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Article: Number theoretic generalization of the Monster denominator formula
Title | Number theoretic generalization of the Monster denominator formula |
---|---|
Authors | |
Keywords | denominator formula moonshine polar harmonic Maass forms |
Issue Date | 2017 |
Publisher | Institute of Physics Publishing Ltd. The Journal's web site is located at http://iopscience.iop.org/1751-8121/ |
Citation | Journal of Physics A: Mathematical and Theoretical, 2017, v. 50 n. 47, p. 473001:1-473001:14 How to Cite? |
Abstract | The denominator formula for the Monster Lie algebra is the product expansion for the modular function $j(z) - j( au)$ in terms of the Hecke system of $SL2(Z)$-modular functions $j_n(z)$. This formula can be reformulated entirely number theoretically. Namely, it is equivalent to the description of the generating function for the $j_n(z)$ as a weight 2 modular form in with a pole at $z$. Although these results rely on the fact that $X_0(1)$ has genus 0, here we obtain a generalization, framed in terms of polar harmonic Maass forms, for all of the $X_0(N)$ modular curves. In this survey of recent work, we discuss this generalization, and we offer an introduction to the theory of polar harmonic Maass forms. We conclude with applications to formulas of Ramanujan and Green's functions. |
Persistent Identifier | http://hdl.handle.net/10722/247471 |
ISSN | 2023 Impact Factor: 2.0 2023 SCImago Journal Rankings: 0.769 |
ISI Accession Number ID |
DC Field | Value | Language |
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dc.contributor.author | Bringmann, K | - |
dc.contributor.author | Kane, BR | - |
dc.contributor.author | Löbrich, S | - |
dc.contributor.author | Rolen, L | - |
dc.contributor.author | Ono, K | - |
dc.date.accessioned | 2017-10-18T08:27:47Z | - |
dc.date.available | 2017-10-18T08:27:47Z | - |
dc.date.issued | 2017 | - |
dc.identifier.citation | Journal of Physics A: Mathematical and Theoretical, 2017, v. 50 n. 47, p. 473001:1-473001:14 | - |
dc.identifier.issn | 1751-8113 | - |
dc.identifier.uri | http://hdl.handle.net/10722/247471 | - |
dc.description.abstract | The denominator formula for the Monster Lie algebra is the product expansion for the modular function $j(z) - j( au)$ in terms of the Hecke system of $SL2(Z)$-modular functions $j_n(z)$. This formula can be reformulated entirely number theoretically. Namely, it is equivalent to the description of the generating function for the $j_n(z)$ as a weight 2 modular form in with a pole at $z$. Although these results rely on the fact that $X_0(1)$ has genus 0, here we obtain a generalization, framed in terms of polar harmonic Maass forms, for all of the $X_0(N)$ modular curves. In this survey of recent work, we discuss this generalization, and we offer an introduction to the theory of polar harmonic Maass forms. We conclude with applications to formulas of Ramanujan and Green's functions. | - |
dc.language | eng | - |
dc.publisher | Institute of Physics Publishing Ltd. The Journal's web site is located at http://iopscience.iop.org/1751-8121/ | - |
dc.relation.ispartof | Journal of Physics A: Mathematical and Theoretical | - |
dc.rights | Journal of Physics A: Mathematical and Theoretical. Copyright © Institute of Physics Publishing Ltd. | - |
dc.rights | This is an author-created, un-copyedited version of an article published in [Journal of Physics A: Mathematical and Theoretical]. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://dx.doi.org/10.1088/1751-8121/aa8f5d | - |
dc.subject | denominator formula | - |
dc.subject | moonshine | - |
dc.subject | polar harmonic Maass forms | - |
dc.title | Number theoretic generalization of the Monster denominator formula | - |
dc.type | Article | - |
dc.identifier.email | Kane, BR: bkane@hku.hk | - |
dc.identifier.authority | Kane, BR=rp01820 | - |
dc.description.nature | postprint | - |
dc.identifier.doi | 10.1088/1751-8121/aa8f5d | - |
dc.identifier.scopus | eid_2-s2.0-85034241930 | - |
dc.identifier.hkuros | 281576 | - |
dc.identifier.volume | 50 | - |
dc.identifier.issue | 47 | - |
dc.identifier.spage | 473001:1 | - |
dc.identifier.epage | 473001:14 | - |
dc.identifier.isi | WOS:000414068400001 | - |
dc.publisher.place | United Kingdom | - |
dc.identifier.issnl | 1751-8113 | - |