Ramanujan differential operator, certain CM elliptic curve and Kummer congruences
by
P. Guerzhoy
Temple University, Department of Mathematics

Consider a point in the complex upper half-plane such that the corresponding elliptic curve can be defined over rationals, and let f be a modular form on the full modular group with rational Fourier coefficients. By applying the Ramanujan differential operator to f repeatedly, we obtain a family of modular forms. We normalize these modular forms in a reasonable way and take their values at the chosen point in the upper half-plane. We obtain in this way a sequence of rational numbers. We study the behavior of these numbers modulo the powers of a prime p > 3. Surprisingly, this depends on the residue of the prime p modulo 3. If this residue is 1, then certain periodicity modulo powers of p (Kummer-type congruences) takes place. If this residue is 2, then the p-adic valuations of these quantities grow arbitrary large. We explain the phenomenon by making a connection with a certain elliptic curve whose reduction modulo p is ordinary or supersingular depending on the residue of p modulo 3. (The entire paper is submitted for publication.)

Date received: November 14, 2003