Jay Jorgenson

Bounds on canonical Green's functions

A fundamental object in the theory of arithmetic surfaces is the Green’s function associated to the canonical metric. Previous expressions for the canonical Green’s function have relied on general functional analysis or, when using specific properties of the canonical metric, the classical Riemann theta function. In this article, we derive a new identity for the canonical Green’s function involving the hyperbolic heat kernel. As an application of our results, we obtain bounds for the canonical Green’s function through covers and for families of modular curves.

Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori

By a discrete torus we mean the Cayley graph associated to a finite product of finite cycle groups with the generating set given by choosing a generator for each cyclic factor. In this article we e ...

Uniform Sup-Norm Bounds on Average for Cusp Forms of Higher Weights

Let \(\Gamma \subset \mathrm{ PSL}_{2}(\mathbb{R})\) be a Fuchsian subgroup of the first kind acting by fractional linear transformations on the upper half-plane \(\mathbb{H}\). Consider the d-dimensional space of cusp forms \(\mathcal{S}_{2k}^{\Gamma }\) of weight 2k for \(\Gamma\), and let {f1, …, f d } be an orthonormal basis of \(\mathcal{S}_{2k}^{\Gamma }\) with respect to the Petersson inner product. In this paper we show that the sup-norm of the quantity \(S_{2k}^{\Gamma }(z):=\sum _{ j=1}^{d}\vert f_{j}(z)\vert ^{2}\,\mathrm{Im}(z)^{2k}\) is bounded as \(O_{\Gamma }(k)\) in the cocompact setting, and as \(O_{\Gamma }(k^{3/2})\) in the cofinite case, where the implied constants depend solely on \(\Gamma\). We also show that the implied constants are uniform if \(\Gamma\) is replaced by a subgroup of finite index.

On the appearance of Eisenstein series through degeneration

Let G be a Fuchsian group of the first kind acting on the hyperbolic upper half plane H, and let M = G \ H be the associated finite volume hyperbolic Riemann surface. If ? is parabolic, there is an associated (parabolic) Eisenstein series, which, by now, is a classical part of mathematical literature. If ? 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 ? ? G corresponds to a degenerating hyperbolic element, then a multiple of the associated hyperbolic Eisenstein series converges to parabolic Eisenstein series on the limit surface.

Kronecker’s Limit Formula, Holomorphic Modular Functions, and q-Expansions on Certain Arithmetic Groups

ABSTRACT For any square-free integer N such that the “moonshine group” Γ0(N)+ has genus zero, the Monstrous Moonshine Conjectures relate the Hauptmodul of Γ0(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 groups Γ0(N)+. For all such arithmetic 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∞ 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.

A Relation Involving Rankin-Selberg L-Functions of Cusp Forms and Maass Forms

In previous articles, an identity relating the canonical metric to the hyperbolic metric associated with any compact Riemann surface of genus at least two has been derived and studied. In this article, this identity is extended to any hyperbolic Riemann surface of finite volume. The method of proof is to study the identity given in the compact case through degeneration and to understand the limiting behavior of all quantities involved. In the second part of the paper, the Rankin-Selberg transform of the noncompact identity is studied, meaning that both sides of the relation after multiplication by a nonholomorphic, parabolic Eisenstein series are being integrated over the Riemann surface in question. The resulting formula yields an asymptotic relation involving the Rankin-Selberg L-functions of weight two holomorphic cusp forms, of weight zero Maass forms, and of nonholomorphic weight zero parabolic Eisenstein series.

Architecture of Generalized Network Service Anomaly and Fault Thresholds

In this paper we introduce GAFT (Generalized Anomaly and Fault Threshold), featuring a novel system architecture that is capable of setting, monitoring and detecting generalized thresholds and soft faults proactively and adaptively. GAFT monitors many network parameters simultaneously, analyzes statistically their performance, combines intelligently the individual decisions and derives an integrated result of compliance for each service class. We have carried out simulation experiments of network resource and service deterioration, when increasingly congested in the presence of class-alien traffic, where GAFT combines intelligently, using a neural network classifier, 12 monitored network performance parameter decisions into a unified result. To this end, we tested five different types of neural network classifiers: Perceptron, BP, PBH, Fuzzy ARTMAP, and RBF. Our results indicate that BP and PBH provide more effective classification than the other neural networks. We also stress tested the entire system, which showed that GAFT can reliably detect class-alien traffic with intensity as low as five to ten percent of typical service class traffic.

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 fixed points for a few small level moonshine groups.RésuméDans cet article nous développons deux applications de la formule limite de Kronecker associée aux series d’Eisenstein elliptiques: Un théorème de factorisation pour des formes modulaires holomorphes et une preuve de la loi de réciprocité de Weil. Plusieurs exemples de notre résultat général de factorisation sont donnés, particulièrement pour quelques groupes de type moonshine, groupes de congruence et, plus généralement, pour des groupes non-cocompactes à une seule pointe. En particulier, nous calculons la fonction limite de Kronecker associée à certains points elliptiques pour des groupes de type moonshine de petit niveau.

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.

Star products of Green's currents and automorphic forms

In previous work, the authors computed archimedian heights of hermitian line bundles on families of polarized, n-dimensional abelian varieties. In this paper, a detailed analysis of the results obtained in the setting of abelian fibrations is given, and it is shown that the proofs can be modified in such a way that they no longer depend on the specific setting of abelian fibrations and hence extend to a quite general situation. Specifically, we let f : X → Y be any family of smooth, projective, ndimensional complex varieties over some base, and consider a line bundle on X equipped with a smooth, hermitian metric. To this data is associated a hermitian line bundle M on Y characterized by conditions on the first Chern class. Under mild additional hypotheses, it is shown that, for generically chosen sections of L, the integral of the (n + 1)-fold star product of Green’s currents associated to the sections, integrated along the fibers of f , is the log-norm of a global section of M . Furthermore, it is proven that in certain general settings the global section of M can be explicitly expressed in terms of point evaluations of the original sections. A particularly interesting example of this general result appears in the setting of polarized Enriques surfaces when M is a moduli space of degree-2 polarizations. In this setting, the global section constructed via Green’s currents is equal to a power of the -function first studied by R. Borcherds. Additional examples and problems are presented.