2-adic properties of traces of singular moduli
by
Matthew Boylan
University of Illinois at Urbana-Champaign

Suppose that d is a positive integer congruent to 0 or 3 mod 4, that Q(x, y)=ax^2+bxy+cy^2 is a positive definite integral binary quadratic form with discriminant -d, that alpha_Q is the unique solution of Q(x, 1)=0 in the upper half-plane, and that j(z) is the usual elliptic modular invariant. Recently, Zagier defined the function t(d), the Hecke trace of the singular modulus with discriminant -d, to be the trace of j(alpha_Q) over a complete set (Q_j) of representatives of PSL_2(Z)-equivalence classes of forms of discriminant -d weighted by the size of the stabilizer of Q_j. In recent work, Ahlgren and Ono showed that if p is an odd prime which splits in an imaginary quadratic field of discriminant -d, then t(p^2*d)=0 mod p. A question of Ono asks for natural generalizations modulo arbitrary prime powers. We provide the answer when p=2 by showing, for all positive integers n and d, that t(4^n*(8d+7))= 0 mod (2*16^n).

Date received: April 30, 2004