$2$-adic properties of Hecke traces of singular moduli

In \cite{Z}, Zagier initiated a study of the function $t_1(d)$, the function which gives the trace of a singular modulus of discriminant $-d<0$. Ahlgren and Ono \cite{A-O, Theorem 1 (1)} proved that if $p$ is an odd prime which splits in $\Q(\sqrt{-d})$, then $t_1(p^2d)\equiv 0\pmod{p}$. A question of Ono \cite{O, Problem 7.30} asks for generalizations modulo arbitrary prime powers. We provide the answer for $p=2$. In particular, we show, for all positive integers $n$ and $d$, that $t_1(4^n\cdot (8d+7))\equiv 0\pmod{2\cdot 16^n}$.

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Published 1 January 2005