Multiplicative independence of modular functions

Guy Fowler

doi:10.5802/jtnb.1167

Guy Fowler ¹

¹ Mathematical Institute University of Oxford Oxford, OX2 6GG, United Kingdom

Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 2, pp. 459-509.

Résumé
Abstract

Nous offrons une nouvelle preuve élémentaire de l’indépendance multiplicative de ${GL}_{2}^{+} (ℚ)$ -translatées, deux à deux distinctes, de la fonction modulaire $j$ , un résultat dû initialement à Pila et Tsimerman. Nous sommes ainsi en mesure de généraliser ce résultat à une classe de fonctions modulaires plus large. Nous montrons que cette classe contient un ensemble composé de fonctions modulaires qui apparaissent naturellement comme des relèvements de Borcherds de certaines formes modulaires faiblement holomorphes. Pour une fonction modulaire $f$ appartenant à cette classe, nous déduisons que pour chaque $n \geq 1,$ le nombre de $n$ -uplets de points $f$ -spéciaux distincts qui sont multiplicativement dépendants et minimaux pour cette propriété est fini. Cela généralise un théorème de Pila et Tsimerman sur les invariants modulaires singuliers. Nous montrons ensuite comment ces résultats sont liés à la conjecture de Zilber–Pink pour les sous-variétés algébriques de la variété de Shimura mixte $Y {(1)}^{n} \times 𝔾_{m}^{n}$ , et nous prouvons quelques cas particuliers de cette conjecture.

We provide a new, elementary proof of the multiplicative independence of pairwise distinct ${GL}_{2}^{+} (ℚ)$ -translates of the modular $j$ -function, a result due originally to Pila and Tsimerman. We are thereby able to generalise this result to a wider class of modular functions. We show that this class includes a set comprising modular functions which arise naturally as Borcherds lifts of certain weakly holomorphic modular forms. For $f$ a modular function belonging to this class, we deduce, for each $n \geq 1$ , the finiteness of $n$ -tuples of distinct $f$ -special points that are multiplicatively dependent and minimal for this property. This generalises a theorem of Pila and Tsimerman on singular moduli. We then show how these results relate to the Zilber–Pink conjecture for subvarieties of the mixed Shimura variety $Y {(1)}^{n} \times 𝔾_{m}^{n}$ and prove some special cases of this conjecture.

Reçu le : 2020-05-27
Révisé le : 2021-03-16
Accepté le : 2021-04-19
Publié le : 2021-09-10

DOI : 10.5802/jtnb.1167

Classification : 11F03, 11G15
Mots clés : Modular functions, multiplicative independence, Zilber–Pink conjecture

Affiliations des auteurs :

Guy Fowler ¹

¹ Mathematical Institute University of Oxford Oxford, OX2 6GG, United Kingdom

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JTNB_2021__33_2_459_0,
     author = {Guy Fowler},
     title = {Multiplicative independence of modular functions},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {459--509},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {33},
     number = {2},
     year = {2021},
     doi = {10.5802/jtnb.1167},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1167/}
}

TY  - JOUR
AU  - Guy Fowler
TI  - Multiplicative independence of modular functions
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2021
SP  - 459
EP  - 509
VL  - 33
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1167/
DO  - 10.5802/jtnb.1167
LA  - en
ID  - JTNB_2021__33_2_459_0
ER  -

%0 Journal Article
%A Guy Fowler
%T Multiplicative independence of modular functions
%J Journal de théorie des nombres de Bordeaux
%D 2021
%P 459-509
%V 33
%N 2
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1167/
%R 10.5802/jtnb.1167
%G en
%F JTNB_2021__33_2_459_0

Guy Fowler. Multiplicative independence of modular functions. Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 2, pp. 459-509. doi : 10.5802/jtnb.1167. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1167/

Bibliographie
Cité par

[1] Janusz Adamus; Serge Randriambololona Tameness of holomorphic closure dimension in a semialgebraic set, Math. Ann., Volume 355 (2013) no. 3, pp. 985-1005 | DOI | MR | Zbl

[2] Yves André Finitude des couples d’invariants modulaires singuliers sur une courbe algébrique plane non modulaire, J. Reine Angew. Math., Volume 505 (1998), pp. 203-208 | DOI | Zbl

[3] James Ax On Schanuel’s conjectures, Ann. Math., Volume 93 (1971), pp. 252-268 | MR | Zbl

[4] Yuri Bilu; Florian Luca; David Masser Collinear CM-points, Algebra Number Theory, Volume 11 (2017) no. 5, pp. 1047-1087 | DOI | MR | Zbl

[5] Yuri Bilu; Florian Luca; Amalia Pizarro-Madariaga Rational products of singular moduli, J. Number Theory, Volume 158 (2016), pp. 397-410 | DOI | MR | Zbl

