Andr\'e-Oort for a Nonholomorphic Modular Function

Abstract

The modular case of the Andr\'e-Oort Conjecture is a theorem of Andre and Pila, having at its heart the well-known modular function j. I give an overview of two other `nonclassical' classes of modular function, namely the quasimodular (QM) and almost holomorphic modular (AHM) functions. These are perhaps less well-known than j, but have been studied by various authors including for example Masser, Shimura and Zagier. It turns out to be sufficient to focus on a particular QM function $\chi$ and its dual AHM function $\chi^*$, since these generate the relevant fields. After discussing some of the properties of these functions, I go on to prove some Ax-Lindemann results for $\chi$ and $\chi^*$. I then combine these with a fairly standard method of o-minimality and point counting to prove the central result of the paper; a natural analogue of the modular Andr\'e-Oort conjecture for the function $\chi^*$.