A thermodynamic formalism approach to the Selberg zeta function for Hecke triangle surfaces of infinite area

We provide an explicit construction of a cross section for the geodesic flow on infinite-area Hecke triangle surfaces which allows us to conduct a transfer operator approach to the Selberg zeta function. Further we construct closely related cross sections for the billiard flow on the associated triangle surfaces and endow the arising discrete dynamical systems and transfer operator families with two weight functions which presumably encode Dirichlet respectively Neumann boundary conditions. The Fredholm determinants of these transfer operator families constitute dynamical zeta functions, which provide a factorization of the Selberg zeta function of the Hecke triangle surfaces.Comment: 23 pages, 6 figure

An equivariant Poincar\'e series of filtrations and monodromy zeta functions

We define a new equivariant (with respect to a finite group $G$ action) version of the Poincar\'e series of a multi-index filtration as an element of the power series ring ${\widetilde{A}}(G)[[t_1, \ldots, t_r]]$ for a certain modification ${\widetilde{A}}(G)$ of the Burnside ring of the group $G$. We give a formula for this Poincar\'e series of a collection of plane valuations in terms of a $G$-resolution of the collection. We show that, for filtrations on the ring of germs of functions in two variables defined by the curve valuations corresponding to the irreducible components of a plane curve singularity defined by a $G$-invariant function germ, in the majority of cases this equivariant Poincar\'e series determines the corresponding equivariant monodromy zeta functions defined earlier

Four Perspectives on Secondary Terms in the Davenport-Heilbronn Theorems

This paper is an expanded version of the author's lecture at the Integers Conference 2011. We discuss the secondary terms in the Davenport-Heilbronn theorems on cubic fields and 3-torsion in class groups of quadratic fields. Such secondary terms had been conjectured by Datskovsky-Wright and Roberts, and proofs of these or closely related secondary terms were obtained independently by Bhargava, Shankar, and Tsimerman; Hough; Zhao; and Taniguchi and the author. In this paper we discuss the history of the problem and highlight the diverse methods used to address it.Comment: 18 pages, survey/overview of recent work. To submit to the proceedings of Integers 2011 shortly. Comments very welcome

Modular and holomorphic graph function from superstring amplitudes

We compare two classes of functions arising from genus-one superstring amplitudes: modular and holomorphic graph functions. We focus on their analytic properties, we recall the known asymptotic behaviour of modular graph functions and we refine the formula for the asymptotic behaviour of holomorphic graph functions. Moreover, we give new evidence of a conjecture which relates these two asymptotic expansions.Comment: 26 Pages. Based on a talk given at the conference "Elliptic integrals, elliptic functions and modular forms in quantum field theory" held at DESY-Zeuthen in October 201

On the zeros of the Epstein zeta function

In this article, we count the number of consecutive zeros of the Epstein zeta-function, associated to a certain quadratic form, on the critical line with ordinates lying in $[0,T], T$ sufficiently large and which are separated apart by a given positive number $V$.Comment: 13 pages, to appear in proceedings of CINTAA, 200

Zeta functions associated to admissible representations of compact p-adic Lie groups

Let $G$ be a profinite group. A strongly admissible smooth representation $\rho$ of $G$ over $\mathbb{C}$ decomposes as a direct sum $\rho \cong \bigoplus_{\pi \in \mathrm{Irr}(G)} m_\pi(\rho) \, \pi$ of irreducible representations with finite multiplicities $m_\pi(\rho)$ such that for every positive integer $n$ the number $r_n(\rho)$ of irreducible constituents of dimension $n$ is finite. Examples arise naturally in the representation theory of reductive groups over non-archimedean local fields. In this article we initiate an investigation of the Dirichlet generating function $$\zeta_\rho (s) = \sum_{n=1}^\infty r_n(\rho) n^{-s} = \sum_{\pi \in \mathrm{Irr}(G)} \frac{m_\pi(\rho)}{(\dim \pi)^s}$$ associated to such a representation $\rho$. Our primary focus is on representations $\rho = \mathrm{Ind}_H^G(\sigma)$ of compact $p$-adic Lie groups $G$ that arise from finite dimensional representations $\sigma$ of closed subgroups $H$ via the induction functor. In addition to a series of foundational results - including a description in terms of $p$-adic integrals - we establish rationality results and functional equations for zeta functions of globally defined families of induced representations of potent pro-$p$ groups. A key ingredient of our proof is Hironaka's resolution of singularities, which yields formulae of Denef-type for the relevant zeta functions. In some detail, we consider representations of open compact subgroups of reductive $p$-adic groups that are induced from parabolic subgroups. Explicit computations are carried out by means of complementing techniques: (i) geometric methods that are applicable via distance-transitive actions on spherically homogeneous rooted trees and (ii) the $p$-adic Kirillov orbit method. Approach (i) is closely related to the notion of Gelfand pairs and works equally well in positive defining characteristic.Comment: 61 pages; small changes, contains an abridged Section 5.3. Final version to be published in Trans. Amer. Math. Soc. (arXiv version contains an additional footnote in Section 5.3

