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Arithmetic of singular moduli and class polynomials

Published online by Cambridge University Press:  10 February 2005

Scott Ahlgren  and
Ken Ono
Scott Ahlgren
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 61801, USAahlgren@math.uiuc.edu
Ken Ono
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706, USAono@math.wisc.edu
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Abstract

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We investigate divisibility properties of the traces and Hecke traces of singular moduli. In particular we prove that, if p is prime, these traces satisfy many congruences modulo powers of p which are described in terms of the factorization of p in imaginary quadratic fields. We also study generalizations of Lehner's classical congruences $j(z)| U_p\equiv 744 \pmod p$ (where $p\leq 11$ and j(z) is the usual modular invariant), and we investigate connections between class polynomials and supersingular polynomials in characteristic p.

Keywords

singular moduliclass polynomialsmodular forms11F3311F37 (primary)
Type
Research Article
Information
Compositio Mathematica , Volume 141 , Issue 2 , March 2005 , pp. 293 - 312
Copyright
Foundation Compositio Mathematica 2005