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.

We show that if the L-function of an irreducible 2-dimensional complex Galois representation over Q is not automorphic then it has infinitely many poles. In particular, the Artin conjecture for a single representation implies the corresponding strong Artin conjecture. Introduction Let ρ : Gal(Q/Q) → GLn(C) be an irreducible continuous representation of the absolute Galois group of Q. Brauer [2]...

We prove new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor–Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the automorphic forms in question contribute to a single degree of cohomology. In practice, this imposes several restrictions – one must be in a Shimura variety settin...

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.

The notion of functoriality arose from the spectral analysis of automorphic forms but its definition was informed by two major theories: the theory of class fields as created by Hilbert, Takagi, and Artin and others; and the representation theory of semisimple Lie groups in the form given to it by Harish-Chandra. In both theories the statements are deep and general and the proofs difficult, hig...

It is proved that certain types of modular cusp forms generate irreducible automorphic representations of the underlying algebraic group. Analogous Archimedean and non-Archimedean local statements are also given. Introduction One of the motivations for this paper was to show that full level cuspidal Siegel eigenforms generate irreducible, automorphic representations of the adelic symplectic sim...

We conjecture that the only irreducible cuspidal automorphic representations for GL(3)/Q of cohomological type and level 1 are (up to twisting) the symmetric square lifts of classical cuspforms on GL(2)/Q of level 1. We present computational evidence for this conjecture. 1. Statement and explanation of a conjecture Arithmetic objects defined over Q and unramified everywhere are rare. For exampl...

In this paper we deal with monogenic and k-hypermonogenic automorphic forms on arithmetic subgroups of the Ahlfors-Vahlen group. Monogenic automorphic forms are exactly the 0-hypermonogenic automorphic forms. In the first part we establish an explicit relation between k-hypermonogenic automorphic forms and Maaß wave forms. In particular, we show how one can construct from any arbitrary non-vani...

We study the subconvexity problem for GL3(R) L-functions in t-aspect using integral representations by combining techniques employed Michel–Venkatesh their of corresponding GL2 with ideas from recent works Munshi, Holowinsky–Nelson and Lin. Our main objective is to give – perspective automorphic representation theory a possible explanation origin “key identity” arising latter series works.

Abstract We prove that a cuspidal automorphic representation of $\mathrm{GL}(3)$ over any number field is determined by the quadratic twists its central value. In case π not Gelbart–Jacquet lift, result conditional on analytic behavior certain Euler product. deduce nonvanishing infinitely many values. This generalizes Chinta and Diaconu was valid only Q explored for lifts.