arXiv subjects

Jay Jorgenson

Publications and source records attributed to Jay Jorgenson.

At least 37 records · Page 2Linked to original sources

arXiv2021

Kronecker limit functions and an extension of the Rohrlich-Jensen formula

In 1984 Rohrlich proved a modular analogue of Jensen's formula. Under certain conditions, the Rohrlich-Jensen formula expresses an integral of the log-norm $\log \Vert f \Vert$ of a $\text{\rm PSL}(2,\ZZ)$ modular form $f$ in terms of the Dedekind Delta function evaluated at the divisor of $f$. Recently, Bringmann-Kane re-interpreted the Rohrlich-Jensen formula as evaluating a regularized inner product of $\log \Vert f \Vert$ and extended the result to compute a regularized inner product of $\log \Vert f \Vert$ with what amounts to powers of the Hauptmoduli of $\text{\rm PSL}(2,\ZZ)$. In the present article, we revisit the Rohrlich-Jensen formula and prove that it can be viewed as a regularized inner product of special values of two Poincaré series, one of which is the Niebur-Poincaré series and the other is the resolvent kernel of the Laplacian. The regularized inner product can be seen as a type of Maass-Selberg relation. In this form, we develop a Rohrlich-Jensen formula associated to any Fuchsian group $Γ$ of the first kind with one cusp by employing a type of Kronecker limit formula associated to the resolvent kernel. We present two examples of our main result: First, when $Γ$ is the full modular group $\text{\rm PSL}(2,\ZZ)$, thus reproving the theorems from \cite{BK19}; and second when $Γ$ is an Atkin-Lehner group $Γ_{0}(N)^+$, where explicit computations are given for certain genus zero, one and two levels.

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arXiv2020

Super-zeta functions and regularized determinants associated to cofinite Fuchsian groups with finite-dimensional unitary representations

Let $M$ be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $χ$ denote a finite dimensional unitary representation of the fundamental group of $M$. Let $Δ$ denote the hyperbolic Laplacian which acts on smooth sections of the flat bundle over $M$ associated to $χ$. From the spectral theory of $Δ$, there are three distinct sequences of numbers: The first coming from the eigenvalues of $L^{2}$ eigenfunctions, the second coming from resonances associated to the continuous spectrum, and the third being the set of negative integers. Using these sequences of spectral data, we employ the super-zeta approach to regularization and introduce two super-zeta functions, $\Z_-(s,z)$ and $\Z_+(s,z)$ that encode the spectrum of $Δ$ in such a way that they can be used to define the regularized determinant of $Δ-z(1-z)I$. The resulting formula for the regularized determinant of $Δ-z(1-z)I$ in terms of the Selberg zeta function, see Theorem 5.3, encodes the symmetry $z\leftrightarrow 1-z$, which could not be seen in previous works, due to a different definition of the regularized determinant.

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arXiv2019

Transformation laws for generalized Dedekind sums associated to Fuchsian groups

We establish transformation laws for generalized Dedekind sums associated to the Kronecker limit function of non-holomorphic Eisenstein series and their higher-order variants. These results apply to general Fuchsian groups of the first kind, and examples are provided in the cases of the Hecke triangle groups, the Hecke congruence groups $Γ_0(N)$, and the non-congruence arithmetic groups $Γ_0(N)^+$.

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arXiv2018

Superzeta functions, regularized products, and the Selberg zeta function on hyperbolic manifolds with cusps

Let $Λ= \{λ_{k}\}$ denote a sequence of complex numbers and assume that that the counting function $#\{λ_{k} \in Λ: | λ_{k}| < T\} =O(T^{n})$ for some integer $n$. From Hadamard's theorem, we can construct an entire function $f$ of order at most $n$ such that $Λ$ is the divisor $f$. In this article we prove, under reasonably general conditions, that the superzeta function $\Z_{f}(s,z)$ associated to $Λ$ admits a meromorphic continuation. Furthermore, we describe the relation between the regularized product of the sequence $z-Λ$ and the function $f$ as constructed as a Weierstrass product. In the case $f$ admits a Dirichlet series expansion in some right half-plane, we derive the meromorphic continuation in $s$ of $\Z_{f}(s,z)$ as an integral transform of $f'/f$. We apply these results to obtain superzeta product evaluations of Selberg zeta function associated to finite volume hyperbolic manifolds with cusps.

