Shintani Zeta Functions Research Articles

Subconvexity of twisted Shintani zeta functions

Previously the authors proved subconvexity of Shintani's zeta function enumerating class numbers of binary cubic forms. Here we return to prove subconvexity of the Maass form twisted version. The method demonstrated here has applications to the subconvexity of some of the twisted zeta functions introduced by F. Sato. The argument demonstrates that the symmetric space condition used by Sato is not necessary to estimate the zeta function in the critical strip.

Arithmetic of Hecke L-functions of quadratic extensions of totally real fields

Deep work by Shintani in the 1970's describes Hecke L-functions associated to narrow ray class group characters of totally real fields F in terms of what are now known as Shintani zeta functions. However, for [F:Q]=n≥3, Shintani's method was ineffective due to its crucial dependence on abstract fundamental domains for the action of totally positive units of F on R+n, so-called Shintani sets. These difficulties were recently resolved in independent work of Charollois, Dasgupta, and Greenberg and Diaz y Diaz and Friedman. For those narrow ray class group characters whose conductor is an inert rational prime in a totally real field F with narrow class number 1, we obtain a natural combinatorial description of these sets, allowing us to obtain a simple description of the associated Hecke L-functions. As a consequence, we generalize earlier work of Girstmair, Hirzebruch, and Zagier, that offer combinatorial class number formulas for imaginary quadratic fields, to real and imaginary quadratic extensions of totally real number fields F with narrow class number 1. For CM quadratic extensions of F, our work may be viewed as an effective affirmative answer to Hecke's Conjecture that the relative class number has an elementary arithmetic expression in terms of the relative discriminant.

On the theory of normalized Shintani L-functions and its application to Hecke L-functions, I: Real quadratic fields

We define the class of normalized Shintani L-functions of several variables. Unlike Shintani zeta functions, the normalized Shintani L-function is a holomorphic function. Moreover it satisfies a good functional equation. We show that Hecke L-function of a real quadratic field can be expressed as a diagonal part of some normalized Shintani L-function of several variables. This gives a good several variables generalization of a Hecke L-function of a real quadratic field.

The dimensions of spaces of Siegel cusp forms of general degree

In this paper, we give a dimension formula for spaces of Siegel cusp forms of general degree with respect to neat arithmetic subgroups. The formula was conjectured before by several researchers (cf. [40,43,45,75]). The dimensions are expressed by special values of Shintani zeta functions for spaces of symmetric matrices at non-positive integers. This formula was given by Shintani for only a small part of the geometric side of the trace formula (see [64]). To be precise, it is the contribution of unipotent elements corresponding to the partitions (2j,12n−2j), where n denotes the degree and 0≤j≤n. Hence, our work is to show that all the other contributions vanish. Therefore, one finds that Shintani's formula means the dimension itself. Combining our formula and an explicit formula of the Shintani zeta functions, which was discovered by Ibukiyama and Saito (cf. [44,42,61]), we can derive an explicit dimension formula for the principal congruence subgroups of level greater than two (cf. [43,45]). In this explicit dimension formula, the dimensions are described by degree n, weight k, level N, and the Bernoulli numbers Bm.

On the Geometric Side of the Arthur Trace Formula for the Symplectic Group of Rank 2

We study the non-semisimple terms in the geometric side of the Arthur trace formula for the split symplectic similitude group or the split symplectic group of rank 2 over any algebraic number field. In particular, we show that the coefficients of unipotent orbital integrals are expressed by the Dedekind zeta function, Hecke L-functions, and the Shintani zeta function for the space of binary quadratic forms.

On a remarkable identity in class numbers of cubic rings

In 1997, Y. Ohno empirically stumbled on an astoundingly simple identity relating the number of cubic rings h(Δ) of a given discriminant Δ, over the integers, to the number of cubic rings hˆ(Δ) of discriminant −27Δ in which every element has trace divisible by 3:(1)hˆ(Δ)={3h(Δ)if Δ>0h(Δ)if Δ<0, where in each case, rings are weighted by the reciprocal of their number of automorphisms. This allows the functional equations governing the analytic continuation of the Shintani zeta functions (the Dirichlet series built from the functions h and hˆ) to be put in self-reflective form. In 1998, J. Nakagawa verified (1). We present a new proof of (1) that uses the main ingredients of Nakagawa's proof (binary cubic forms, recursions, and class field theory), as well as one of Bhargava's celebrated higher composition laws, while aiming to stay true to the stark elegance of the identity.

Identities for field extensions generalizing the Ohno–Nakagawa relations

In previous work, Ohno conjectured, and Nakagawa proved, relations between the counting functions of certain cubic fields. These relations may be viewed as complements to the Scholz reflection principle, and Ohno and Nakagawa deduced them as consequences of ‘extra functional equations’ involving the Shintani zeta functions associated to the prehomogeneous vector space of binary cubic forms. In the present paper, we generalize their result by proving a similar identity relating certain degree-$\ell$ fields to Galois groups $D_{\ell }$ and $F_{\ell }$, respectively, for any odd prime $\ell$; in particular, we give another proof of the Ohno–Nakagawa relation without appealing to binary cubic forms.

