p-ADIC LIFTINGS OF THE SUPERSINGULAR j-INVARIANTS AND j-ZEROS OF CERTAIN EISENSTEIN SERIES

Let p > 3 be a prime. We consider j-zeros of Eisenstein series Ek of weights k = p−1+Mp(p−1) with M,a ≥ 0 as elements of Qp. If M = 0, the j-zeros of Ep−1 belong to Qp(ζp2−1) by Hensel’s Lemma. Call these j-zeros p-adic liftings of supersingular j-invariants. We show that for every such lifting u there is a j-zero r of Ek such that ordp(r − u) > a. Applications of this result are considered. The proof is based on the techniques of formal groups. 1. Statement and discussion of results Zeros of modular forms is an interesting subject, and there has been a big amount of research connected to this subject during the past several decades (see [1, 2, 3, 5, 14] to name a few). Zeros of Eisenstein series attract special attention. For an even integer k ≥ 4 denote by Ek the weight k Eisenstein series Ek = 1− 2k Bk ∑ n≥1 ∑ d|n dk−1  q, q = exp(2πiτ), =τ > 0, where theBk are Bernoulli numbers defined by the power series x/(exp(x)− 1) = ∑ k≥0Bk xk k! . Following the terminology of [5], we define j-zeros to be the j-invariants of zeros of Ek. Denote by Ψk(X) the polynomial that encodes the j-zeros of Ek: Ψk(X) = ∏ j=j(τ), where Ek(τ)=0 (X − j) Let p > 3 be a prime. The coefficients of Ψp−1 are p-integral. It is a well-known observation of Deligne (see [9] for a full exposition) that Ψ̃p−1(X), the modulo p reduction of Ψp−1(X), is the supersingular 1991 Mathematics Subject Classification. 11F11,11F12. ∗Supported by NSF grant DMS-0700933. 1 2 P. GUERZHOY AND Z. KENT polynomial at p. The roots of Ψ̃p−1(X) over Fp are supersingular jinvariants. This polynomial, considered as a polynomial over Fp, splits into a product of factors over Fp,