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The arithmetic of the values of modular functions and the divisors of modular forms

Published online by Cambridge University Press:  04 December 2007

Jan H. Bruinier ,
Winfried Kohnen  and
Ken Ono
Jan H. Bruinier
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706, USAbruinier@math.uni-koeln.de, ono@math.wisc.edu Mathematisches Institut, Universität Köln, Im Weyertal 86-90, D-50931 Köln, Germany
Winfried Kohnen
Affiliation:
Mathematisches Institut, Universität Heidelberg, INF 288, D-69120 Heidelberg, Germanywinfried@mathi.uni-heidelberg.de
Ken Ono
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706, USAbruinier@math.uni-koeln.de, ono@math.wisc.edu
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Abstract

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We investigate the arithmetic and combinatorial significance of the values of the polynomials jn(x) defined by the q-expansion \[\sum_{n=0}^{\infty}j_n(x)q^n:=\frac{E_4(z)^2E_6(z)}{\Delta(z)}\cdot\frac{1}{j(z)-x}.\] They allow us to provide an explicit description of the action of the Ramanujan Theta-operator on modular forms. There are a substantial number of consequences for this result. We obtain recursive formulas for coefficients of modular forms, formulas for the infinite product exponents of modular forms, and new p-adic class number formulas.

Keywords

Borcherds productssingular modulimodular forms and functions11F0311F1111F33
Type
Research Article
Information
Compositio Mathematica , Volume 140 , Issue 3 , May 2004 , pp. 552 - 566
Copyright
Foundation Compositio Mathematica 2004

Footnotes

The first and third authors thank the Number Theory Foundation for its generous support, and the third author is grateful for the support of an Alfred P. Sloan Fellowship, a David and Lucile Packard Fellowship, an H. I. Romnes Fellowship, a John Guggenheim Fellowship and a grant from the National Science Foundation.