Ample Divisors, Automorphic Forms and Shafarevich's Conjecture

In this article we give a general approach to the following analogue of Shafarevich's conjecture for some polarized algebraic varieties; suppose that we fix a type of an algebraic variety and look at families of such type of varieties over a fixed Riemann surface with fixed points over which we have singular varieties, then one can ask if the set of such families, up to isomorphism, is finite. In this paper we give a general approach to such types of problems. The main observation is the following; suppose that the moduli space of a fixed type of algebraic polarized variety exists and suppose that in some projective smooth compactification of the coarse moduli the discriminant divisor supports an ample one, then it is not difficult to see that this fact implies the analogue of Shafarevich's conjecture. In this article we apply this method to certain polarized algebraic K3 surfaces and also to Enriques surfaces

On the distribution of zeros of the derivative of Selberg's zeta function associated to finite volume Riemann surfaces

W. Luo has investigated the distribution of zeros of the derivative of the Selberg zeta function associated to compact hyperbolic Riemann surfaces. In essence, the main results in Luo's article involve the following three points: Finiteness for the number of zeros in the half plane to the left of the critical line; an asymptotic expansion for the counting function measuring the vertical distribution of zeros; and an asymptotic expansion for the counting function measuring the horizontal distance of zeros from the critical line. In the present article, we study the more complicated setting of distribution of zeros of the derivative of the Selberg zeta function associated to a non-compact, finite volume hyperbolic Riemann surface. There are numerous difficulties which exist in the non-compact case that are not present in the compact setting, beginning with the fact that in the non-compact case the Selberg zeta function does not satisfy the analogue of the Riemann hypothesis. To be more specific, we actually study the zeros of the derivative of ZH, where Z is the Selberg zeta function and H is the Dirichlet series component of the scattering matrix, both associated to an arbitrary finite-volume hyperbolic Riemann surface. Our main results address finiteness of zeros in the half plane to the left of the critical line, an asymptotic count for the vertical distribution of zeros, and an asymptotic count for the horizontal distance of zeros

Convolution Dirichlet series and a Kronecker limit formula for second-order Eisenstein series

The classical Kronecker limit formula gives the constant term of the non-holomorphic Eisenstein series E(z,s) for SL(2,Z) at s=1 in terms of the Dedekind eta function. Here we compute the analagous formula for an Eisenstein series twisted by modular symbols (periods of weight two holomorphic cusp forms) for general Fuchsian groups of the first kind.Comment: 36 page

Spectral asymptotics on sequences of elliptically degenerating Riemann surfaces

This is the second in a series of two articles where we study various aspects of the spectral theory associated to families of hyperbolic Riemann surfaces obtained through elliptic degeneration. In the first article, we investigate the asymptotics of the trace of the heat kernel both near zero and infinity and we show the convergence of small eigenvalues and corresponding eigenfunctions. Having obtained necessary bounds for the trace, this second article presents the behavior of several spectral invariants. Some of these invariants, such as the Selberg zeta function and the spectral counting functions associated to small eigenvalues below 1/4, converge to their respective counterparts on the limiting surface. Other spectral invariants, such as the spectral zeta function and the logarithm of the determinant of the Laplacian diverge. In these latter cases, we identify diverging terms and remove their contributions, thus regularizing convergence of these spectral invariants. Our study is motivated by a result from \cite{He 83}, which D. Hejhal attributes to A. Selberg, proving spectral accumulation for the family of Hecke triangle groups. In this article, we obtain a quantitative result to Selberg's remark

Unipotent vector bundles and higher-order non-holomorphic Eisenstein series

Higher-order non-holomorphic Eisenstein series associated to a Fuchsian group $\Gamma$ are defined by twisting the series expansion for classical non-holomorphic Eisenstein series by powers of modular symbols. Their functional identities include multiplicative and additive factors, making them distinct from classical Eisenstein series. Here we prove the meromorphic continuation of these series and establish their functional equations. In addition, we construct high rank vector bundles $\cal V$ from certain unipotent representations $\pi$ of $\Gamma$ and show that higher-order non-holomorphic Eisenstein series can be viewed as components of certain eigensections, $\mathbb E$, of $\cal V$. With this viewpoint the functional identities of these higher-order series are formally identical to the classical case. Going further, we prove bounds for the Fourier coefficients of the higher-order non-holomorphic Eisenstein series.Comment: 23 page

Heat kernel asymptotics on sequences of elliptically degenerating Riemann surfaces

This is the first of two articles in which we define an elliptically degenerating family of hyperbolic Riemann surfaces and study the asymptotic behavior of the associated spectral theory. Our study is motivated by a result from \cite{He 83}, which Hejhal attributes to Selberg, proving spectral accumulation for the family of Hecke triangle groups. In this article, we prove various results regarding the asymptotic behavior of heat kernels and traces of heat kernels for both real and complex time. In \cite{GJ 16}, we will use the results from this article and study the asymptotic behavior of numerous spectral functions through elliptic degeneration, including spectral counting functions, Selberg's zeta function, Hurwitz-type zeta functions, determinants of the Laplacian, wave kernels, spectral projections, small eigenfunctions, and small eigenvalues. The method of proof we employ follows the template set in previous articles which study spectral theory on degenerating families of finite volume Riemann surfaces (\cite{HJL 95}, \cite{HJL 97}, \cite{JoLu 97a}, and \cite{JoLu 97b}) and on degenerating families of finite volume hyperbolic three manifolds (\cite{DJ 98}). Although the types of results developed here and in \cite{GJ 16} are similar to those in existing articles, it is necessary to thoroughly present all details in the setting of elliptic degeneration in order to uncover all nuances in this setting.Comment: arXiv admin note: text overlap with arXiv:1603.0149

