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Reduction of CM elliptic curves and modular function congruences

Noam D. Elkies·Ken Ono·Tonghai Yang
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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.

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Noam D. Elkies, Ken Ono, Tonghai Yang. 2005-12-15. Reduction of CM elliptic curves and modular function congruences. https://arxiv.org/abs/math/0512350

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