Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8896906 | Journal of Number Theory | 2018 | 17 Pages |
Abstract
Let pâ¥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 Fâ¾p, or equivalently supersingular Legendre λ-invariants. Let Fp(X,Y)âZ[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(Î0(p)â©Î(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.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Adel Betina, Emmanuel Lecouturier,