Tsuzuki Masao

Abstract In this paper, we give an explicit formula for the Shintani double zeta functions with any ramification in the most general setting of adeles over an arbitrary number field. Three applications of the explicit formula are given. First, we obtain a functional equation satisfied by the Shintani double zeta functionsin addition to Shintani’s functional equations.Second, we establish the holomorphicity of a certain Dirichlet series generalizing a result by Ibukiyama and Saito. This Dirichlet series occurs in the study of unipotent contributions of the geometric side of the Arthur–Selberg trace formula of the symplectic group.Third, we prove an asymptotic formula of the weighted average of the central values of quadratic Dirichlet L-functions.

For a given stale quadratic algebra E over a p-adic field F, we establish a transfer of unramified test functions on the symmetric space GL(2, F)\GL(2, E) to those on a unitary hyperbolic space so that the orbital integrals match. This is an important step toward a comparison of relative trace formulas of these symmetric spaces, which would yield an example of a non-tempered analogue of a refined global Gross-Prasad conjecture.

We develop a derivative version of the relative trace formula on PGL(2) studied in our previous work, and derive an asymptotic formula of an average of central values (derivatives) of automorphic L-functions for Hilbert cusp forms. As an application, we prove the existence of Hilbert cusp forms with non-vanishing central values (derivatives) such that the absolute degrees of their Hecke fields are arbitrarily large.

Given a maximal even integral lattice L of signature (m+, 2-) (m >= 3), we consider an orthonormal Hecke eigen basis B-1 of the holomorphic cusp forms of weight I on a tube domain with respect to the discriminant subgroup of the orthogonal group O(L). We construct a certain adelic holomorphic Poincare series whose spectral expansion is described by the central standard L-value of F is an element of B-1 and the square of a Fourier-Bessel coefficient of F. (C) 2012 Elsevier Inc. All rights reserved.

Let (G, H) = (U(p,q),U (p - 1, q) x U(1)) and {Gamma(n)} a tower of congruence uniform lattices in G. By the period integrals of automorphic forms on Gamma\G along Gamma(n) boolean AND H\H, we introduce a certain discrete measure d mu(H)(Gamma n) on the H-spherical unitary dual of G. It is shown that the sequence of measures d mu(H)(Gamma n) with growing n converges in a weak sense to the Plancherel measure d(mu)(H) for the symmetric space H\G.

We study the period integrals of Laplace eigenfunctions on an arithmetic quotient X of the d-dimensional hyperbolic space along a fixed eigenfunction on an arithmetic quotient of (d - 1)-dimensional hyperbolic space embedded in X. We introduce a certain counting function for period integrals and prove its asymptotic law. (c) 2009 Elsevier Inc. All rights reserved.

We consider the real rank one unitary group G and its subgroup H obtained as the stabilizer of an anisotropic vector in the skew-hermitian space defining G. We compute the inner-product of an Eisenstein series on H and a non-holomorphic cuspidal Hecke eigenform on G restricted to H to obtain an integral representation of the standard L-function of the eigenform. We also discuss sonic consequences of the integral representation.

Given a modular embedding j : Delta\H/H boolean AND K -> Gamma\G/K associated with an equivariant embedding (H, H/H boolean AND K) -> (G, G/K) of symmetric domains with actions of a semisimple Lie group G and a reductive subgroup H, both defined over Q compatibly, together with some other conditions. Starting from a certain harmonic left H-invariant "spherical" current on G/K with sigularity along HK/K, we can define a Poincare series. Apply partial derivative(partial derivative) over bar operator to the analytic continutation of this with respect to the parameter of eigenvalues of the "Laplacian". Then as a analogue of the Kronecker limit formula, we can construct a Green current for the cycle defined by j. This is a continuation of the previous paper [26], and here we treat the case of higher-codimensional cycles with compact Gamma\G/K.

Let G be the unitary group of a non-degenerate Hermitian space and H the stabilizer of a one-dimensional positive definite subspace of the Hermitian space. For a uniform lattice Gamma in G such that Gamma boolean AND H is a uniform lattice of H, we introduce the (averaged) H-period integrals of automorphic forms on Gamma\G; we study their behavior as Gamma shrinks to the identity along a tower of lattices in G and prove a limit formula of the H-period integrals. (C) 2008 Elsevier Inc. All rights reserved.

We obtain a Green current in the sense of Gillet-Soule on an arithmetic quotient of a complex hyperball for the modular cycle stemming from a complex subhyperball of codimension greater than one, generalizing the classical construction of the automorphic Green function for the modular curves.