On the p-adic geometry of traces of singular moduli

The aim of this article is to show that p-adic geometry of modular curves is useful in the study of p-adic properties of traces of singular moduli. In order to do so, we partly answer a question by Ono ([5, Problem 7.30]). As our goal is just to illustrate how p-adic geometry can be used in this context, we focus on a relatively simple case, in the hope that others will try to obtain the strongest and most general results. For example, for p = 2, a result stronger than Thm. 2 is proved in [2], and a result on some modular curves of genus zero can be found in [6] . It should be easy to apply our method, because of its local nature, to modular curves of arbitrary level, as well as to Shimura curves.