We study the properties of one-dimensional hypergeometric integral solutions of the q-difference ("quantum") analogue of the Knizhnik-Zamolodchikov-Bernard equations on tori. We show that they also obey a di...
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We study the properties of one-dimensional hypergeometric integral solutions of the q-difference ("quantum") analogue of the Knizhnik-Zamolodchikov-Bernard equations on tori. We show that they also obey a difference KZB heat equation in the modular parameter, give formulae for modular transformations, and prove a completeness result, by showing that the associated Fourier transform is invertible. These results are based on SL(3, Z) transformation properties parallel to those of elliptic gamma functions. (C) 2002 Elsevier Seience (USA).
Holomorphic almost modular forms are holomorphic functions of the complex upper half plane that can be approximated arbitrarily well (in a suitable sense) by modular forms of congruence subgroups of large index in SL(...
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Holomorphic almost modular forms are holomorphic functions of the complex upper half plane that can be approximated arbitrarily well (in a suitable sense) by modular forms of congruence subgroups of large index in SL(2, Z). It is proved that such functions have a rotation-invariant limit distribution when the argument approaches the real axis. AD example of a holomorphic almost modular form is the logarithm of Pi(n=1)(infinity) (1 - exp (2pi i n(2)z)). The paper is motivated by the author's previous studies [Int. Math. Res. Not. 39 (2003) 2131-2151] on the connection between almost modular functions and the distribution of the sequence n(2)x modulo one.
We study boundary conditions for extended topological quantum field theories (TQFTs) and their relation to topological anomalies. We introduce the notion of TQFTs with moduli level m, and describe extended anomalous t...
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We study boundary conditions for extended topological quantum field theories (TQFTs) and their relation to topological anomalies. We introduce the notion of TQFTs with moduli level m, and describe extended anomalous theories as natural transformations of invertible field theories of this type. We show how in such a framework anomalous theories give rise naturally to homotopy fixed points for n-characters on infinity-groups. By using dimensional reduction on manifolds with boundaries, we show how boundary conditions for n + 1-dimensional TQFTs produce n-dimensional anomalous field theories. Finally, we analyse the case of fully extended TQFTs, and show that any fully extended anomalous theory produces a suitable boundary condition for the anomaly field theory.
Let p be a prime and f (z) = Sigma(n) a(n)q(n) be a weakly holomorphicmodular function for Gamma(0)*(p(2)) with a(0) = 0. We use Bruinier and Funke's work to find the generating series of modular traces of f (z) a...
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Let p be a prime and f (z) = Sigma(n) a(n)q(n) be a weakly holomorphicmodular function for Gamma(0)*(p(2)) with a(0) = 0. We use Bruinier and Funke's work to find the generating series of modular traces of f (z) as Jacobi forms. And as an application we construct Borcherds products related to the Hauptmoduln j(p2)* for genus zero groups Gamma(0)*(p(2)).
We investigate two kinds of Fricke families, those consisting of Fricke functions and those consisting of Siegel functions. In terms of their special values we then generate ray class fields of imaginary quadratic fie...
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We investigate two kinds of Fricke families, those consisting of Fricke functions and those consisting of Siegel functions. In terms of their special values we then generate ray class fields of imaginary quadratic fields over the Hilbert class fields, which are related to the Lang-Schertz conjecture.
We construct a new system of three terms arithmetic geometric mean (we say AGM). Our system is an extension of the classical Gauss AGM. It is induced from an isogeny formula of the modular forms defined on the complex...
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We construct a new system of three terms arithmetic geometric mean (we say AGM). Our system is an extension of the classical Gauss AGM. It is induced from an isogeny formula of the modular forms defined on the complex 2-dimensional hyperball with respect to a principal congruence subgroup of the Picard modular group U(2, 1;Z vertical bar root-1 vertical bar). Our AGM function is expressed via the Appell hypergeometric function. These results are the extension of the properties discovered by Gauss himself.. Our result is based on the work of K. Matsumoto in 1989 [K. Matsumoto, On modular functions in 2 variables attached to a family of hyperelliptic curves of genus 3, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 16 (4) (1989) 557-578]. And our present paper is a continuation of the previous work [K. Koike, H. Shiga, Isogeny formulas for the Picard modular form and a three terms arithmetic geometric mean, J. Number Theory 124 (2007) 123-141. [Ko-Shi]]. (C) 2008 Elsevier Inc. All rights reserved.
The partition function p([1cld])(n) can be defined using the generating function Sigma(infinity)(n=0)p([1cld])(n)q(n) = Pi(infinity)(n=1)1/(1-q(n))(c)(1-q(ln))(d). In Mestrige (Res Number Theory 6(1), Paper No. 5, 202...
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The partition function p([1cld])(n) can be defined using the generating function Sigma(infinity)(n=0)p([1cld])(n)q(n) = Pi(infinity)(n=1)1/(1-q(n))(c)(1-q(ln))(d). In Mestrige (Res Number Theory 6(1), Paper No. 5, 2020), we proved an infinite family of congruences for this partition function for l = 11. In this paper, we extend the ideas that we have used in Mestrige (2020) to prove infinite families of congruences for the partition function p([1cld])(n) modulo powers of l for any integers c and d, for primes 5 <= l <= 17. This generalizes Atkin, Gordon and Hughes' congruences for powers of the partition function. The proofs use an explicit basis for the vector space of modular functions on the congruence subgroup Gamma(0)(l). Finally we use these congruences to prove congruences and incongruences for l-colored generalized Frobenius partitions, l-regular partitions, and l-core partitions for l = 5, 7, 11, and 13. We also prove incongruences for 17-regular, and 17-core partitions.
We attach p-adic L-functions to critical modular forms and study them. We prove that those L-functions fit in a two-variables p-adic L-function defined locally everywhere on the eigencurve.
We attach p-adic L-functions to critical modular forms and study them. We prove that those L-functions fit in a two-variables p-adic L-function defined locally everywhere on the eigencurve.
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