Congruence formulas for Legendre modular polynomials

Abstract

Let p ≥ 5 be a prime number. We generalize the results of E. de Shalit [13] 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 F p (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) :=We give a formula for R(λ) if λ is a supersingular. This formula is related to the Manin-Drinfeld pairing used in the p-adic uniformization of the modular curve X(Γ 0 (p) ∩ Γ(2)). This pairing was computed explicitly modulo principal units in a previous work of both authors. Furthermore, if λ is supersingular and lives in F p , then we also express R(λ) in terms of a CM lift (which are showed to exist) of the Legendre elliptic curve associated to λ.1 2 ADEL BETINA AND EMMANUEL LECOUTURIER By combining our previous work (cf.[1]) on the p-adic uniformization of our modular curve and the present results, we obtain an elementary formula for values taken by F p (X, X) modulo p 2 (which does not however give us a formula for the polynomial itself). We now give more precise details about ours results.Let M Γ 0 (p)∩Γ(2) be the stack over Z[1/2] whose S-points are the isomorphism classes of generalized elliptic curves E/S, endowed with a locally free subgroup A of rank p such that A + E[2] meets each irreducible component of any geometric fiber of E (E[2] is the subgroup of 2-torsion points of E) and a basis of the 2-torsion (i.e. an isomorphism α 2 : E[2] ≃ (Z/2Z) 2 ). Deligne and Rapoport proved in [2] that M Γ 0 (p)∩Γ(2) is a regular algebraic stack, proper, of pure dimension 2 and flat over Z[1/2].Let M Γ 0 (p)∩Γ(2) be the coarse space of the algebraic stack M Γ 0 (p)∩Γ(2) over Z[1/2]. Deligne-Rapoport proved that M Γ 0 (p)∩Γ(2) is a normal scheme and proper flat of relative dimension one over Z[1/2]. Moreover, Deligne-Rapoport proved that M Γ 0 (p)∩Γ (2) is smooth over Z[1/2] outside the points associated to supersingular elliptic curves in characteristic p and that M Γ 0 (p)∩Γ(2) is a semi-stable regular scheme (cf. [2, V.1.14, Variante] and [1, Proposition 2.1] for more details).Let K be the unique quadratic unramified extension of Q p , O K be the ring of integers of K and k be the residual field. Let X be the base change M Γ 0 (p)∩Γ(2) ⊗ O K ; it is the coarse moduli space of the base change M Γ 0 (p)∩Γ(2) ⊗O K (because the formation of coarse moduli space commutes with flat base change).Let M Γ(2) be the model over Z[1/2] of the modular curve X(2) introduced by Igusa [9]. The special fiber of the scheme X is the union of two copies of M Γ(2) ⊗k meeting transversally at the supersingular points, and such that a supersingular point x of the first copy is identified with the point x p = Frob p (x) of the second copy (the supersingular points of the special fiber of X are k-rational). Moreover, we have M Γ(2) ⊗k ≃ P 1 k (cf. [1, Proposition 2.1]).The cusps of M Γ 0 (p)∩Γ(2) correspond to Néron 2-gons or 2p-...

Higher Eisenstein elements, higher Eichler formulas and rank of Hecke algebras

“…By (5) the (7), we get Ann T( Ĩ) • T = 0. As above, this implies Ann T( Ĩ) = 0, so T = T. This contradicts hypothesis (iv).…”