Let G = GSp(2) be the symplectic group with similitude of degree two, which is defined over Q. For a generic cusp form F on the adelized group GA which generates a large discrete series representation at infinity, we show that its spinor Lfunction is continued to an entire function and satisfies the functional equation. 0. Introduction. Let G = GSp(2) be the symplectic group with similitude of ...

The space of elliptic modular forms of fixed weight and level can be identified with a space of intertwining operators, from a holomorphic discrete series representation of SL2(R) to a space of automorphic forms. Moreover, multiplying elliptic modular forms corresponds to a branching problem involving tensor products of holomorphic discrete series representations. In this paper, we explicitly c...

The purpose of this paper is to develop a scattering theory for twisted automorphic functions on the hyperbolic plane, defined by a cofinite (but not cocompact) discrete group Γ with an irreducible unitary representation ρ and satisfying u(γz) = ρ(γ)u(z). The Lax-Phillips approach is used with the wave equation playing a central role. Incoming and outgoing subspaces are employed to obtain corre...

In this paper, we study the period map of a certain one-parameter family of quartic K3 surfaces with an S5-action. We construct automorphic forms on the period domain as the pull-backs of theta constants of genus 2 by a modular embedding. Using these automorphic forms, we give an explicit presentation of the inverse period map.

2 Modular forms and Automorphic forms 2 2.1 Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 The Adele group of GL2 over Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Automorphic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Hecke act...

The automorphic A-chromatic index of a graph Γ is the minimum integer m for which Γ has a proper edge-coloring with m colors which is preserved by a given subgroup A of the full automorphism group of Γ. We compute the automorphic A-chromatic index of each generalized Petersen graph when A is the full automorphism group.

We show that the irregular connection on Gm constructed by Frenkel-Gross ([FG]) and the one constructed by Heinloth-Ngô-Yun ([HNY]) are the same, which confirms the Conjecture 2.14 of [HNY]. The proof is simple, modulo the big machinery of quantization of Hitchin’s integrable systems as developed by Beilinson-Drinfeld ([BD]). The idea is as follows. Let E be the irregular connection on Gm as co...

Let T be a discrete subgroup of automorphisms of H", with fundamental polyhedron of finite volume, finite number of sides, and N cusps. Denote by Ar the Laplace-Beltrami operator acting on functions automorphic with respect to T. We give a new short proof of the fact that Ar has absolutely continuous spectrum of uniform multiplicity N on (-00,«11 l)/2)2), plus a standard discrete spectrum. We s...

Let F ∈ Sk(Sp(4,Z)) be a cuspidal Hecke eigenform of degree 2. Let μ(n), n > 0, be the Hecke eigenvalues. It is known that F is in the Maaß space (i.e., F is a Saito-Kurokawa lifting), if and only if μ(n) > 0 for all n; see [2]. On the other hand, Kohnen has recently proved (see [6]) that if F is not in the Maaß space, then the sequence μ(n), n > 0, has infinitely many sign changes. The proof m...