Traces of singular moduli of arbitrary level modular functions

Dohoon Choi, Daeyeol Jeon, Soon Yi Kang, Chang Heon Kim

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

Generalizing Zagier's work, Bruinier and Funke recently proved that for modular curves of arbitrary genus, the generating series for the traces of the CM values of a weakly holomorphic modular function is the holomorphic part of a harmonic weak Maass form of weight 3/2. The present article shows that by adding a suitable linear combination of weight 3/2 Eisenstein series, one can always obtain a generating series that is weakly holomorphic. In particular, the modular traces of a Hauptmodul on Γ*0(4) are found to be either Fourier coefficients of a weakly holomorphic modular form of weight 3/2 or constantmultiples of class numbers. As an application, we obtain congruence properties for the traces of singular moduli of a weakly holomorphic modular function of arbitrary level.

Original languageEnglish
Article numberrnm110
JournalInternational Mathematics Research Notices
Volume2007
DOIs
Publication statusPublished - 2007 Dec 1
Externally publishedYes

Fingerprint

Modular Functions
Modulus
Trace
Analytic function
Arbitrary
Modular Curves
Eisenstein Series
Series
Class number
Modular Forms
Fourier coefficients
Congruence
Linear Combination
Genus
Harmonic

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Traces of singular moduli of arbitrary level modular functions. / Choi, Dohoon; Jeon, Daeyeol; Kang, Soon Yi; Kim, Chang Heon.

In: International Mathematics Research Notices, Vol. 2007, rnm110, 01.12.2007.

Research output: Contribution to journalArticle

Choi, Dohoon ; Jeon, Daeyeol ; Kang, Soon Yi ; Kim, Chang Heon. / Traces of singular moduli of arbitrary level modular functions. In: International Mathematics Research Notices. 2007 ; Vol. 2007.
@article{4b42b2db007f47b0b0367e486678471f,
title = "Traces of singular moduli of arbitrary level modular functions",
abstract = "Generalizing Zagier's work, Bruinier and Funke recently proved that for modular curves of arbitrary genus, the generating series for the traces of the CM values of a weakly holomorphic modular function is the holomorphic part of a harmonic weak Maass form of weight 3/2. The present article shows that by adding a suitable linear combination of weight 3/2 Eisenstein series, one can always obtain a generating series that is weakly holomorphic. In particular, the modular traces of a Hauptmodul on Γ*0(4) are found to be either Fourier coefficients of a weakly holomorphic modular form of weight 3/2 or constantmultiples of class numbers. As an application, we obtain congruence properties for the traces of singular moduli of a weakly holomorphic modular function of arbitrary level.",
author = "Dohoon Choi and Daeyeol Jeon and Kang, {Soon Yi} and Kim, {Chang Heon}",
year = "2007",
month = "12",
day = "1",
doi = "10.1093/imrn/rnm110",
language = "English",
volume = "2007",
journal = "International Mathematics Research Notices",
issn = "1073-7928",
publisher = "Oxford University Press",

}

TY - JOUR

T1 - Traces of singular moduli of arbitrary level modular functions

AU - Choi, Dohoon

AU - Jeon, Daeyeol

AU - Kang, Soon Yi

AU - Kim, Chang Heon

PY - 2007/12/1

Y1 - 2007/12/1

N2 - Generalizing Zagier's work, Bruinier and Funke recently proved that for modular curves of arbitrary genus, the generating series for the traces of the CM values of a weakly holomorphic modular function is the holomorphic part of a harmonic weak Maass form of weight 3/2. The present article shows that by adding a suitable linear combination of weight 3/2 Eisenstein series, one can always obtain a generating series that is weakly holomorphic. In particular, the modular traces of a Hauptmodul on Γ*0(4) are found to be either Fourier coefficients of a weakly holomorphic modular form of weight 3/2 or constantmultiples of class numbers. As an application, we obtain congruence properties for the traces of singular moduli of a weakly holomorphic modular function of arbitrary level.

AB - Generalizing Zagier's work, Bruinier and Funke recently proved that for modular curves of arbitrary genus, the generating series for the traces of the CM values of a weakly holomorphic modular function is the holomorphic part of a harmonic weak Maass form of weight 3/2. The present article shows that by adding a suitable linear combination of weight 3/2 Eisenstein series, one can always obtain a generating series that is weakly holomorphic. In particular, the modular traces of a Hauptmodul on Γ*0(4) are found to be either Fourier coefficients of a weakly holomorphic modular form of weight 3/2 or constantmultiples of class numbers. As an application, we obtain congruence properties for the traces of singular moduli of a weakly holomorphic modular function of arbitrary level.

UR - http://www.scopus.com/inward/record.url?scp=77951978819&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77951978819&partnerID=8YFLogxK

U2 - 10.1093/imrn/rnm110

DO - 10.1093/imrn/rnm110

M3 - Article

VL - 2007

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

M1 - rnm110

ER -