Adjoint Selmer groups of automorphic Galois representations of unitary type

Abstract

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

Modularity of Galois Representations and Langlands Functoriality

“…Allen 2 was able to generalize this to prove vanishing of an adjoint Bloch-Kato Selmer group for self-dual automorphic Galois representations ρ with just an assumption that the image of ρ(G Q(ζ p ) ) is sufficiently large. AE The author and Thorne recently proved a similar vanishing result replacing this large image assumption with a (much milder) large image assumption on the characteristic 0 representation ρ itself 75 . We use an idea due to Lue Pan (it appears in the work we have already mentioned on the Fontaine-Mazur conjecture in the residually reducible case) which allows us to carry out a version of the Taylor-Wiles method up to a bounded p-power torsion error term, which disappears when we invert p. Thorne subsequently improved our result to only require irreducibility of ρ| G Q(ζ p ∞ ) 93 .…”

“…In what remains of this survey, we will discuss some of the ideas of the works 73,74 which establish symmetric power functoriality (Conjecture 2.6.1) for holomorphic modular forms. Crucial inputs come from some other works 3,5,75 .…”

“…The key step in the proof is to use vanishing of an adjoint Selmer group 75 to show that y ′ defines a regular point of Spec (P patch ). ) is in the support of N Sym n m .…”

“…We do crucially use our results 75 on vanishing of adjoint Selmer groups which in turn are proved using the Taylor-Wiles method.…”

Reciprocity in the Langlands program since Fermat's Last Theorem

“…Taylor's argument proves theorems of the form R[1/p] red = T[1/p] rather than R = T. This is still perfectly sufficient for proving modularity lifting results, but not always other interesting corollaries associated to R = T theorems like finiteness of the corresponding adjoint Selmer groups (though see[3,133]). …”