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.

1.1. Let X be a smooth, complete, geometrically connected curve over Fq. Denote by F the field of rational functions on X , by A the ring of adeles of F , and by Gal(F/F ) the Galois group of F . The present paper may be considered as a step towards understanding the geometric Langlands correspondence between n–dimensional `–adic representations of Gal(F/F ) and automorphic forms on the group G...

Let π1, π2 be cuspidal automorphic representations of PGL2(R) of conductor 1 and Hecke eigenvalues λπ1,2 (n), and let h > 0 be an integer. For any smooth compactly supported weight functions W1,2 : R → C and any Y > 0 a spectral decomposition of the shifted convolution sum

We show how to deduce the standard sign conjecture (a weakening of the Künneth standard conjecture) for Shimura varieties from some statements about discrete automorphic representations (Arthur’s conjectures plus a bit more). We also indicate what is known (to us) about these statements. 1)

Let E/F be a quadratic extension of p-adic fields. We compute the multiplicity of the space of SL2(F )-invariant linear forms on a representation of SL2(E). This multiplicity varies inside an L-packet similar in spirit to the multiplicity formula for automorphic representations due to Labesse and Langlands.

Drinfeld in 1974, in his seminal paper [10], revolutionized the contribution to arithmetic of the area of global function fields. He introduced a function field analog of elliptic curves over number fields. These analogs are now called Drinfeld modules. For him and for many subsequent developments in the theory of automorphic forms over function fields, their main use was in the exploration of ...

Abstract We prove some qualitative results about the p -adic Jacquet–Langlands correspondence defined by Scholze, in $\operatorname {\mathrm {GL}}_2(\mathbb{Q}_p )$ residually reducible case, using a vanishing theorem proved Judith Ludwig. In particular, we show that cases under consideration, global can also deal with automorphic forms principal series representations at nontrivial way, unlike...