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

  • Jan H. Bruinier (a1) (a2), Winfried Kohnen (a3) and Ken Ono (a1)

Abstract

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.

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Copyright

Footnotes

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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.

Footnotes

MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

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

  • Jan H. Bruinier (a1) (a2), Winfried Kohnen (a3) and Ken Ono (a1)

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