Effective Andr\'e-Oort Type Results for Almost Holomorphic Modular Funcions

Abstract

In this short paper we discuss a number of effective and/or explicit results of Andre-Oort type for the nonholomorphic function "$\chi^*$", which I have discussed in a number of other papers. After working in a rather ad-hoc manner to get some good estimates on the tails of the $q$-expansions involved, we prove weak effective Andr\'e-Oort results for $\chi^*$, which mimic but are not full analogues of effective Andr\'e-Oort results known due to K\"uhne/Bilu-Masser-Zannier for the classical modular function $j$. Then we go on to discuss what we call an "explicit" result; that certain triples of special points cannot often be collinear, looking for an analogue of results known for $j$. Again we cannot get a perfect analogy, but we do prove a weaker result and discuss what remains to be proved to complete this. An important result which arises as a side-effect of the explicit calculation done here is "Corollary 2.4", which affirms a conjecture I made in earlier papers, that for a quadratic point $\tau$ we have $\mathbb{Q}(j(\tau)) = \mathbb{Q}(\chi^*(\tau))$. Although it appears here somewhat tangentially, it may be the most significant result in the paper.