We generalize a beautiful method of Blasius and Ramakrishnan, that in order to exhibit particular instances of the Langlands functorial correspondence, it is enough to show that the correspondence holds in the semistable case, provided the arithmetic data we are considering is closed with respect to cyclic base change and descent.

In this paper, we consider a novel version of the classical Jacquet-Langlands correspondence, explore a p-adic extension of the correspondence , and as an explicit example we find an overconvergent automorphic form of weight 1 which corresponds to a classical modular form of weight 1, using both experimental and theoretical methods.

Two of the primary methods of constructing automorphic forms are the Langlands program and Howe's theory of dual pairs. The Langlands program concerns a reductive linear group G deened over a number eld. Associated to G is its dual group L G. The con-jectural principle of functoriality says that a homomorphism L H ! L G should provide a \transfer" of automorphic representations from H to those ...

We use the p-adic local Langlands correspondence for GL2(Qp) to explicitly compute the reduction modulo p of certain 2-dimensional crystalline representations of small slope, and give applications to modular forms.

We study quadratic periods on quaternionic Shimura varieties and formulate an integral refinement of Shimura's conjecture regarding Petersson inner products automorphic forms that are related by the Jacquet-Langlands correspondence. The main result is this implied another (Conjecture D below) integrality theta lifts between certain unitary groups.

Let K be a local non-Archimedean field of positive characteristic and let L be the degree-n unramified extension of K. Let θ be a smooth character of L× such that for each nontrivial γ ∈ Gal(L/K), θ and θ/θ have the same level. Via the local Langlands and Jacquet-Langlands correspondences, θ corresponds to an irreducible representation ρθ of D×, where D is the central division algebra over K wi...

In the late 1960s Robert Langlands launched what has become known as the Langlands Program with the ambitious goal of relating deep questions in Number Theory to Harmonic Analysis [L]. In particular, Langlands conjectured that Galois representations and motives can be described in terms of the more tangible data of automorphic representations. A striking application of this general principle is...