Elkies, NoamOno, KenYang, Tonghai2009-04-212005Elkies, Noam D., Ken Ono, and Tonghai Yang. 2005. Reduction of CM elliptic curves and modular function congruences. International Mathematics Research Notices (44): 2695-2707.1073-7928http://nrs.harvard.edu/urn-3:HUL.InstRepos:2797455We 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.en-USquadratic formssingular modulihalf-integral weight