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Ax-Lindemann and André-Oort for a nonholomorphic modular function. (English. French summary) Zbl 1441.11164

Summary: The modular case of the André-Oort Conjecture is a theorem of André 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 (together with \(j\)) generate the relevant fields. After discussing some of the properties of these functions, I go on to prove some Ax-Lindemann results about \(\chi\) 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 André-Oort conjecture for the function \(\chi^*\).

MSC:

11G18 Arithmetic aspects of modular and Shimura varieties
03C64 Model theory of ordered structures; o-minimality
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