The next steps after this prehistoric Part I will be directed to a description of the Howe operators introduced by Kudla and Millson and their special Schwartz forms and classes. This has as attractor the fact that the modular and automorphic forms arising naturally in context with these classes find very nice geometric interpretations of their Fourier coefficients and thus lead to an intriguin...

Our aim in this paper is to present a framework in which the results of Waldspurger and Gross–Zagier can be viewed simultaneously. This framework may also be useful in understanding recent work of Zhang, Xue, Cornut, Vatsal, and Darmon. It involves a blending of techniques from representation theory and automorphic forms with those from the arithmetic of modular curves. I hope readers from one ...

A result of Chai–Faltings on Satake parameters of Siegel cusp forms together with the classification of unitary, unramified, irreducible, admissible representations of GSp4 over a p-adic field, imply that the local components of the automorphic representation of GSp4 attached to a cuspidal Siegel eigenform of degree 2 must lie in certain families. Applications include estimates on Hecke eigenva...

The Jacobi group is the semi-direct product of the symplectic group and the Heisenberg group. The Jacobi group is an important object in the framework of quantum mechanics, geometric quantization and optics. In this paper, we study the Weil representations of the Jacobi group and their properties. We also provide their applications to the theory of automorphic forms on the Jacobi group and repr...

In this paper, we present a refinement of the global Langlands correspondence for discrete symplectic motives of rank 2n over Q. To such a motive Langlands conjecturally associates a generic, automorphic representation π of the split orthogonal group SO2n+1 over Q, which appears with multiplicity one in the cuspidal spectrum. Using the local theory of generic representations of odd orthogonal g...

We consider the Fourier expansions of automorphic forms on general Lie groups, with a particular emphasis on exceptional groups. After describing some principles underlying known results on GL(n), Sp(4), and G2, we perform an analysis of the expansions on split real forms of E6 and E7 where simplifications take place for automorphic realizations of real representations which have small Gelfand-...

Irreducible representations are the building blocks of general, semisimple Galois representations ρ, and cuspidal representations are the building blocks of automorphic forms π of the general linear group. It is expected that when an object of the former type is associated to one of the latter type, usually in terms of an identity of L-functions, the irreducibility of the former should imply th...

In this paper, we define the concept of Jacobi forms of half-integral weight using Takase’s automorphic factor of weight 1/2 for a two-fold covering group of the symplectic group on the Siegel upper half plane and find covariant maps for the SchrödingerWeil representation. Using these covariant maps, we construct Jacobi forms of half integral weight with respect to an arithmetic subgroup of the...

We formulate a Serre-type conjecture for n-dimensional Galois representations that are tamely ramified at p. The weights are predicted using a representation-theoretic recipe. For n = 3 some of these weights were not predicted by the previous conjecture of Ash, Doud, Pollack, and Sinnott. Computational evidence for these extra weights is provided by calculations of Doud and Pollack. We obtain t...

Let [Formula: see text] be a congruence subgroup of text]. Using Poincaré series text]-finite matrix coefficients integrable discrete representations text], we construct spanning set for the space Siegel cusp forms weight We prove non-vanishing certain elements this using Muić’s integral criterion on locally compact Hausdorff groups. Moreover, representation theory study Petersson inner product...