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.

In this paper we review some connections between harmonic analysis and the modern theory of automorphic forms. We indicate in some examples how the study of problems of harmonic analysis brings us to the important objects of the theory of automorphic forms, and conversely. We consider classical groups and their unitary, tempered, automorphic and unramified duals. The most important representati...

In this paper, we shall deal with μ-pseudo almost automorphic solutions to a semi-linear fractional differential equation by a new concept of μ-pseudo almost automorphic functions presented recently. First we establish some new properties of μ-pseudo almost automorphic functions, and then apply the obtained results to prove some existence theorems combined with the Leray-Schauder alternative th...