representations attached to classical holomorphic modular forms under some mild conditions on the residual representation
Allen [All16] proved a result similar to Theorem A, but assuming a stronger condition on the residual representation rπ,ι, requiring in particular that it be irreducible

Summary

Notation and preliminaries

If F is a field of characteristic zero, we generally fix an algebraic closure F /F and write GF for the absolute Galois group of F with respect to this choice. If G is a locally profinite group and U ⊂ G is an open compact subgroup, we write H(G, U ) for the set of compactly supported, U -biinvariant functions f : G → Z It is a Z-algebra, where convolution is defined using the left-invariant Haar measure normalized to give U measure 1; see [NT16, §2.2]. If ρ : GK → GLn(Qp) is a continuous representation (assumed to be de Rham if p equals the residue characteristic of K), we write WD(ρ) = (r, N ) for the associated Weil–Deligne representation, and WD(ρ)F −ss for its Frobenius semisimplification.

Pseudocharacters

A result about Hecke algebras

Patching

Applications