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

In this paper we study reducibility of those representations of quasi-split unitary p-adic groups which are parabolically induced from supercuspidal representations of general linear groups. For a supercuspidal representation associated via Howe’s construction to an admissible character, we show that in many cases a criterion of Goldberg for reducibility of the induced representation reduces to...

We prove a version of the weight part Serre's conjecture for mod $p$ Galois representations attached to automorphic forms on rank 2 unitary groups which are non-split at $p$. More precisely, let $F/F^+$ denote CM extension totally real field such that every place $F^+$ above is unramified and inert in $F$, $\overline{r}: \textrm{Gal}(\overline{F^+}/F^+) \longrightarrow {}^C\mathbf{U}_2(\overlin...

For certain mod p Galois representations ρ̄, arising from modular forms on definite unitary groups in two variables, we express the ρ̄-part of completed cohomology Ĥ ρ̄ (away from Σ = Σp ∪Σ0) as a tensor product Πp⊗ΠΣ0 . Here Πp is attached to the universal deformation ρ via the p-adic local Langlands correspondence for GL2(Qp), and ΠΣ0 is given by the local Langlands correspondence in families, o...

Let F be a p-adic field of characteristic 0, and let A be an F -central division algebra of dimension dA over F . In the paper we first develop the representation theory of GL(m,A), assuming that holds the conjecture which claims that unitary parabolic induction is irreducible for GL(m,A)’s. Among others, we obtain the formula for characters of irreducible unitary representations of GL(m,A) in ...