Ax–Lindemann and André–Oort for a Nonholomorphic Modular Function

Spence, Haden

Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 743-779.

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 ${\chi }^{*}$.

Le cas modulaire de la Conjecture d’André–Oort est un théorème démontré par André et Pila, qui concerne la fonction modulaire bien connue $j$. Je décris deux autres classes « non classiques » de la fonction modulaire, à savoir les fonctions quasimodulaires (QM) et presque holomorphes modulaires (AHM). Celles-ci sont peut-être moins connues que $j$, mais divers auteurs, y compris Masser, Shimura et Zagier, les ont étudiées. Il suffit de se concentrer sur une fonction QM précise $\chi$ et sa fonction AHM duale ${\chi }^{*}$, car celles-ci (avec $j$) engendrent les corps concernés. Après avoir discuté certaines des propriétés de ces fonctions, je montre par la suite quelques résultats de type Ax–Lindemann sur $\chi$ et ${\chi }^{*}$. Je les combine ensuite avec une méthode ordinaire de o-minimalité et de comptage de points pour démontrer le résultat central de l’article ; une analogique naturelle de la conjecture d’André–Oort modulaire qui s’applique à la fonction ${\chi }^{*}$.

Classification: 11G18, 03C64
Keywords: Definable set, rational point, André–Oort conjecture, nonholomorphic modular function, Pila–Wilkie Theorem

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