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

Let τ be a point in the upper half-plane such that the elliptic curve corresponding to τ can be defined over 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 Dl(f)(τ) modulo the powers of a prime p > 3. We show that for p ≡ 1 mod 3 the quantities Dl(f)(τ), suitably normalized, satisfy Kummer-type congruences, and that for p ≡ 2 mod 3 the p-adic valuations of Dl(f)(τ) grow arbitrarily large. We prove these congruences by making a connection with a certain elliptic curve whose reduction modulo p is ordinary if p ≡ 1 mod 3 and supersingular otherwise.