Compositio Mathematica

Arithmetic of singular moduli and class polynomials

Scott Ahlgren a1 and Ken Ono a2
a1 Department of Mathematics, University of Illinois, Urbana, IL 61801, USA
a2 Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA

Article author query
ahlgren s   [Google Scholar] 
ono k   [Google Scholar] 


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.

(Received April 29 2003)
(Accepted May 14 2004)
(Published Online February 10 2005)

Key Words: singular moduli; class polynomials; modular forms.

Maths Classification

11F33; 11F37 (primary).