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arXiv2018

Effective sup-norm bounds on average for cusp forms of even weight

Let $Γ\subset\mathrm{PSL}_{2}(\mathbb{R})$ be a Fuchsian subgroup of the first kind acting on the upper half-plane $\mathbb{H}$. Consider the $d_{2k}$-dimensional space of cusp forms $\mathcal{S}_{2k}^Γ$ of weight $2k$ for $Γ$, and let $\{f_{1},\ldots,f_{d_{2k}}\}$ be an orthonormal basis of $\mathcal{S}_{2k}^Γ$ with respect to the Petersson inner product. In this paper we will give effective upper and lower bounds for the supremum of the quantity $S_{2k}^Γ(z):=\sum_{j=1}^{d_{2k}}\vert f_{j}(z)\vert^{2}\,\mathrm{Im}(z)^{2k}$ as $z$ ranges through $\mathbb{H}$.

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arXiv2017

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 $Γ_{0}(N)^{+}$ has genus one. In a previous article, we constructed canonical generators $x_{N}$ and $y_{N}$ of the holomorphic function field associated to $Γ_{0}(N)^{+}$ 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 $x_{N}(τ)$ and $y_{N}(τ)$ for any CM point $τ$ in the upper half plane $\mathbb{H}$. If $τ$ is any CM point which is not equivalent to an elliptic point of $Γ_{0}(N)^{+}$, we prove that the complex numbers $x_{N}(τ)$ and $y_{N}(τ)$ are algebraic integers. Going further, we characterize the algebraic nature of $x_{N}(τ)$ as the generator of a certain ring class field of $\mathbb{Q}(τ)$ of prescribed order and discriminant depending on properties of $τ$ and level $N$. The theoretical considerations are supplemented by computational examples. As a result, several explicit evaluations are given for various $N$ and $τ$, 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 $6.6\times 10^{73}$.

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arXiv2017

The Hauptmodul at elliptic points of certain arithmetic groups

Let $N$ be a square-free integer such that the arithmetic group $Γ_0(N)^+$ has genus zero; there are $44$ such groups. Let $j_N$ denote the associated Hauptmodul normalized to have residue equal to one and constant term equal to zero in its $q$-expansion. In this article we prove that the Hauptmodul at any elliptic point of the surface associated to $Γ_0(N)^+$ is an algebraic integer. Moreover, for each such $N$ and elliptic point $e$, we show how to explicitly evaluate $j_{N}(e)$ and provide the list of generating polynomials (with small coefficients) of the class fields or their subfields corresponding to the orders over the imaginary quadratic extension of rationals stemming from the elliptic points under consideration.

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arXiv2017

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 $Γ_0(N)^+$, where $N>1$ is a square-free integer. After we prove that $Γ_0(N)^+$ 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 $Γ_0(5)^+$ than for $Γ_0(6)^+$. 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 $Γ_0(5)^+$ and the first $12474$ eigenvalues of $Γ_0(6)^+$. With this information, we empirically verify some conjectured distributional properties of the eigenvalues.

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arXiv2016

Modular Dedekind symbols associated to Fuchsian groups and higher-order Eisenstein series

Let $E(z,s)$ be the non-holomorphic Eisenstein series for the modular group $SL(2,{\mathbb Z})$. The classical Kronecker limit formula shows that the second term in the Laurent expansion at $s=1$ of $E(z,s)$ is essentially the logarithm of the Dedekind eta function. This eta function is a weight $1/2$ modular form and Dedekind expressed its multiplier system in terms of Dedekind sums. Building on work of Goldstein, we extend these results from the modular group to more general Fuchsian groups $Γ$. The analogue of the eta function has a multiplier system that may be expressed in terms of a map $S:Γ\to {\mathbb R}$ which we call a modular Dedekind symbol. We obtain detailed properties of these symbols by means of the limit formula. Twisting the usual Eisenstein series with powers of additive homomorphisms from $Γ$ to ${\mathbb C}$ produces higher-order Eisenstein series. These series share many of the properties of $E(z,s)$ though they have a more complicated automorphy condition. They satisfy a Kronecker limit formula and produce higher-order Dedekind symbols $S^*:Γ\to {\mathbb R}$. As an application of our general results, we prove that higher-order Dedekind symbols associated to genus one congruence groups $Γ_0(N)$ are rational.

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arXiv2016

An evaluation of the central value of the automorphic scattering determinant

Let $M$ be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $ϕ(s)$ denote the automorphic scattering determinant. From the known functional equation $ϕ(s)ϕ(1-s)=1$ one concludes that $ϕ(1/2)^{2} = 1$. However, except for the relatively few instances when $ϕ(s)$ is explicitly computable, one does not know $ϕ(1/2)$. In this article we address this problem and prove the following result. Let $N$ and $P$ denote the number of zeros and poles, respectively, of $ϕ(s)$ in $(1/2,\infty)$, counted with multiplicities. Let $d(1)$ be the coefficient of the leading term from the Dirichlet series component of $ϕ(s)$. Then $ϕ(1/2)=(-1)^{N+P} \cdot \mathrm{sgn}(d(1))$.