Multinomial distributions in Shintani zeta class

Multidimensional stochastic models in mathematical finance and so on are now well-studied. As to obtain more properties of them, we focus on some multidimensional discrete distributions in relation to a class of multiple zeta functions. The class called “multidimensional Shintani zeta functions” was first introduced in Aoyama and Nakamura (Tokyo J Math 36:521–538, 2013), where a class of probability distributions called “multidimensional Shintani zeta distributions associated with the zeta functions is definable. In this paper, we show that this class includes many kinds of multidimensional discrete distributions. We pick up some cases of multidimensional Shintani zeta functions and introduce some classes of distributions which contain multinomial and negative multinomial distributions as their generalizations. More precisely, we give some necessary and sufficient conditions for the functions to generate probability distributions in view of zeta functions and consider their infinite divisibilities as well.

A converse theorem for double Dirichlet series and Shintani zeta functions

The main aim of this paper is to obtain a converse theorem for double Dirichlet series and use it to show that the Shintani zeta functions which arise in the theory of prehomogeneous vector spaces are actually linear combinations of Mellin transforms of metaplectic Eisenstein series on GL (2). The converse theorem we prove will apply to a very general family of double Dirichlet series which we now define.

Shintani’s zeta function is not a finite sum of Euler products

We prove that the Shintani zeta function associated to the space of binary cubic forms cannot be written as a finite sum of Euler products. Our proof also extends to several closely related Dirichlet series. This answers a question of Wright in the negative.

Multidimensional Shintani Zeta Functions and Zeta Distributions on $\mathbb{R}^d$

The class of Riemann zeta distribution is one of the classical classes of probability distributions on $\mathbb{R}$. Multidimensional Shintani zeta function is introduced and its definable probability distributions on $\mathbb{R}^d$ are studied. This class contains some fundamental probability distributions such as binomial and Poisson distributions. The relation with multidimensional polynomial Euler product, which induces multidimensional infinitely divisible distributions on $\mathbb{R}^d$, is also studied.

On Yoshida’s conjecture on the derivative of Shintani zeta functions

The purpose of this paper is to prove a conjecture in Yoshida’s book [2, p.33] on the higher derivative of Shintani zeta functions at $s=0$. We use multivariable Shintani zeta functions to prove the conjecture.

Shintani zeta functions and Gross-Stark units for totally real fields

Let F be a totally real number field, and let p be a finite prime of F such that p splits completely in the finite abelian extension H of F. Tate has proposed a conjecture [22, Conjecture 5.4] stating the existence of a p-unit u in H with absolute values at the places above p specified in terms of the values at zero of the partial zeta functions associated to H/F. This conjecture is an analogue of Stark's conjecture, which Tate called the Brumer-Stark conjecture. Gross [12, Conjecture 7.6] proposed a refinement of the Brumer-Stark conjecture that gives a conjectural formula for the image of u in Fp×/E∧, where Fp denotes the completion of F at p and E∧ denotes the topological closure of the group of totally positive units E of F. We present a further refinement of Gross's conjecture by proposing a conjectural formula for the exact value of u in Fp×

Explicit Formula for the Hyperbolic Scattering Determinant

We prove the explicit formula for the hyperbolic scattering determinant in the case of a general subgroup Γ of PSL (2,ℝ). The class of test functions involved (not necessarily odd nor continuous) is much broader than that previously known. As an application of the technique, a new representation of the Millson–Shintani zeta function is obtained.

Shintani–Barnes zeta and gamma functions

We show that Shintani's work on multiple zeta and gamma functions can be simplified and extended by exploiting difference equations. We re-prove many of Shintani's formulas and prove several new ones. Among the latter is a generalization to the Shintani–Barnes gamma functions of Raabe's 1843 formula ∫ 0 1 log Γ(x) dx= log 2π , and a further generalization to the Shintani zeta functions. These explicit formulas can be interpreted as “vanishing period integral” side conditions for the ladder of difference equations obeyed by the multiple gamma and zeta functions. We also relate Barnes’ triple gamma function to the elliptic gamma function appearing in connection with certain integrable systems.

On the Shintani Zeta Function for the Space of Pairs of Binary Hermitian Forms

In this paper we determine the principal part of the adjusted zeta function for the space of pairs of binary Hermitian forms.

On the Shintani zeta function for the space of binary tri-Hermitian forms

In this paper, we consider the most non-split parabolic D_4 type prehomogeneous vector space. The vector space is an analogue of the space of Hermitian forms. We determine the principal part of the zeta function.

On the Shintani zeta function for the space of binary quadratic forms

On the Shintani zeta function for the space of binary quadratic forms

The theory of zeta functions of several complex variables, I

The paper introduces a general class of Tate-like zeta functions and proves an analytic continuation and a general formula for the values of such zeta functions at negative integral arguments. In particular these zeta functions discussed include the Shintani zeta functions and the results generalize his formula for the value of such zeta functions at negative integral arguments.