p-Adic Properties of Values of the Modular j-Function

Abstract

As usual, let q:=e^2πizand letj(z) be the classical modular function

$$ j(z) = \sum\limits_{n = - 1}^\infty {c(n){q^n} = {q^{ - 1}} + 744 + 196884q + \cdot \cdot \cdot } $$

The first author thanks the National Science Foundation, the Alfred P. Sloan Foundation, the David and Lucile Packard Foundation, and an H. I. Romnes Fellowship for their generous research support. Both authors thank the referee for his/her helpful suggestions and comments. The first author thanks Professors K. Hashimoto, K. Miyake and H. Nakamura for their kind hospitality during the 1999 and 2001 meetings. These were two wonderful conferences!