2-adic properties of Hecke traces of singular moduli. (English) Zbl 1088.11031
Let \(d \in {\mathbb N}\) be such that \(-d\) is a discriminant and denote by \({\mathcal Q}_d\) the set of equivalence classes of integral positive definite binary quadratic forms \(Q(x,y) = ax^2+bxy+cy^2\) of discriminant \(-d = b^2-4ac\). Writing \(\Gamma = \text{PSL}_2({\mathbb Z})\) the \(m\)-th Hecke trace of the singular modulus of discriminant \(-d\) is defined as
\[
t_m(d) = \sum_{Q\in {\mathcal Q}_{d/\Gamma}} {{(j(z)-744)| T(m) (\alpha_Q)}\over{\omega_Q}},
\]
where \(\alpha_Q\in {\mathbb H}\) is the CM point of \({\mathbb H}\) associated to \({\mathbb Q}\) and \(q\omega_Q\) the number of integral automorphs of \(Q\). Congruence properties of the \(t_m(d)\) have recently received some attention. In this paper the author proves
\[
t_m(4^nd) \equiv 0 \bmod 2 \cdot 16^n
\]
for odd \(m\) and \(d \equiv 7 \bmod 8\).
Reviewer: Rainer Schulze-Pillot (Saarbrücken)
MSC:
11F33 | Congruences for modular and \(p\)-adic modular forms |
11F03 | Modular and automorphic functions |
11F25 | Hecke-Petersson operators, differential operators (one variable) |