On zeta functions and families of Siegel modular forms

Let p be a prime, and let G = \textS\textp_g( \mathbbZ ) \Gamma = {\text{S}}{{\text{p}}_g}\left( \mathbb{Z} \right) be the Siegel modular group of genus g. This paper is concerned with p-adic families of zeta functions and L-functions of Siegel modular forms; the latter are described in terms of motivic L-functions attached to Sp _g ; their analytic properties are given. Critical values for the spinor L-functions are discussed in relation to p-adic constructions. Rankin’s lemma of higher genus is established. A general conjecture on a lifting of modular forms from GSp_2m   × GSp_2m to GSp_4m (of genus g = 4 m) is formulated. Constructions of p-adic families of Siegel modular forms are given using Ikeda–Miyawaki constructions.     

On a unit group generated by special values of Siegel modular functions

There has been important progress in constructing units and -units associated to curves of genus 2 or 3. These approaches are based mainly on the consideration of properties of Jacobian varieties associated to hyperelliptic curves of genus 2 or 3. In this paper, we construct a unit group of the ray class field of modulo 6 with full rank by special values of Siegel modular functions and circular units. We note that . Our construction of units is number theoretic, and closely based on Shimura's work describing explicitly the Galois actions on the special values of theta functions.

     

Rapidly convergent series representations of symmetric Tornheim double zeta functions

Acta Mathematica Hungarica - For $$s,t,u in {mathbb{C}}$$ , we show rapidly (or globally) convergent series representations of the Tornheim double zeta function T(s, t, u) and (desingularized)...     

On geometric representations of modular ortholattices

A pre-orthogonality on a projective geometry is a symmetric binary relation, ⊥, such that for each point ${p, p^{\perp} = \{q | p \perp q \}}$ is a subspace. An orthogonality is a pre-orthogonality such that each p ^⊥ is a hyperplane. Such ⊥ is called anisotropic iff it is irreflexive. For projective geometries with an anisotropic pre-orthogonality, we show how to find a (large) projective subgeometry with a natural embedding for the lattices of subspaces and with an orthogonality induced by the given pre-orthogonality. We also discuss (faithful) representations of modular ortholattices within this context and derive a condition which allows us to transform a representation by means of an anisotropic pre-orthogonality into an anisotropic orthogeometry by means of an anisotropic orthogonality.     

On a certain class of modular functions

We give a characterization of those meromorphic modular functions on a subgroup of finite index of the full modular group whose divisors are supported at the cusps, in terms of the growth of the exponents of their infinite product expansions.

     

On unitary deformations of smooth modular representations

Let G be a lo/cally ? _p -analytic group and K a finite extension of ? _p with residue field k. Adapting a strategy of B. Mazur (cf. [Maz89]) we use deformation theory to study the possible liftings of a given smooth G-representation ρ over k to unitary G-Banach space representations over K. The main result proves the existence of a universal deformation space in case ρ admits only scalar endomorphisms. As an application we let G = GL₂(? _p ) and compute the fibers of the reduction map in principal series representations.     

On some series representations of the Hurwitz zeta function

A variety of infinite series representations for the Hurwitz zeta function are obtained. Particular cases recover known results, while others are new. Specialization of the series representations apply to the Riemann zeta function, leading to additional results. The method is briefly extended to the Lerch zeta function. Most of the series representations exhibit fast convergence, making them attractive for the computation of special functions and fundamental constants.     

On zeta functions and Iwasawa modules

We study the relation between zeta-functions and Iwasawa modules. We prove that the Iwasawa modules for almost all determine the zeta function when is a totally real field. Conversely, we prove that the -part of the Iwasawa module is determined by its zeta-function up to pseudo-isomorphism for any number field Moreover, we prove that arithmetically equivalent CM fields have also the same -invariant.

     

On the poles of topological zeta functions

We study the topological zeta function associated to a polynomial with complex coefficients. This is a rational function in one variable, and we want to determine the numbers that can occur as a pole of some topological zeta function; by definition these poles are negative rational numbers. We deal with this question in any dimension. Denote has a pole in . We show that is a subset of ; for and , the last two authors proved before that these are exactly the poles less than . As the main result we prove that each rational number in the interval is contained in .

     

Congruence of Siegel modular forms and special values of their standard zeta functions

In this paper, we consider the relationship between the congruence of cuspidal Hecke eigenforms with respect to Sp _n (Z) and the special values of their standard zeta functions. In particular, we propose a conjecture concerning the congruence between Saito-Kurokawa lifts and non-Saito-Kurokawa lifts, and prove it under certain condition. Partially supported by Grant-in-Aid for Scientific Research C-17540003, JSPS.     

On Igusa zeta functions of monomial ideals

We show that the real parts of the poles of the Igusa zeta function of a monomial ideal can be computed from the torus-invariant divisors on the normalized blow-up of the affine space along the ideal. Moreover, we show that every such number is a root of the Bernstein-Sato polynomial associated to the monomial ideal.

     

On holomorphic representations of discrete Heisenberg groups

We define a class of monomial not necessary unitary representations of discrete Heisenberg groups. Their moduli space is a complex-analytic manifold. There exist characters of the representations which are automorphic forms on the moduli space.     

On irreducible representations of discrete Heisenberg groups

We prove that the irreducible representations with finite weight of discrete nilpotent groups of class 2, including the discrete Heisenberg groups, are monomial representations and obtain their classification for the group Heis(3, ?).