Related papers: On analytic properties of the standard zeta functi…

In this note, we consider discriminant forms that are given by the norm form of real quadratic fields and their induced Weil representations. We prove that there exists an isomorphism between the space of vector-valued modular forms for the…

We study analytic properties of the representation zeta functions of arithmetic groups of type $\mathsf{A}_2$, such as $\textrm{SL}_3(\mathbb{Z})$. In particular, we uncover further poles of these functions and determine a natural boundary…

Let G_2 be the exceptional Lie group of automorphisms of the complex Cayley algebra and C be a generic, smooth, connected, projective curve over $\mathbb{C}$ of genus at least 2. For a complex Lie group G, let H^0(M(G),L^k) be the space of…

It is a classical fact that the elliptic modular functions satisfies an algebraic differential equation of order 3, and none of lower order. We show how this generalizes to Siegel modular functions of arbitrary degree. The key idea is that…

Lichtenbaum has conjectured the existence of a Grothendieck topology for an arithmetic scheme $X$ such that the Euler characteristic of the cohomology groups of the constant sheaf $\mathbb{Z}$ with compact support at infinity gives, up to…

Let C be a smooth projective complex curve of genus at least 2. For a simply-connected complex Lie group G the vector space of global sections H^0(M(G), L^l) of the l-th power of the ample generator L of the Picard group of the moduli stack…

We determine the unipotent orbits attached to degenerate Eisenstein series on general linear groups. This confirms a conjecture of David Ginzburg. This also shows that any unipotent orbit of general linear groups does occur as the unipotent…

We define a generalized class of modified zeta series transformations generating the partial sums of the Hurwitz zeta function and series expansions of the Lerch transcendent function. The new transformation coefficients we define within…

We study Selberg zeta functions $Z(s,\sigma)$ associated to locally homogeneous vector bundles over the unit-sphere bundle of a complete odd-dimensional hyperbolic manifold of finite volume. We assume a certain condition on the fundamental…

The finite symplectic group Sp(2g) over the field of two elements has a natural representation on the vector space of Siegel modular forms of given weight for the principal congruence subgroup of level two. In this paper we decompose this…

We prove certain general forms of functional relations among Witten multiple zeta-functions in several variables (or zeta-functions of root systems). The structural background of those functional relations is given by the symmetry with…

We study the injectivity of the Kudla-Millson lift of genus 2 Siegel cusp forms, vector-valued with respect to the Weil representation associated to an even lattice L. We prove that if L splits off two hyperbolic planes and is of…

In this paper we generalise the notion of Drinfeld modular form for the group $\Gamma$ := GL2(Fq[$\theta$]) to a vector-valued setting, where the target spaces are certain modules over positive characteristic Banach algebras over which are…

Based on our previous work on an arithmetic analogue of Christol's theorem, this paper studies in more detail the structure of the lambda-ring $E_K = K \otimes W_{O_K}^a (O_{\bar{K}})$ of algebraic Witt vectors for number fields $K$. First…

In this note, we extend Euler's transformation formula from the alternating series to more general series. Then we give new expressions for the Riemann zeta function $\zeta(s)$ by the generalized difference operator $\Delta_{c}$, which…

We obtain a Weyl law with power savings for the universal families of cuspidal automorphic representations, ordered by analytic conductor, of $\mathrm{GL}_2$ over $\mathbb{Q}$, as well as for Hecke characters over any number field. The…

We formulate a conjecture that describes the vector-valued Siegel modular forms of degree 2 and level 2 of weight Sym^j det^2 and provide some evidence for it. We construct such modular forms of weight (j,2) via covariants of binary sextics…

The subgroup zeta function and the normal zeta function of a finitely generated virtually nilpotent group can be expressed as finite sums of Dirichlet series admitting Euler product factorization. We compute these series except for a finite…

We prove the expected algebraicity property for the critical values of character twists of the standard $L$-function associated to vector-valued holomorphic Siegel cusp forms of archimedean type $(k_1, k_2, \ldots, k_n)$, where $k_n \geq…

In Sarnak's paper, it was proved that the Selberg zeta function for SL(2,Z) is expressed in terms of the fundamental units and the class numbers of the primitive indefinite binary quadratic forms. The aim of this paper is to obtain similar…