Universität zu Köln

Ono, Ken, Rolen, Larry and Trebat-Leder, Sarah (2015). Classical and umbral moonshine: Connections and p-adic properties. J. Ramanujan Math. Soc., 30 (2). S. 135 - 160. MYSORE: RAMANUJAN MATHEMATICAL SOC. ISSN 2320-3110

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Abstract

The classical theory of monstrous moonshine describes the unexpected connection between the representation theory of the monster group M, the largest of the sporadic simple groups, and certain modular functions, called Hauptmoduln. In particular, the n-th Fourier coefficient of Klein's j-function is the dimension of the grade n part of a special infinite dimensional representation Vth of the monster group. More generally the coefficients of Hauptmoduln are graded traces T-g of g epsilon M acting on Vu. Similar phenomena have been shown to hold for the Mathieu group M-24, but instead of modular functions, mock modular forms must be used. This has been conjecturally generalized even further, to umbral moonshine, which associates to each of the 23 Niemeier lattices a finite group, infinite dimensional representation, and mock modular form. We use generalized Borcherds products to relate monstrous moonshine and umbral moonshine. Namely, we use mock modular forms from umbral moonshine to construct via generalized Borcherds products rational functions of the Hauptmoduln T-g from monstrous moonshine. This allows us to associate to each pure A-type Niemeier lattice a conjugacy class g of the monster group, and gives rise to identities relating dimensions of representations from umbral moonshine to values of T-g. We also show that the logarithmic derivatives of the Borcherds products are p-adic modular forms for certain primes p and describe some of the resulting properties of their coefficients modulo p.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Ono, Ken UNSPECIFIED UNSPECIFIED UNSPECIFIED Rolen, Larry UNSPECIFIED UNSPECIFIED UNSPECIFIED Trebat-Leder, Sarah UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-402466
Journal or Publication Title:	J. Ramanujan Math. Soc.
Volume:	30
Number:	2
Page Range:	S. 135 - 160
Date:	2015
Publisher:	RAMANUJAN MATHEMATICAL SOC
Place of Publication:	MYSORE
ISSN:	2320-3110
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language HALF-INTEGRAL WEIGHT; MODULAR-FORMS; K3 SURFACE; SERIES; VALUES Multiple languages Mathematics Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/40246