Ax–Lindemann and André–Oort for a Nonholomorphic Modular Function
Journal de Théorie des Nombres de Bordeaux, Tome 30 (2018) no. 3, pp. 743-779.

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 χ et sa fonction AHM duale χ * , 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 χ et χ * . 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 χ * .

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 χ and its dual AHM function χ * , 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 χ and χ * . 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 χ * .

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.1048
Classification : 11G18,  03C64
Mots clés : Definable set, rational point, André–Oort conjecture, nonholomorphic modular function, Pila–Wilkie Theorem
@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},
     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
DA  - 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/
UR  - https://doi.org/10.5802/jtnb.1048
DO  - 10.5802/jtnb.1048
LA  - en
ID  - JTNB_2018__30_3_743_0
ER  - 
Haden Spence. Ax–Lindemann and André–Oort for a Nonholomorphic Modular Function. Journal de Théorie des Nombres de Bordeaux, Tome 30 (2018) no. 3, pp. 743-779. doi : 10.5802/jtnb.1048. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1048/

[1] Fred Diamond; Jerry Shurman A first course in modular forms, Graduate Texts in Mathematics, Volume 228, Springer, 2005, xv+436 pages | Zbl 1062.11022

[2] Guy Diaz Transcendance et indépendance algébrique: liens entre les points de vue elliptique et modulaire, Ramanujan J., Volume 4 (2000) no. 2, pp. 157-199 | Zbl 1115.11312

[3] Lou van den Dries Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, Volume 248, Cambridge University Press, 1998, x+180 pages | Zbl 0953.03045

[4] David S. Dummit; Richard M. Foote Abstract algebra, John Wiley & Sons, 2004 | Zbl 1037.00003

[5] Bas Edixhoven On the André–Oort conjecture for Hilbert modular surfaces, Moduli of abelian varieties (Texel Island, 1999) (Progress in Mathematics) Volume 195, Birkhäuser, 2001, pp. 133-155 | Zbl 1029.14007

[6] Bas Edixhoven; Andrei Yafaev Subvarieties of Shimura varieties, Ann. Math., Volume 157 (2003) no. 2, pp. 621-645 | Zbl 1053.14023

[7] Bruno Klingler; Andrei Yafaev The André–Oort conjecture, Ann. Math., Volume 180 (2014) no. 3, pp. 867-925 | Zbl 1377.11073

[8] Serge Lang Elliptic functions, Graduate Texts in Mathematics, Volume 112, Springer, 1987, xi+326 pages | Zbl 0615.14018

[9] David Masser Elliptic functions and transcendence, Lecture Notes in Mathematics, Volume 437, Springer, 1975, xiv+143 pages | Zbl 0312.10023

[10] Ya’acov Peterzil; Sergei Starchenko Uniform definability of the Weierstrass functions and generalized tori of dimension one, Sel. Math., New Ser., Volume 10 (2004) no. 4, pp. 525-550 | Zbl 1071.03022

[11] Jonathan Pila Rational points of definable sets and results of André–Oort–Manin–Mumford type, Int. Math. Res. Not., Volume 2009 (2009) no. 13, pp. 2476-2507 | Zbl 1243.14021

[12] Jonathan Pila O-minimality and the André–Oort conjecture for n , Ann. Math., Volume 173 (2011) no. 3, pp. 1779-1840 | Zbl 1243.14022

[13] Jonathan Pila Modular Ax–Lindemann–Weierstrass with derivatives, Notre Dame J. Formal Logic, Volume 54 (2013) no. 3-4, pp. 553-565 | Zbl 1355.11078

[14] Jonathan Pila O-minimality and Diophantine Geometry, Proceedings of the ICM 2014, KM Kyung Moon Sa, 2014, pp. 547-572 | Zbl 1373.11050

[15] Jonathan Pila; Jacob Tsimerman Ax–Lindemann for 𝒜 g , Ann. Math., Volume 179 (2014) no. 2, pp. 659-681 | Zbl 1305.14020

[16] Theodor Schneider Einführung in die transzendenten Zahlen, Grundlehren der Mathematischen Wissenschaften, Volume 81, Springer, 1957 | Zbl 0077.04703

[17] Carl Ludwig Siegel Über die Klassenzahl quadratischer Zahlkörper, Acta Arith., Volume 1 (1935), pp. 83-86 | Zbl 0011.00903

[18] Jacob Tsimerman The André–Oort conjecture for 𝒜 g , Ann. Math., Volume 187 (2018) no. 2, pp. 379-390 | Zbl 06841543

[19] Emmanuel Ullmo; Andrei Yafaev Galois orbits of special subvarieties of Shimura varieties (2006) (preprint)

[20] Don Zagier Elliptic modular forms and their applications, The 1-2-3 of modular forms (Universitext), Springer, 2008, pp. 1-103 | Zbl 1259.11042

[21] Umberto Zannier Some problems of unlikely intersections in arithmetic and geometry, Annals of Mathematics Studies, Volume 181, European Mathematical Society, 2012, xi+160 pages | Zbl 1246.14003

Cité par Sources :