Let π be a cuspidal, automorphic representation of GSp4 attached to a Siegel modular form of degree 2. We refine the method of Furusawa [9] to obtain an integral representation for the degree-8 L-function L(s, π × τ), where τ runs through certain cuspidal, automorphic representation of GL2. Our calculations include the case of square-free level for the p-adic components of τ , and a wide class ...

Conjecture 1 (Langlands). To each reductive group G over a number field K, each automorphic (complex) representation π of G, and each finite dimensional representation r of the (complex) group G, there is defined an automorphic L-function L(s, π, r), which enjoys an analytic continuation and functional equation generalizing the Riemann zeta function πΓ(s/2)ζ(s) (or Artin’s L-function L(s, σ), w...

The Selberg-Arthur trace formula is one of the tools available for approaching the conjecture of global functoriality in the Langlands program. Global functoriality is described within this volume in [Kn2]. We start with reductive groups G and H, say over the rationals Q for simplicity. We assume that G is quasisplit, and we suppose that we are given an L homomorphism ψ : H → G. From an automor...

Let L(s, π) be the principal L-function attached to an irreducible unitary cuspidal automorphic representation π of GLm(AQ). The aim of the paper is to give a simple method to show the lower bounds of mean value for automorphic L-functions over short intervals.

Let ∧ : GLn(C) −→ GLN (C), where N = n(n−1) 2 , be the map given by the exterior square. Then Langlands’ functoriality predicts that there is a map from cuspidal representations of GLn to automorphic representations of GLN , which satisfies certain canonical properties. To explain, let F be a number field, and let A be its ring of adeles. Let π = ⊗ v πv be a cuspidal (automorphic) representatio...

I present a general theory of overconvergent p-adic automorphic forms and eigenvarieties for connected reductive algebraic groups G whose real points are compact modulo centre, extending earlier constructions for forms of GLn due to Buzzard, Chenevier and Yamagami. This leads to some new phenomena, including the appearance of intermediate spaces of “semiclassical” automorphic forms; this gives ...

This paper describes the lifting of automorphic characters of O(3)(A) to S̃L2(A). It does so by matching the image of this lift with the lift of automorphic characters from O(1)(A) to S̃L2(A). Our matching actually gives a matching of individual automorphic forms, and not just of representation spaces. Let V be a 3− dimensional quadratic vector space and U a certain 1− dimensional quadratic space...

The method of L functions is one of the major methods for analyzing automorphic forms. For example, the Hecke Converse Theorem gives an equivalence via the Mellin transform between holomorphic modular forms on the upper half plane and certain L functions associated to Dirichlet series, which have analytic continuation and functional equation. The classical theory of automorphic forms on the gro...

We obtain explicit formulas for the test vector in the Bessel model and derive the criteria for existence and uniqueness for Bessel models for the unramified, quadratic twists of the Steinberg representation π of GSp4(F ), where F is a non-archimedean local field of characteristic zero. We also give precise criteria for the Iwahori spherical vector in π to be a test vector. We apply the formula...

Goal of Conference 2 1. Matt Emerton: Classical Modular Forms to Automorphic Forms 2 1.1. The Growth Condition 3 1.2. Passage to Representation Theory 4 2. David Nadler: Real Lie Groups 5 2.1. Basic Notions 5 2.2. Examples 5 2.3. Classification 6 2.4. Useful Decompositions 7 3. Jacob Lurie: Lie Theory and Algebraic Groups 8 3.1. Classification 9 4. Jacob Lurie: Representations of algebraic grou...