On the evaluation of singular invariants for canonical generators of certain genus one arithmetic groups

Let $N$ be a positive square-free integer such that the discrete group $\Gamma_{0}(N)^{+}$ has genus one. In a previous article, we constructed canonical generators $x_{N}$ and $y_{N}$ of the holomorphic function field associated to $\Gamma_{0}(N)^{+}$ as well as an algebraic equation $P_{N}(x_{N},y_{N}) = 0$ with integer coefficients satisfied by these generators. In the present paper, we study the singular moduli problem corresponding to $x_{N}$ and $y_{N}$, by which we mean the arithmetic nature of the numbers $x_{N}(\tau)$ and $y_{N}(\tau)$ for any CM point $\tau$ in the upper half plane $\mathbb{H}$. If $\tau$ is any CM point which is not equivalent to an elliptic point of $\Gamma_{0}(N)^{+}$, we prove that the complex numbers $x_{N}(\tau)$ and $y_{N}(\tau)$ are algebraic integers. Going further, we characterize the algebraic nature of $x_{N}(\tau)$ as the generator of a certain ring class field of $\mathbb{Q}(\tau)$ of prescribed order and discriminant depending on properties of $\tau$ and level $N$. The theoretical considerations are supplemented by computational examples. As a result, several explicit evaluations are given for various $N$ and $\tau$, and further arithmetic consequences of our analysis are presented. In one example, we explicitly construct a set of minimal polynomials for the Hilbert class field of $\mathbb{Q}(\sqrt{-74})$ whose coefficients are less than $2.2\times 10^{4}$, whereas the minimal polynomials obtained from the Hauptmodul of $\textrm{PSL}(2,\mathbb{Z})$ has coefficients as large as $6.6\times 10^{73}$

On the distribution of eigenvalues of Maass forms on certain moonshine groups

In this paper we study, both analytically and numerically, questions involving the distribution of eigenvalues of Maass forms on the moonshine groups $\Gamma_0(N)^+$, where $N>1$ is a square-free integer. After we prove that $\Gamma_0(N)^+$ has one cusp, we compute the constant term of the associated non-holomorphic Eisenstein series. We then derive an "average" Weyl's law for the distribution of eigenvalues of Maass forms, from which we prove the "classical" Weyl's law as a special case. The groups corresponding to $N=5$ and $N=6$ have the same signature; however, our analysis shows that, asymptotically, there are infinitely more cusp forms for $\Gamma_0(5)^+$ than for $\Gamma_0(6)^+$. We view this result as being consistent with the Phillips-Sarnak philosophy since we have shown, unconditionally, the existence of two groups which have different Weyl's laws. In addition, we employ Hejhal's algorithm, together with recently developed refinements from [31], and numerically determine the first $3557$ of $\Gamma_0(5)^+$ and the first $12474$ eigenvalues of $\Gamma_0(6)^+$. With this information, we empirically verify some conjectured distributional properties of the eigenvalues.Comment: metadata updated, no changes in the articl

On the Asymptotic Behavior of Counting Functions Associated to Degenerating Hyperbolic Riemann Surfaces

We develop an asymptotic expansion of the spectral measures on a degenerating family of hyperbolic Riemann surfaces of finite volume. As an application of our results, we study the asymptotic behavior of weighted counting functions, which, if $M$ is compact, is defined for $w \geq 0$ and $T > 0$ by $$N_{M,w}(T) = \sum\limits_{\lambda_n \leq T}(T-\lambda_n)^w$$ where $\{\lambda_n\}$ is the set of eigenvalues of the Laplacian which acts on the space of smooth functions on $M$. If $M$ is non-compact, then the weighted counting function is defined via the inverse Laplace transform. Now let $M_{\ell}$ denote a degenerating family of compact or non-compact hyperbolic Riemann surfaces of finite volume which converges to the non-compact hyperbolic surface $M_{0}$. As an example of our results, we have the following theorem: There is an explicitly defined function $G_{\ell,w}(T)$ which depends solely on $\ell$, $w$, and $T$ such that for $w > 3/2$ and $T>0$, we have $$N_{M_{\ell},w}(T) = G_{\ell,w}(T) +N_{M_{0},w}(T) +o(1)$$ for $\ell \to 0$. We also consider the setting when $w < 3/2$, and we obtain a new proof of the continuity of small eigenvalues on degenerating hyperbolic Riemann surfaces of finite volume

Heat kernels on regular graphs and generalized Ihara zeta function formulas

We establish a new formula for the heat kernel on regular trees in terms of classical I-Bessel functions. Although the formula is explicit, and a proof is given through direct computation, we also provide a conceptual viewpoint using the horocyclic transform on regular trees. From periodization, we then obtain a heat kernel expression on any regular graph. From spectral theory, one has another expression for the heat kernel as an integral transform of the spectral measure. By equating these two formulas and taking a certain integral transform, we obtain several generalized versions of the determinant formula for the Ihara zeta function associated to finite or infinite regular graphs. Our approach to the Ihara zeta function and determinant formula through heat kernel analysis follows a similar methodology which exists for quotients of rank one symmetric spaces.Comment: 15 page