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We define a multiple Dirichlet series associated with quadrics which is the zero locus of a quadratic form. This multiple Dirichlet series is linked to a Shintani zeta function associated with a prehomogeneous vector space. To obtain the…
We study the class numbers of integral binary cubic forms. For each $SL_2(Z)$ invariant lattice $L$, Shintani introduced Dirichlet series whose coefficients are the class numbers of binary cubic forms in $L$. We classify the invariant…
Enumerating integral orbits in prehomogeneous vector spaces plays an important role in arithmetic statistics. We describe a method of proving subconvexity of the zeta function enumerating the integral orbits, illustrated by proving a…
We introduce the zeta function of the prehomogenous vector space of binary cubic forms, twisted by the real analytic Eisenstein series. We prove the meromorphic continuation of this zeta function and identify its poles and their residues.…
As a continuation of the authors and Wakatsuki's previous paper [5], we study relations among Dirichlet series whose coefficients are class numbers of binary cubic forms. We show that for any integral models of the space of binary cubic…
We study the space of binary cubic and quadratic forms over the ring of integers $O$ of an algebraic number field $k$. By applying the theory of prehomogeneous vector spaces founded by M. Sato and T. Shintani, we can associate the zeta…
In this paper we study the analytic properties of a multiple Dirichlet series associated to the prehomogeneous vector space of binary cubic forms.
In this paper, we give an explicit formula of 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…
In this paper, we consider the prehomogeneous vector space for pair of simple algebras which are $k$-forms of the $D_4$ type and the $E_6$ type. We mainly study the non-split cases. The main purpose of this paper is to determine the…
In this paper, we investigate Weng zeta functions associated with curves of genus 2 over finite fields. Building upon Weng's framework for non-abelian zeta functions, we establish that, as the rank n tends to infinity, the Riemann…
We prove a general convergence result for zeta functions of prehomogeneous vector spaces extending results of H. Saito, F. Sato and Yukie. Our analysis points to certain subspaces which yield boundary terms. We study it further in the setup…
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.
This is the second of two papers introducing and investigating two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. In the first part, we proved some of their properties such as…
Attached to a vector space V is a vertex algebra S(V) known as the beta-gamma system or algebra of chiral differential operators on V. It is analogous to the Weyl algebra D(V), and is related to D(V) via the Zhu functor. If G is a connected…
We study the hyperk\"ahler analogues of moduli spaces of semistable n-gons in complex projective space. We prove that the hyperk\"ahler Kirwan map is surjective and produce a formula that recursively calculates the Betti numbers of these…
We revisit the Hitchin integrable system whose phase space is the bundle cotangent to the moduli space $N$ of holomorphic $SL_2$-bundles over a smooth complex curve of genus two. $N$ may be identified with the 3-dimensional projective space…
Let $k$ be a number field and $O$ the ring of integers. In the previous paper [T06] we study the Dirichlet series counting discriminants of cubic algebras of $O$ and derive some density theorems on distributions of the discriminants by…
In this paper, we study the Dirichlet series that enumerates proper equivalence classes of full-rank sublattices of a given quadratic lattice in a hyperbolic plane -- that is, a nondegenerate isotropic quadratic space of dimension $2$. We…
It is known that the Selberg zeta function for the modular group has an expression in terms of the class numbers and the fundamental units of the indefinite binary quadratic forms. In the present paper, we generalize such a expression to…
Every four-dimensional ${\cal N}=2$ superconformal field theory comes equipped with an intricate algebraic invariant, the associated vertex operator algebra. The relationships between this invariant and more conventional protected…