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We prove that certain archimedean integrals arising in global zeta integrals involving holomorphic discrete series on unitary groups are predictable powers of π times rational or algebraic numbers. In some cases we can compute the integral exactly in terms of values of gamma functions, and it is plausible that the value in the most general case is given by the corresponding expression. Non-vanishing of the algebraic factor is readily demonstrated via the explicit expression. Regarding analytical aspects of such integrals, whether archimedean or p-adic, a recent systematic treatment is [Lapid-Rallis 2005] in the Rallis conference volume. In particular, the results of Lapid and Rallis allow us to focus on the arithmetic aspects of the special values of the integrals. This is implicit in (4.4) (iv) in [Harris 2006]. Roughly, the integrals here are those that arise in the so-called doubling method if the Siegel-type Eisenstein series is differentiated transversally before being restricted to the smaller group. In traditional settings, details involving Fourier expansions would be apparent, but such details are inessential. Specifically, many Fourier-expansion details concerning classical Maaß-Shimura operators are spurious, referring, in fact, only to the structure of holomorphic discrete series representations. Of course, the translation to and from rationality issues in spaces of automorphic forms should not be taken lightly. [ABSTRACT FROM AUTHOR]

We explain an important result of Hee Oh [20] on bounding matrix coefficients of semi-simple groups and survey some applications. [ABSTRACT FROM AUTHOR]

This is a semi-expository article concerning Langlands functoriality and Deligne's conjecture on the special values of L-functions. The emphasis is on symmetric power L-functions associated to a holomorphic cusp form, while appealing to a recent work of Mahnkopf on the special values of automorphic L-functions. [ABSTRACT FROM AUTHOR]

We review some recent results on the arithmetic of the theta correspondence for certain symplectic-orthogonal dual pairs and some applications to periods and congruences of modular forms. We also propose an integral version of a conjecture on Petersson inner products of modular forms on quaternion algebras over totally real fields. [ABSTRACT FROM AUTHOR]

We prove meromorphic continuation of Eisenstein series for smooth but not necessarily K-finite sections. [ABSTRACT FROM AUTHOR]

We survey recent extensions of the classical Siegel-Weil formula linking special values of certain Eisenstein series with integrals of theta functions to relations between residues of Eisenstein series and regularized integrals of theta functions. [ABSTRACT FROM AUTHOR]

In this paper, we discuss an approach which determines the residues of certain families of Eisenstein series in terms of periods on the cuspidal data. By contrast, the traditional approach is to determine the residues of Eisenstein series in terms of certain L-functions attached to the cuspidal data. The relation between these two methods is the general conjecture which relates periods on the cuspidal data to the existence of poles or the special values of L-functions attached to the cuspidal data. [ABSTRACT FROM AUTHOR]

We study rationality properties of Eisenstein series on quasi-split unitary similitude groups. [ABSTRACT FROM AUTHOR]

We discuss the construction, characterization and classification of the Saito-Kurokawa representations of PGS{inp4} and its inner forms, interpreting them in the framework of Arthur's conjectures. These Saito-Kurokawa representations are among the first examples of the so-called CAP representations or shadows of Eisenstein series. [ABSTRACT FROM AUTHOR]

We construct a topological model for certain Eisenstein cohomology of congruence subgroups. [ABSTRACT FROM AUTHOR]

Weyl group multiple Dirichlet series were associated with a root system Φ and a number field F containing the n-th roots of unity by Brubaker, Bump, Chinta, Friedberg, and Hoffstein [2]. Brubaker, Bump, and Friedberg [4] provided for when n is sufficiently large; the coefficients involve n-th order Gauss sums and reflect the combinatorics of the root system. Conjecturally, these functions coincide with Whittaker coefficients of metaplectic Eisenstein series, but they are studied in these papers by a method that is independent of this fact. The assumption that n is large is called stability and allows a simple description of the Dirichlet series. "Twisted" Dirichet series were introduced in Brubaker, Bump, Friedberg, and Hoffstein [5] without the stability assumption, but only for root systems of type A{inr}. Their description is given differently, in terms of Gauss sums associated to Gelfand-Tsetlin patterns. In this paper, we reimpose the stability assumption and study the twisted multiple Dirichlet series for general Φ by introducing a description of the coefficients in terms of the root system similar to that given in the untwisted case in [4]. We prove the analytic continuation and functional equation of these series, and when Φ = A{inr} we also relate the two different descriptions of multiple Dirichlet series given here and in [5] for the stable case. [ABSTRACT FROM AUTHOR]

We discuss several applications of the recent developments in the Langlands functoriality conjecture such as the automorphy of the symmetric powers of 2-dimensional complex representations of Galois groups of number fields, lattice point problems, Ramanujan- Selberg and Sato-Tate conjectures.We conclude by explaining how these recent developments are established. [ABSTRACT FROM AUTHOR]

In this article we study the discriminant of those K3 surfaces with involution which were introduced and investigated by Matsumoto, Sasaki, and Yoshida. We extend several classical results on the discriminant of elliptic curves to the discriminant of Matsumoto- Sasaki-Yoshida's K3 surfaces. [ABSTRACT FROM AUTHOR]

This paper describes our method of pairing automorphic distributions.We present a third technique for obtaining the analytic properties of automorphic L-functions, in addition to the existing methods of integral representations (Rankin-Selberg) and Fourier coefficients of Eisenstein series (Langlands-Shahidi). We recently used this technique to establish new cases of the full analytic continuation of the exterior square L-functions. The paper here gives an exposition of our method in two special, yet representative cases: the Rankin-Selberg tensor product L-functions for PGL(2,Z)∖PGL(2,R), as well as for the exterior square L-functions for GL(4,Z)∖GL(4,R). [ABSTRACT FROM AUTHOR]

The complex analytic methods have found a wide range of applications in the study of multiplicity-free representations. This article discusses, in particular, its applications to the question of restricting highest weight modules with respect to reductive symmetric pairs. We present a number of multiplicity-free branching theorems that include the multiplicity-free property of some of known results such as the Clebsh-Gordan-Pieri formula for tensor products, the Plancherel theorem for Hermitian symmetric spaces (also for line bundle cases), the Hua-Kostant-Schmid K-type formula, and the canonical representations in the sense of Vershik-Gelfand-Graev. Our method works in a uniform manner for both finite and infinite dimensional cases, for both discrete and continuous spectra, and for both classical and exceptional cases. [ABSTRACT FROM AUTHOR]

In this article, we discuss a lifting of elliptic cusp forms to higher dimensional symmetric spaces. We will consider two cases. The first case is the Siegel modular case, and the second case is the hermitian modular case. The Fourier coefficients of our liftings are closely related to those of Eisenstein series. When the degree is 2, our lifting reduces to the classical Saito-Kurokawa lifting or hermitian Maass lifting. [ABSTRACT FROM AUTHOR]

Suppose ρ is an n-dimensional representation of the absolute Galois group of Q which is associated, via an identity of L-functions, with an automorphic representation π of GL(n) of the adele ring of Q. It is expected that π is cuspidal if and only if ρ is irreducible, though nothing much is known in either direction in dimensions > 2. The object of this article is to show for n < 6 that the cuspidality of a regular algebraic π is implied by the irreducibility of ρ. For n < 5, it suffices to assume that π is semi-regular. [ABSTRACT FROM AUTHOR]