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‾p, or equivalently supersingular Legendre λ-invariants. Let Fp(X,Y)∈Z[X,Y] be the p-th modular polynomial for λ-invariants. A simple generalization of Kronecker's classical congruence shows that R(X):=Fp(X,Xp)p is in Z[X]. We give a formula for R(λ) if λ is supersingular. This formula is related to the Manin-Drinfeld pairing used in the p-adic uniformization of the modular curve X(Γ0(p)∩Γ(2)). This pairing was computed explicitly modulo principal units in a previous work of both authors. Furthermore, if λ is supersingular and is in Fp, then we also express R(λ) in terms of a CM lift (which is shown to exist) of the Legendre elliptic curve associated to λ.