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Arithmetic of singular moduli and class polynomials

Published online by Cambridge University Press:  10 February 2005

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.

Type
Research Article
Copyright
Foundation Compositio Mathematica 2005