[6] Enrico Bombieri; Walter Gubler Heights in Diophantine geometry, New Mathematical Monographs, 4, Cambridge University Press, 2006 | MR | Zbl

[7] Enrico Bombieri; David Masser; Umberto Zannier Anomalous subvarieties – structure theorems and applications, Int. Math. Res. Not., Volume 2007 (2007) no. 19, rnm057, 33 pages | MR | Zbl

[8] Richard Borcherds Automorphic forms on $O_{s + 2, 2} {(ℝ)}^{+}$ and generalized Kac–Moody algebras, Proceedings of the international congress of mathematicians, Vol. II (Zürich, 1994) (1994), pp. 744-752

[9] Richard Borcherds Automorphic forms on $O_{s + 2, 2} (ℝ)$ and infinite products, Invent. Math., Volume 120 (1995) no. 1, pp. 161-213 | DOI | MR | Zbl

[10] David Brink On alternating and symmetric groups as Galois groups, Isr. J. Math., Volume 142 (2004), pp. 47-60 | DOI | MR | Zbl

[11] François Charles; Bjorn Poonen Bertini irreducibility theorems over finite fields, J. Am. Math. Soc., Volume 29 (2016) no. 1, pp. 81-94 corrigendum in ibid. 32 (2019), no. 2, p. 606-607 | DOI | MR | Zbl

[12] David Cox Primes of the form $x^{2} + n y^{2}$ , John Wiley & Sons, 1989 | Zbl

[13] Lou van den Dries Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, 248, London Mathematical Society, 1998 | MR | Zbl

[14] Lou van den Dries; C. Miller On the real exponential field with restricted analytic functions, Isr. J. Math., Volume 85 (1994) no. 1, pp. 1-3 corrigendum in ibid. 92 (1995), no. 1-3, p. 427 | MR | Zbl

[15] Benedict Gross; Don Zagier On singular moduli, J. Reine Angew. Math., Volume 355 (1985), pp. 191-220 | MR | Zbl

[16] Philipp Habegger Singular moduli that are algebraic units, Algebra Number Theory, Volume 9 (2015) no. 7, pp. 1515-1524 | DOI | MR | Zbl

[17] Philipp Habegger; Jonathan Pila Some unlikely intersections beyond André–Oort, Compos. Math., Volume 148 (2012) no. 1, pp. 1-27 | DOI | Zbl

[18] Philipp Habegger; Jonathan Pila O-minimality and certain atypical intersections, Ann. Sci. Éc. Norm. Supér., Volume 49 (2016) no. 4, pp. 813-858 | DOI | MR | Zbl

[19] Marc Hindry; Joseph H. Silverman Diophantine geometry. An introduction, Graduate Texts in Mathematics, 201, Springer, 2000 | Zbl

[20] Thomas Loher; David Masser Uniformly counting points of bounded height, Acta Arith., Volume 111 (2004) no. 3, pp. 277-297 | DOI | MR | Zbl

[21] Ken Ono The web of modularity: arithmetic of the coefficients of modular forms and $q$ -series, CBMS Regional Conference Series in Mathematics, 102, American Mathematical Society, 2004 | MR | Zbl

[22] Jonathan Pila On the algebraic points of a definable set, Sel. Math., New Ser., Volume 15 (2009) no. 1, pp. 151-170 | DOI | MR | Zbl

[23] Jonathan Pila Special point problems with elliptic modular surfaces, Mathematika, Volume 60 (2014) no. 1, pp. 1-31 | DOI | MR | Zbl

[24] Jonathan Pila; Jacob Tsimerman Independence of CM points in elliptic curves to appear in J. Eur. Math. Soc. (JEMS)

[25] Jonathan Pila; Jacob Tsimerman Ax–Schanuel for the $j$ -function, Duke Math. J., Volume 165 (2016) no. 13, pp. 2587-2605 | MR | Zbl

[26] Jonathan Pila; Jacob Tsimerman Multiplicative relations among singular moduli, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 17 (2017) no. 4, pp. 1357-1382 | MR | Zbl

[27] Jonathan Pila; Alex J. Wilkie The rational points of a definable set, Duke Math. J., Volume 133 (2006) no. 3, pp. 591-616 | MR | Zbl

[28] Richard Pink A combination of the conjectures of Mordell–Lang and André–Oort, Geometric methods in algebra and number theory (Progress in Mathematics), Volume 235, Birkhäuser, 2005, pp. 251-282 | DOI | Zbl

[29] Haden Spence A note on the degree of field extensions involving classical and nonholomorphic singular moduli (2017) (https://arxiv.org/abs/1702.01950)

[30] Boris Zilber Exponential sums equations and the Schanuel conjecture, J. Lond. Math. Soc., Volume 65 (2002) no. 1, pp. 27-44 | DOI | MR | Zbl

Cité par Sources :