The modular case of the André–Oort Conjecture is a theorem of André and Pila, having at its heart the well-known modular function . 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 , 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 and its dual AHM function , since these (together with ) generate the relevant fields. After discussing some of the properties of these functions, I go on to prove some Ax–Lindemann results about and . 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 .
Le cas modulaire de la Conjecture d’André–Oort est un théorème démontré par André et Pila, qui concerne la fonction modulaire bien connue . 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 , mais divers auteurs, y compris Masser, Shimura et Zagier, les ont étudiées. Il suffit de se concentrer sur une fonction QM précise et sa fonction AHM duale , car celles-ci (avec ) 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 et . 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 .
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Accepted:
Published online:
DOI: 10.5802/jtnb.1048
Keywords: Definable set, rational point, André–Oort conjecture, nonholomorphic modular function, Pila–Wilkie Theorem
Haden Spence 1
@article{JTNB_2018__30_3_743_0, author = {Haden Spence}, title = {Ax{\textendash}Lindemann and {Andr\'e{\textendash}Oort} for a {Nonholomorphic} {Modular} {Function}}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {743--779}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {30}, number = {3}, year = {2018}, doi = {10.5802/jtnb.1048}, zbl = {1441.11164}, mrnumber = {3938625}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1048/} }
TY - JOUR AU - Haden Spence TI - Ax–Lindemann and André–Oort for a Nonholomorphic Modular Function JO - Journal de théorie des nombres de Bordeaux PY - 2018 SP - 743 EP - 779 VL - 30 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1048/ DO - 10.5802/jtnb.1048 LA - en ID - JTNB_2018__30_3_743_0 ER -
%0 Journal Article %A Haden Spence %T Ax–Lindemann and André–Oort for a Nonholomorphic Modular Function %J Journal de théorie des nombres de Bordeaux %D 2018 %P 743-779 %V 30 %N 3 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1048/ %R 10.5802/jtnb.1048 %G en %F JTNB_2018__30_3_743_0
Haden Spence. Ax–Lindemann and André–Oort for a Nonholomorphic Modular Function. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 743-779. doi : 10.5802/jtnb.1048. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1048/
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