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

2004; Cambridge University Press; Volume: 140; Issue: 03 Linguagem: Inglês

10.1112/s0010437x03000721

1570-5846

Jan Hendrik Bruinier, Winfried Kohnen, Ken Ono,

Analytic Number Theory Research

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.