We develop the orbit method in a quantitative form, along lines of microlocal analysis, and apply it to analytic theory automorphic forms. Our main global application is an asymptotic formula for averages Gan--Gross--Prasad periods arbitrary rank. The form on larger group held fixed, while that smaller varies over family size roughly fourth root conductors corresponding $L$-functions. Ratner's ...

For two positive integers m and n, we let Pn be the open convex cone in R consisting of positive definite n × n real symmetric matrices and let R be the set of all m×n real matrices. In this article, we investigate differential operators on the non-reductive manifold Pn × R (m,n) that are invariant under the natural action of the semidirect product group GL(n,R)⋉R on the MinkowskiEuclid space P...

A variant of Hadamard’s notion of partie finie is applied to the theory of automorphic functions on arithmetic quotients of the upper half-plane. As a consequence, conceptually simple proofs of the volume formula and the Maass-Selberg relations are given. This technique interprets Zagier’s idea of renormalization (Jour. Fac. Sci. Univ. Tokyo 28 (1982), 415–437) so that it can be generalized eas...

Automorphic distributions for SL(2) arise as boundary values of modular forms and, in a more subtle manner, from Maass forms. In the case of modular forms of weight one or of Maass forms, the automorphic distributions have continuous first antiderivatives. We recall earlier results of one of us on the Hölder continuity of these continuous functions and relate them to results of other authors; t...

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...

The Lewis-Zagier correspondence, which attaches period functions to Maaß wave forms, is extended to wave forms of higher order, which are higher invariants of the Fuchsian group in question. The key ingredient is an identification of Higher order invariants with ordinary invariants of unipotent twists. This makes it possible to apply standard methods of automorphic forms to higher order forms.