For a fairly general reductive group G/Qp , we explicitly compute the space of locally algebraic vectors in the Breuil-Herzig construction Π(ρ)ord, for a potentially semistable Borel-valued representation ρ of Gal(Q̄p/Qp). The point being we deal with the whole representation, not just its socle – and we go beyond GLn(Qp). In the case of GL2(Qp), this relation is one of the key properties of the...

We generalise Coleman’s construction of Hecke operators to define an action of GL2(Ql) on the space of finite slope overconvergent p-adic modular forms (l 6= p). In this way we associate to any Cp-valued point on the tame level N Coleman-Mazur eigencurve an admissible smooth representation of GL2(Ql) extending the classical construction. Using the Galois theoretic interpretation of the eigencur...

In [7], results about the global Jacquet-Langlands correspondence, (weak and strong) multiplicity-one theorems and the classification of automorphic representations for inner forms of the general linear group over a number field are established, under the condition that the local inner forms are split at archimedean places. In this paper, we extend the main local results of [7] to archimedean p...

The goal of this paper is to generalize a theorem of Fujiwara (formerly Deligne’s conjecture) to the situation appearing in a joint work [KV] with David Kazhdan on the global Langlands correspondence over function fields. Moreover, our proof is much simpler than the original one and applies to more general situations like algebraic spaces or Deligne–Mumford stacks without any changes. We also i...

In this paper we construct certain moduli spaces, which we call moduli spaces of (principal) F -bundles, and study their basic properties. These spaces are associated to triples consisting of a smooth projective geometrically connected curve over a finite field, a split reductive group G, and an irreducible algebraic representation ω of (Ǧ)n/Z(Ǧ). Our spaces generalize moduli spaces of F -sheav...

In this paper, we study modular forms on two simply connected groups of type D4 over Q. One group, Gs, is a globally split group of type D4, viewed as the group of isotopies of the split rational octonions. The other, Gc, is the isotopy group of the rational (non-split) octonions. We study automorphic forms on Gs in analogy to the work of Gross, Gan, and Savin on G2; namely we study automorphic...

Introduction 1 1. Modular forms, congruences and the adjoint L-function 2 2. Quaternion algebras and the Jacquet-Langlands correspondence 6 3. Integral period relations for quaternion algebras over Q 8 4. The theta correspondence 12 5. Arithmetic of the Shimizu lift and Waldspurger’s formula 16 6. Hilbert modular forms, Shimura’s conjecture and a refined version 19 7. Unitary groups and Harris’...

By the geometric Satake correspondence, the number of components of certain fibres of the affine Grassmannian convolution morphism equals the tensor product multiplicity for representations of the Langlands dual group. On the other hand, in the case of GLn, combinatorial objects called hives also count tensor product multiplicities. The purpose of this paper is to give a simple bijection betwee...

For a fairly general reductive group G/Qp , we explicitly compute the space of locally algebraic vectors in the Breuil–Herzig construction (ρ)ord , for a potentially semistableBorel-valued representationρ ofGal(Q̄p/Qp). The point beingwedealwith thewhole representation, not just its socle—and we go beyond GLn(Qp). In the case of GL2(Qp), this relation is one of the key properties of the p-adic l...

Abstract: There exists an intriguing relation between genus zero correlation functions in the H 3 WZNW model and in Liouville field theory. We provide a path integral derivation of the correspondence and then use our new approach to generalize the relation to surfaces of arbitrary genus g. In particular we determine the correlation functions of N primary fields in the WZNWmodel explicitly throu...