CLASSICAL AND p-ADIC MODULAR FORMS ARISING FROM THE BORCHERDS EXPONENTS OF OTHER MODULAR FORMS

نویسنده

  • JAYCE GETZ
چکیده

Abstract. Let f(z) = q ∏∞ n=1(1−q) be a modular form on SL2(Z). Formal logarithmic differentiation of f yields a q-series g(z) := h −∞n=1 ∑ d|n c(d)dq n whose coefficients are uniquely determined by the exponents of the original form. We provide a formula, due to Bruinier, Kohnen, and Ono for g(z) in terms of the values of the classical j-function at the zeros and poles of f(z). Further, we give a variety of cases in which g(z) is additionally a p-adic modular form in the classical sense of Serre. As an application, we derive some p-adic formulae, due to Bruinier, Ono, and Papanikolas, in which the class numbers of a family of imaginary quadratic fields are written in terms of special values of the j-function at imaginary quadratic arguments.

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تاریخ انتشار 2004