Compositio Mathematica



The Ramanujan differential operator, a certain CM elliptic curve and Kummer congruences


P. Guerzhoy a1
a1 Department of Mathematics, Temple University, 1805 N. Broad Street, Philadelphia, PA 19122, USA pasha@math.temple.edu

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Abstract

Let $\tau$ be a point in the upper half-plane such that the elliptic curve corresponding to $\tau$ can be defined over $\mathbb{Q}$, and let f be a modular form on the full modular group with rational Fourier coefficients. By applying the Ramanujan differential operator D to f, we obtain a family of modular forms Dlf. In this paper we study the behavior of $D^l(f)(\tau)$ modulo the powers of a prime p > 3. We show that for $p \equiv 1 \bmod 3$ the quantities $D^l(f)(\tau)$, suitably normalized, satisfy Kummer-type congruences, and that for $p \equiv 2 \bmod 3$ the p-adic valuations of $D^l(f)(\tau)$ grow arbitrarily large. We prove these congruences by making a connection with a certain elliptic curve whose reduction modulo p is ordinary if $p \equiv 1 \bmod 3$ and supersingular otherwise.

(Received September 26 2003)
(Accepted February 24 2004)
(Published Online April 21 2005)


Key Words: modular forms; formal groups; congruences.

Maths Classification

11F33 (primary); 11F25 (secondary).