Reduction of CM elliptic curves and modular function congruences

Details Reduction of CM elliptic curves and modular function congruences Reduction of CM elliptic curves and modular function congruences

2005-12-15

arxiv.org/abs/math/0512350v1

International Math. Research Notices, Vol. 2005, #44, pages 2695--2707

Mathematics

Number Theory

11 pages

Scientific paper

We study congruences of the form F(j(z)) | U(p) = G(j(z)) mod p, where U(p) is the p-th Hecke operator, j is the basic modular invariant 1/q+744+196884q+... for SL2(Z), and F,G are polynomials with integer coefficients. Using the interplay between singular (a.k.a. CM) j-invariants in characteristic zero and supersingular ones in characteristic p, we obtain such congruences in which F is the minimal polynomial of a CM j-invariant, and give a sufficient condition for G to be a constant polynomial in these congruences.

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