Zeta-Functions for Families of Calabi--Yau n-folds with Singularities

We consider families of Calabi-Yau n-folds containing singular fibres and study relations between the occurring singularity structure and the decomposition of the local Weil zeta-function. For 1-parameter families, this provides new insights into the combinatorial structure of the strong equivalence classes arising in the Candelas - de la Ossa - Rodrigues-Villegas approach for computing the zeta-function. This can also be extended to families with more parameters as is explored in several examples, where the singularity analysis provides correct predictions for the changes of degree in the decomposition of the zeta-function when passing to singular fibres. These observations provide first evidence in higher dimensions for Lauder's conjectured analogue of the Clemens-Schmid exact sequence.Comment: 22 pages, LaTeX, (including 4 tables

Twisted Gromov and Lefschetz invariants associated with bundles

Given a closed symplectic 4-manifold $(X,\omega)$, we define a twisted version of the Gromov-Taubes invariants for $(X,\omega)$, where the twisting coefficients are induced by the choice of a surface bundle over $X$. Given a fibered 3-manifold $Y$, we similarly construct twisted Lefschetz zeta functions associated with surface bundles: we prove that these are essentially equivalent to the Jiang's Lefschetz zeta functions of $Y$, twisted by the representations of $\pi_1(Y)$ that are induced by monodromy homomorphisms of surface bundles over $Y$. This leads to an interpretation of the corresponding twisted Reidemeister torsions of $Y$ in terms of products of "local" commutative Reidemeister torsions. Finally we relate the two invariants by proving that, for any fixed closed surface bundle $\mathcal{B}$ over $Y$, the corresponding twisted Lefschetz zeta function coincides with the Gromov-Taubes invariant of $S^1 \times Y$ twisted by the bundle over $S^1 \times Y$ naturally induced by $\mathcal{B}$.Comment: 31 pages, 1 figure. Version 2: minor changes, typographical correction

Local functional equations for submodule zeta functions associated to nilpotent algebras of endomorphisms

We give a sufficient criterion for generic local functional equations for submodule zeta functions associated to nilpotent algebras of endomorphisms defined over number fields. This allows us, in particular, to prove various conjectures on such functional equations for ideal zeta functions of nilpotent Lie lattices. Via the Mal'cev correspondence, these results have corollaries pertaining to zeta functions enumerating normal subgroups of finite index in finitely generated nilpotent groups, most notably finitely generated free nilpotent groups of any given class.Comment: 28 pages, final version, to appear in Int. Math. Res. Not. IMR

Aspects of Zeta-Function Theory in the Mathematical Works of Adolf Hurwitz

Adolf Hurwitz is rather famous for his celebrated contributions to Riemann surfaces, modular forms, diophantine equations and approximation as well as to certain aspects of algebra. His early work on an important generalization of Dirichlet's $L$-series, nowadays called Hurwitz zeta-function, is the only published work settled in the very active field of research around the Riemann zeta-function and its relatives. His mathematical diaries, however, provide another picture, namely a lifelong interest in the development of zeta-function theory. In this note we shall investigate his early work, its origin and its reception, as well as Hurwitz's further studies of the Riemann zeta-function and allied Dirichlet series from his diaries. It turns out that Hurwitz already in 1889 knew about the essential analytic properties of the Epstein zeta-function (including its functional equation) 13 years before Paul Epstein.Comment: 32 pages, 2 figure