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Jay Jorgenson

Publications and source records attributed to Jay Jorgenson.

46 records · Page 3Linked to original sources
arXiv2013

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.

math.NT↗
arXiv2013

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.

math.CO↗
arXiv2011

Complexity and heights of tori

We prove detailed asymptotics for the number of spanning trees, called complexity, for a general class of discrete tori as the parameters tend to infinity. The proof uses in particular certain ideas and techniques from an earlier paper. Our asymptotic formula provides a link between the complexity of these graphs and the height of associated real tori, and allows us to deduce some corollaries on the complexity thanks to certain results from analytic number theory. In this way we obtain a conjectural relationship between complexity and regular sphere packings.

math-ph↗
arXiv2010

An effective bound for the Huber constant for cofinite Fuchsian groups

Let $Γ$ be a cofinite Fuchsian group acting on hyperbolic two-space $\HH.$ Let $M=Γ\setminus \HH $ be the corresponding quotient space. For $γ,$ a closed geodesic of $M$, let $l(γ)$ denote its length. The prime geodesic counting function $π_{M}(u)$ is defined as the number of $Γ$-inconjugate, primitive, closed geodesics $γ$ such that $e^{l(γ)} \leq u.$ The \emph{prime geodesic theorem} implies: $$π_{M}(u)=\sum_{0 \leq λ_{M,j} \leq 1/4} \text{li}(u^{s_{M,j}}) + O_{M}(\frac{u^{3/4}}{\log{u}}), $$ where $0=λ_{M,0} < λ_{M,1} <...$ are the eigenvalues of the hyperbolic Laplacian acting on the space of smooth functions on $M$ and $s_{M,j} = \frac{1}{2}+\sqrt{\frac{1}{4} - λ_{M,j}}. $ Let $C_{M}$ be smallest implied constant so that $$|π_{M}(u)-\sum_{0 \leq λ_{M,j} \leq 1/4} \text{li}(u^{s_{M,j}})|\leq C_{M}\frac{u^{3/4}}{\log{u}} \quad \text{\text{for all} $u > 1.$}$$ We call the (absolute) constant $C_{M}$ the Huber constant. The objective of this paper is to give an effectively computable upper bound of $C_{M}$ for an arbitrary cofinite Fuchsian group. As a corollary we estimate the Huber constant for $\PSL(2,\ZZ),$ we obtain $C_{M} \leq 16,607,349,020,658 \approx \exp(30.44086643)$.

math.NT↗
arXiv2008

On the appearance of Eisenstein series through degeneration

Let $Γ$ be a Fuchsian group of the first kind acting on the hyperbolic upper half plane $\mathbb H$, and let $M = Γ\backslash \mathbb H$ be the associated finite volume hyperbolic Riemann surface. If $γ$ is parabolic, there is an associated (parabolic) Eisenstein series, which, by now, is a classical part of mathematical literature. If $γ$ is hyperbolic, then, following ideas due to Kudla-Millson, there is a corresponding hyperbolic Eisenstein series. In this article, we study the limiting behavior of parabolic and hyperbolic Eisenstein series on a degenerating family of finite volume hyperbolic Riemann surfaces. In particular, we prove the following result. If $γ\in Γ$ corresponds to a degenerating hyperbolic element, then a multiple of the associated hyperbolic Eisenstein series converges to parabolic Eisenstein series on the limit surface.

math.NT↗
arXiv2006

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

Higher-order non-holomorphic Eisenstein series associated to a Fuchsian group $Γ$ 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 $π$ of $Γ$ 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.

math.NT↗
arXiv2000

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.

math.AG↗
arXiv1998

On an Analog of Selberg's Eigenvalue Conjecture for SL_3(Z)

Let H be the homogeneous space associated to the group PGL_3(R). Let X=Γ/H where Γ=SL_3(Z) and consider the first non-trivial eigenvalue λ_1 of the Laplacian on L^2(X). Using geometric considerations, we prove the inequality λ_1<pi^2/10. Since the continuous spectrum is represented by the band [1,\infty), our bound on λ_1 can be viewed as an analogue of Selberg's eigenvalue conjecture for quotients of the hyperbolic half space.

math.SP↗
arXiv1994

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 $$N_{M,w}(T) = \sum\limits_{λ_n \leq T}(T-λ_n)^w $$ where $\{λ_n\}$ 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.

math.DG↗