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arXiv2016

The determinant of the Lax-Phillips scattering operator

Let $M$ denote a finite volume, non-compact Riemann surface without elliptic points, and let $B$ denote the Lax-Phillips scattering operator. Using the superzeta function approach due to Voros, we define a Hurwitz-type zeta function $ζ^{\pm}_{B}(s,z)$ constructed from the resonances associated to $zI -[ (1/2)I \pm B]$. We prove the meromorphic continuation in $s$ of $ζ^{\pm}_{B}(s,z)$ and, using the special value at $s=0$, define a determinant of the operators $zI -[ (1/2)I \pm B]$. We obtain expressions for Selberg's zeta function and the determinant of the scattering matrix in terms of the operator determinants.

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arXiv2016

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.

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arXiv2016

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.

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arXiv2015

Certain aspects of holomorphic function theory on some genus zero arithmetic groups

There are a number of fundamental results in the study of holomorphic function theory associated to the discrete group PSL(2,Z) including the following statements: The ring of holomorphic modular forms is generated by the holomorphic Eisenstein series of weight four and six; the smallest weight cusp form Delta has weight twelve and can be written as a polynomial in E4 and E6; and the Hauptmodul j can be written as a multiple of E4 cubed divided by Delta. The goal of the present article is to seek generalizations of these results to some other genus zero arithmetic groups, namely those generated by Atkin-Lehner involutions of level N with square-free level N.

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arXiv2015

Applications of Kronecker's limit formula for elliptic Eisenstein series

We develop two applications of the Kronecker's limit formula associated to elliptic Eisenstein series: A factorization theorem for holomorphic modular forms, and a proof of Weil's reciprocity law. Several examples of the general factorization results are computed, specifically for certain moonshine groups, congruence subgroups, and, more generally, non-compact subgroups with one cusp. In particular, we explicitly compute the Kronecker limit function associated to certain elliptic points for a few small level moonshine groups.

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arXiv2015

On the wave representation of hyperbolic, elliptic, and parabolic Eisenstein series

We develop a unified approach to the construction of the hyperbolic and elliptic Eisenstein series on a finite volume hyperbolic Riemann surface. Specifically, we derive expressions for the hyperbolic and elliptic Eisenstein series as integral transforms of the kernel of a wave operator. Established results in the literature relate the wave kernel to the heat kernel, which admits explicit construction from various points of view. Therefore, we obtain a sequence of integral transforms which begins with the heat kernel, obtains a Poisson and wave kernel, and then yields the hyperbolic and elliptic Eisenstein series. In the case of a non-compact finite volume hyperbolic Riemann surface, we finally show how to express the parabolic Eisenstein series in terms of the integral transform of a wave operator.

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arXiv2014

Kronecker's limit formula, holomorphic modular functions and $q$-expansions on certain moonshine groups

For any square-free integer $N$ such that the "moonshine group" $Γ_0(N)^+$ has genus zero, the Monstrous Moonshine Conjectures relate the Hauptmoduli of $Γ_0(N)^+$ to certain McKay-Thompson series associated to the representation theory of the Fischer-Griess monster group. In particular, the Hauptmoduli admits a $q$-expansion which has integer coefficients. In this article, we study the holomorphic function theory associated to higher genus moonshine groups $Γ_0(N)^+$. For all moonshine groups of genus up to and including three, we prove that the corresponding function field admits two generators whose $q$-expansions have integer coefficients, has lead coefficient equal to one, and has minimal order of pole at infinity. As corollary, we derive a polynomial relation which defines the underlying projective curve, and we deduce whether $i\infty$ is a Weierstrass point. Our method of proof is based on modular forms and includes extensive computer assistance, which, at times, applied Gauss elimination to matrices with thousands of entries, each one of which was a rational number whose numerator and denominator were thousands of digits in length.

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arXiv2013

Uniform sup-norm bounds on average for cusp forms of higher weights

Let $Γ\subseteq\mathrm{PSL}_{2}(\mathbb{R})$ be a Fuchsian subgroup of the first kind acting on the upper half-plane $\mathbb{H}$. Consider the $d$-dimensional space of cusp forms $\mathcal{S}_{k}^Γ$ of weight $2k$ for $Γ$, and let $\{f_{1},\ldots,f_{d}\}$ be an orthonormal basis of $\mathcal{S}_{k}^Γ$ with respect to the Petersson inner product. In this paper we show that the sup-norm of the quantity $S_{k}^Γ(z):=\sum_{j=1}^{d}| f_{j}(z)|^{2}\,\mathrm{Im}(z)^{2k}$ is bounded as $O_Γ(k)$ in the cocompact setting, and as $O_Γ(k^{3/2})$ in the cofinite case, where the implied constants depend solely on $Γ$. We also show that the implied constants are uniform if $Γ$ is replaced by a subgroup of finite index.

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