Multiplicative independence of modular functions
Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 2, pp. 459-509.

We provide a new, elementary proof of the multiplicative independence of pairwise distinct GL 2 + ()-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 n1, 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 ×𝔾 m n and prove some special cases of this conjecture.

Nous offrons une nouvelle preuve élémentaire de l’indépendance multiplicative de GL 2 + ()-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 n1, 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 ×𝔾 m n , et nous prouvons quelques cas particuliers de cette conjecture.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1167
Classification: 11F03, 11G15
Keywords: Modular functions, multiplicative independence, Zilber–Pink conjecture
Fowler, Guy 1

1 Mathematical Institute University of Oxford Oxford, OX2 6GG, United Kingdom
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Fowler, Guy. Multiplicative independence of modular functions. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 2, pp. 459-509. doi : 10.5802/jtnb.1167. http://archive.numdam.org/articles/10.5802/jtnb.1167/

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

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

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

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

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

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

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

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

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

[11] Charles, François; Poonen, Bjorn 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] Cox, David Primes of the form x 2 +ny 2 , John Wiley & Sons, 1989 | Zbl

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

[14] van den Dries, Lou; Miller, C. 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] Gross, Benedict; Zagier, Don On singular moduli, J. Reine Angew. Math., Volume 355 (1985), pp. 191-220 | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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