Reduction of CM elliptic curves and modular function congruences
Number Theory
2007-05-23 v1
Abstract
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.
Cite
@article{arxiv.math/0512350,
title = {Reduction of CM elliptic curves and modular function congruences},
author = {Noam D. Elkies and Ken Ono and Tonghai Yang},
journal= {arXiv preprint arXiv:math/0512350},
year = {2007}
}
Comments
11 pages