Dedekind symbols with polynomial reciprocity laws

A Dedekind symbol is a generalization of the classical Dedekind symbol (sum). A Dedekind symbol is characterized by its reciprocity law. Dedekind symbols with polynomial reciprocity laws are of special interest and importance, as such symbols are known to correspond bijectively to cusp forms for the full modular group, and to period polynomials. However, explicit forms of such Dedekind symbols are not yet known. In this article we construct Dedekind symbols explicitly by means of Poincaré series, and then show that these symbols satisfy polynomial reciprocity laws and that they form a spanning set for the space of Dedekind symbols with polynomial reciprocity laws. That is, we show that any Dedekind symbol with polynomial reciprocity law can be expressed as a linear combination of these symbols.Mathematics Subject Classification (2000): 11F20; 11F11, 33E05The author wishes to thank Professor N. Yui for her helpful advice.     

Dedekind symbols with plus reciprocity laws

Dedekind symbols are generalizations of the classical Dedekind sums (symbols), and the symbols are determined uniquely by their reciprocity laws, up to an additive constant. For Dedekind symbols D and F, we can consider two kinds of reciprocity laws: D(p,q)−D(q,−p)=R(p,q) and F(p,q)+F(q,−p)=T(p,q). The first type, which we call minus reciprocity laws, have been studied extensively. On the contrary, the second type, which we call plus reciprocity laws, have not yet been investigated. In this note we study fundamental properties of Dedekind symbols with plus reciprocity law F(p,q)+F(q,−p)=T(p,q). We will see that there is a fundamental difference between Dedekind symbols with minus and plus reciprocity laws.     

3-Cocycles, symbols and reciprocity laws on curves

We introduce a new approach for the study of two-dimensional symbols, F^*×F^*×F^*→G, where F is a discrete valuation field and G is a commutative group. From central extensions of groups we obtain a three-cocycle and the symbol is a differentiated element of the cohomology class . Our construction generalizes well-known two-dimensional symbols, such as the Parshin symbol on a surface, and we offer a proof and a conjecture for reciprocity laws on curves related to these symbols.     

Generalized Dedekind symbols associated with the Eisenstein series

We have shown recently that the space of modular forms, the space of generalized Dedekind sums, and the space of period polynomials are all isomorphic. In this paper, we will prove, under these isomorphisms, that the Eisenstein series correspond to the Apostol generalized Dedekind sums, and that the period polynomials are expressed in terms of Bernoulli numbers. This gives us a new more natural proof of the reciprocity law for the Apostol generalized Dedekind sums. Our proof yields as a by-product new polylogarithm identities.

     

Derivatives of Dedekind sums and their reciprocity law

In this paper derivatives of Dedekind sums are defined, and their reciprocity laws are proved. They are obtained from values at non-positive integers of the first derivatives of Barnes’ double zeta functions. As special cases, they give finite product expressions of the Stirling modular form and the double gamma function at positive rational numbers.     

Invertibility of Bergman Toeplitz operators with harmonic polynomial symbols

Let p be an analytic polynomial on the unit disk. We obtain a necessary and sufficient condition for Toeplitz operators with the symbol z + p to be invertible on the Bergman space when all coefficients of p are real numbers. Furthermore, we establish several necessary and sufficient, easy-to-check conditions for Toeplitz operators with the symbol z + p to be invertible on the Bergman space when some coefficients of p are complex numbers.     

On hyponormal Toeplitz operators with polynomial and circulant-type symbols

This paper characterises those hyponormal Toeplitz operators on the Hardy space of the unit circle among all Toeplitz operators that have polynomial symbols with circulant-type sets of coefficients.Supported in part by The Natural Sciences and Engineering Research Council of CanadaSupported in part by BSRI-96-1420 and KOSEF 94-0701-02-01-3     

On hyponormal toeplitz operators with polynomial and symmetric-type symbols

This paper treats the hyponormality of Toeplitz operators that have polynomial symbols with symmetric-type sets of coefficients.     

Algebraic functions with even monodromy

Let be a compact Riemann surface of genus and be an integer. We show that admits meromorphic functions with monodromy group equal to the alternating group

     

The recovery of even polynomial potentials

We consider the problem of reconstructing an even polynomial potential from one set of spectral data of a Sturm-Liouville problem. We show that we can recover an even polynomial of degree 2m from m+1 given Taylor coefficients of the characteristic function whose zeros are the eigenvalues of one spectrum. The idea here is to represent the solution as a power series and identify the unknown coefficients from the characteristic function. We then compute these coefficients by solving a nonlinear algebraic system, and provide numerical examples at the end. Because of its algebraic nature, the method applies also to non self-adjoint problems.     

Hyperdeterminants associated with multiple even functions

In this paper, we introduce a new method to give the formulae for the hyperdeterminants of higher-dimensional matrices associated with multiple even function (modr) on the gcd-closed sets. Our result generalizes the result of Yamasaki obtained in 2010 and also extends the results of Haukkanen and Hong obtained in 1992 and 2002, respectively.     

The structure of polynomial conservation laws

We consider periodic closures of integrable chains. We establish a compact formula for the generating function of the conservation laws. This generating function is common to classical integrable chains related to various second-order spectral problems. We study the stabilization problem for the form of the conservation laws in the limit as the closure period tends to infinity.     

Theory and numerical evaluation of oddoids and evenoids: Oscillatory cuspoid integrals with odd and even polynomial phase functions

The properties of oscillating cuspoid integrals whose phase functions are odd and even polynomials are investigated. These integrals are called oddoids and evenoids, respectively (and collectively, oddenoids). We have studied in detail oddenoids whose phase functions contain up to three real parameters. For each oddenoid, we have obtained its Maclaurin series representation and investigated its relation to Airy–Hardy integrals and Bessel functions of fractional orders. We have used techniques from singularity theory to characterise the caustic (or bifurcation set) associated with each oddenoid, including the occurrence of complex whiskers. Plots and short tables of numerical values for the oddenoids are presented. The numerical calculations used the software package CUSPINT [N.P. Kirk, J.N.L. Connor, C.A. Hobbs, An adaptive contour code for the numerical evaluation of the oscillatory cuspoid canonical integrals and their derivatives, Comput. Phys. Commun. 132 (2000) 142–165].