We prove a commutative algebra result which has consequences for congruences between automorphic forms modulo prime powers. If C denotes the congruence module for a fixed automorphic Hecke eigenform π0 we prove an exact relation between the p-adic valuation of the order of C and the sum of the exponents of p-power congruences between the Hecke eigenvalues of π0 and other automorphic forms. We a...

We introduce the concept of bi-square-mean almost automorphic functions and Stepanov-like square-mean pseudo almost automorphic functions for stochastic processes. Using the results, we also study the existence and uniqueness theorems for Stepanov-like square-mean pseudo almost automorphic mild solutions to a class of nonautonomous stochastic functional evolution equations in a real separable H...

By Langlands duality one usually understands a correspondence between automorphic representations of a reductive group G over the ring of adels of a field F , and homomorphisms from the Galois group Gal(F/F ) to the Langlands dual group GL. It was originally introduced in the case when F is a number field or the field of rational functions on a curve over a finite field [1]. Recently A. Beilins...

The method of L functions is one of the major methods for analyzing automorphic forms. For example, the Hecke Converse Theorem gives an equivalence via the Mellin transform between holomorphic modular forms on the upper half plane and certain L functions associated to Dirichlet series, which have analytic continuation and functional equation. The classical theory of automorphic forms on the gro...

The purpose of this note is to describe a method for computing general automorphic forms. I have carried out only limited computational tests so far, and have not discovered any new automorphic forms using it. However, the method does identify some lifted cusp forms on GL(3) and until recently was the only general method to compute an automorphic form on a higher rank group. It generalizes the ...

A well-known formula of Whipple relates certain hypergeometric values $$_7F_6(1)$$ and $$_4F_3(1)$$ . In this paper, we revisit relation from the viewpoint underlying data HD, to which there are also associated character sums Galois representations. We explain a special structure behind Whipple’s when HD primitive self-dual. If defined over $$\mathbb Q$$ , by work Katz, Beukers, Cohen, Mellit, ...

We study generalized special cycles on Hermitian locally symmetric spaces $$\Gamma \backslash D$$ associated to the groups $$G={{\mathrm {U}}}(p,q)$$ , $${{\mathrm {Sp}}}(2n,{\mathbb {R}}) $$ and {O}}}^*(2n) . These are algebraic covered by subgroups of G which same type. show that Poincaré duals these can be viewed as Fourier coefficients a theta series. This gives new cases lifts from cohomol...

It is well known that Frobenius reciprocity is one of the central tools in the representation theory. In this paper, we discuss Frobenius reciprocity in the theory of automorphic functions. This Frobenius reciprocity was discovered by Gel’fand, Fomin, and PiatetskiShapiro in the 1960s as the basis of their interpretation of the classical theory of automorphic functions in terms of the represent...

Let K/Q be a number field. Let π and π′ be cuspidal automorphic representations of GLd(AK) and GLd′(AK), and suppose that either both d and d′ are at most 2 or at least one of π and π′ is self-dual. We prove an effective log-free zero density estimate for the Rankin-Selberg L-function L(s, π ⊗ π′,K) when at least one of L(s, π,K) and L(s, π′,K) satisfies the generalized Ramanujan conjecture. Wi...