[1] Despite occasional contrary assertions in the literature, rewriting Eisenstein series, as opposed to more general automorphic forms, to make sense on adele groups is not about Strong Approximation. Strong Approximation does make precise the relation between general automorphic forms on adele groups and automorphic forms on SLn, but rewriting these Eisenstein series does not need this compar...

We present a new method to estimate trilinear period for automorphic representations of SL2(R). The method is based on the uniqueness principle in representation theory. We show how to separate the exponentially decaying factor in the triple period from the essential automorphic factor which behaves polynomially. We also describe a general method which gives an estimate on the average of the au...

In the theory of automorphic forms, two classes of rank one reductive Lie groups O(n, 1) and U(n, 1) are the important objects. Automorphic forms on O(n, 1) have been intensively studied. In this paper we study the automorphic forms on U(n, 1). We construct infinitely many modular forms and non-holomorphic automorphic forms on U(n, 1) with respect to a discrete subgroup of infinite covolume. Mo...

This paper describes the lifting of automorphic characters of O(3)(A) to S̃L2(A). It does so by matching the image of this lift with the lift of automorphic characters from O(1)(A) to S̃L2(A). Our matching actually gives a matching of individual automorphic forms, and not just of representation spaces. Let V be a 3− dimensional quadratic vector space and U a certain 1− dimensional quadratic space...

We study almost automorphic (mild) solutions of the semilinear fractional equation ∂ t u = Au + ∂ α−1 t f(·, u), 1 < α < 2, considered in a Banach space X, where A is a linear operator of sectorial type ω < 0. We prove the existence and uniqueness of an almost automorphic mild solution assuming f(t, x) is almost automorphic in t for each x ∈ X, satisfies some Lipschitz type conditions and takes...

We prove a companion forms theorem for ordinary n-dimensional automorphic Galois representations, by use of automorphy lifting theorems developed by the second author, and a technique for deducing companion forms theorems due to the first author. We deduce results about the possible Serre weights of mod l Galois representations corresponding to automorphic representations on unitary groups. We ...

A loop is automorphic if its inner mappings are automorphisms. Using socalled associated operations, we show that every commutative automorphic loop of odd prime power order is centrally nilpotent. Starting with suitable elements of an anisotropic plane in the vector space of 2 × 2 matrices over the field of prime order p, we construct a family of automorphic loops of order p with trivial center.

In [59], we introduced an invariant of K3 surfaces with involution, which we obtained using equivariant analytic torsion. This invariant gives rise to a function on the moduli space of K3 surfaces with involution and is expressed as the Petersson norm of an automorphic form characterizing the discriminant locus. In this paper, we study the structure of this automorphic form. Under certain assum...

This is an introduction to the theory of Shimura varieties, or, in other words, to the arithmetic theory of automorphic functions and holomorphic automorphic forms.

In this paper, we introduce the notion of Maass-Jacobi forms and investigate some properties of these new automorphic forms. We also characterize these automorphic forms in several ways.