We find defining equations for the Shimura curve of discriminant 15 over Z[1/15]. We then determine the graded ring of automorphic forms over the 2-adic integers, as well as the higher cohomology. We apply this to calculate the homotopy groups of a spectrum of “topological automorphic forms” associated to this curve, as well as one associated to a quotient by an Atkin-Lehner involution.

We here formulate and prove several new results concerning the Eichler cohomology of automorphic forms on finitely generated Fuchsian groups of the first kind and, in particular, on //-groups. The Eichler cohomology we introduce is connected with automorphic forms of arbitrary real degree (as opposed to integral degree), with a suitably chosen underlying space of functions analytic in the upper...

We apply a theorem of J. Lurie to produce cohomology theories associated to certain Shimura varieties of type U(1, n−1). These cohomology theories of topological automorphic forms (TAF ) are related to Shimura varieties in the same way that TMF is related to the moduli space of elliptic curves. We study the cohomology operations on these theories, and relate them to certain Hecke algebras. We c...

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...

A lifting map from cuspidal automorphic representations of the Jacobi group G J to cuspidal auto-morphic representations of the group PGL(2) is constructed. The lifting also has a local deenition, and the local and global versions are compatible. The local lifts can be described in terms of equivalence classes of local representations. The main idea in the construction is to exploit the close r...

where A t for t ∈ R is a family of closed linear operators with domains D A t satisfying Acquistapace-Terreni conditions, and the function f : R × X → X is almost automorphic in t ∈ R uniformly in the second variable, was studied. For that, the author made extensive use of techniques utilized in 2 , exponential dichotomy tools, and the Schauder fixed point theorem. In this paper we study the ex...

We study limits of discrete series with infinitesimal character zero for Sp4: their n-cohomology and their contribution to “nonclassical” automorphic cohomology of the period domain for Hodge structures of mirror quintic type. As an application, we obtain the first generalization beyond SU(2, 1) of a result of [C1], showing that this cohomology can be reached by cup products of pairs of “classi...

Part I. The origins of the Langlands Program 9 1. The Langlands correspondence over number fields 9 1.1. Galois group 9 1.2. Abelian class field theory 10 1.3. Frobenius automorphisms 13 1.4. Rigidifying ACFT 14 1.5. Non-abelian generalization? 15 1.6. Automorphic representations of GL2(AQ) and modular forms 18 1.7. Elliptic curves and Galois representations 22 2. From number fields to function...