Congruence formulas for Legendre modular polynomials

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