In this paper, we obtain the central limit theorem of Hecke eigenvalues in very general setting of split simple algebraic groups over $\mathbb{Q}$, using irreducible characters of compact Lie groups.
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 obtain a functional equation satisfied by the Shintani double zeta functions in 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.
In this paper, a zeta integral for the space of binary cubic forms is associated with the subregular unipotent contribution to the geometric side of the Arthur trace formula for the split exceptional group $G_2$.
We study the zeta functions for the space of binary cubic forms introduced by Shintani. The zeta function is defined for each invariant lattice. We classify the invariant lattices, and investigate explicit relationships between the zeta functions associated with those lattices. We also study the analytic properties of the zeta functions, and rewrite Shintani's functional equation in a self dual form using an explicit relation.
In this paper, we give some non-vanishing results on the central values of prime twists of modular $L$-functions by imaginary quadratic fields for specific elliptic modular forms. In particular, we show that the central values of prime twists of $L$-functions of some elliptic modular forms are always non-vanishing whenever the root number is positive.
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 functions in 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.
In this paper, we explicitly determine the Igusa local zeta functions of several variables for all but one type regular 2-simple prehomogeneous vector spaces of type I with universally transitive open orbits. As for the remaining one type of space, we give the explicit forms of the Igusa local zeta functions of one variable for each of the basic relative invariants. In [4], [5] and [8], the irreducible, simple or 2-simple regular prehomogeneous vector spaces with universally transitive open orbits were classifled. As for the irreducible reduced regular prehomogeneous vector spaces with universally transitive open orbits, J. Igusa gave explicitly their Igusa local zeta functions in [4]. And as for the simple regular prehomogeneous vector spaces with universally transitive open orbits, their Igusa local zeta functions were given explicitly in H. Hosokawa [3] and the author [17]. These results indicate that their p-adic i-factors are expressed by the Tate local factor and the b-functions. Here we treat the Igusa local zeta functions of regular 2-simple prehomogeneous vector spaces of type I with universally transitive open orbits, which were classifled into the following nine spaces:
We prove an equidistribution theorem for a family of holomorphic Siegel cusp forms for $GSp_4/\mathbb{Q}$ in various aspects. A main tool is Arthur's invariant trace formula. While Shin and Shin-Templier used Euler-Poincar\'e functions at infinity in the formula, we use a pseudo-coefficient of a holomorphic discrete series to extract holomorphic Siegel cusp forms. Then the non-semisimple contributions arise from the geometric side, and this provides new second main terms $A, B_1$ in the main theorem which have not been studied and a mysterious second term $B_2$ also appears in the second main term coming from the semisimple elements. Furthermore our explicit study enables us to treat more general aspects in the weight. We also give several applications including the vertical Sato-Tate theorem, the unboundedness of Hecke fields and low-lying zeros for degree 4 spinor $L$-functions and degree 5 standard $L$-functions of holomorphic Siegel cusp forms.
