Congruence formulas for Legendre modular polynomials

Let p >= 5 be a prime number. We generalize the results of E. de Shalit [4] about supersingular j-invariants in characteristic p. We consider supersingular elliptic curves with a basis of 2-torsion over (F) over barp, or equivalently supersingular Legendre lambda-invariants. Let F-p(X,Y) is an element of Z[X,Y] be the p-th modular polynomial for lambda-invariants. A simple generalization of Kronecker's classical congruence shows that R(X) := F-p(X,X-p)/p is in Z[X]. We give a formula for R(lambda) if lambda is supersingular. This formula is related to the Manin-Drinfeld pairing used in the p-adic uniformization of the modular curve X(Gamma(0)(p) boolean AND Gamma(2)). This pairing was computed explicitly modulo principal units in a previous work of both authors. Furthermore, if lambda is supersingular and is in F-p, then we also express R(lambda) in terms of a CM lift (which is shown to exist) of the Legendre elliptic curve associated to lambda. (C) 2018 The Authors. Published by Elsevier Inc.