I will first formulate a problem in the theory of group representations and show how to solve it; then I will discuss the relation of this problem to the theory of automorphic forms. Since there is no point in striving for maximum generality, I start with a connected semisimple group G with finite center. An irreducible unitary representaiton π of G on the Hilbert space H is said to be square-i...

This article gives an introduction to the multiple Dirichlet series arising from sums of twisted automorphic L-functions. We begin by explaining how such series arise from Rankin-Selberg constructions. Then more recent work, using Hartogs’ continuation principle as extended by Bochner in place of such constructions, is described. Applications to the nonvanishing of Lfunctions and to other probl...

on Γ\H, parametrized as λw = w(w−1). Haas listed the w-values. Haas thought he was solving the differential equation (∆ − λ)u = 0. Stark and Hejhal observed zeros of ζ and of an L-function on Haas’ list. This suggested an approach to proving the Riemann Hypothesis, since it seemed that zeros w of ζ might give eigenvalues λ = w(w − 1) of ∆. Since ∆ is a self-adjoint, nonpositive operator, these ...

We prove a variety of results on the existence of automorphic Galois representations lifting a residual automorphic Galois representation. We prove a result on the structure of deformation rings of local Galois representations, and deduce from this and the method of Khare and Wintenberger a result on the existence of modular lifts of specified type for Galois representations corresponding to Hi...

For a division quaternion algebra D with the discriminant two, we construct Maass cusp forms on GL2(D) explicitly by some lifting from Maass cusp forms for Γ0(2). This is an analogue of the Saito-Kurokawa lifting. We expect that this lifting leads to a construction of a CAP representation of GL2(D), which is an inner form of GL(4). The aim of the talk is to report recent progress of our study o...

1. There has recently been much interest, if not a tremendous amount of progress, in the arithmetic theory of automorphic forms. In this lecture I would like to present the views not of a number theorist but of a student of group representations on those of its problems that he finds most fascinating. To be more precise, I want to formulate a series of questions which the reader may, if he like...

In Carayol’s note [4], a geometric construction of the Galois representations associated to Hilbert modular forms and the compatibility with the local Langlands correspondence are discussed. In loc. cit., the compatibility is established in the case = p where the Galois representation is an -adic representation and p is the prime divided by the prime p of the totally real field where the restri...

We give a geometrical construction of the canonical automorphic factor for the Jacobi group and construct new vector valued modular forms from Jacobi forms by differentiating them with respect to toroidal variables and then evaluating at zero.

We introduce a “geometric” method to bound periods of automorphic forms. The key features of this method are the use of equidistribution results in place of mean value theorems, and the systematic use of mixing and the spectral gap. Applications are given to equidistribution of sparse subsets of horocycles and to equidistribution of CM points; to subconvexity of the triple product period in the...