Effective André–Oort for non-compact curves in Hilbert modular varieties

Gal Binyamini; David Masser

doi:10.5802/crmath.177

Number theory

Effective André–Oort for non-compact curves in Hilbert modular varieties

1. Weizmann Institute of Science, Rehovot, Israel
2. Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, 4051 Basel, Switzerland

Comptes Rendus. Mathématique, Volume 359 (2021) no. 3, pp. 313-321.

Abstract
Résumé

In the proofs of most cases of the André–Oort conjecture, there are two different steps whose effectivity is unclear: the use of generalizations of Brauer–Siegel and the use of Pila–Wilkie. Only the case of curves in $ℂ^{2}$ is currently known effectively (by other methods).

We give an effective proof of André–Oort for non-compact curves in every Hilbert modular surface and every Hilbert modular variety of odd genus (under a minor generic simplicity condition). In particular we show that in these cases the first step may be replaced by the endomorphism estimates of Wüstholz and the second author together with the specialization method of André via G-functions, and the second step may be effectivized using the Q-functions of Novikov, Yakovenko and the first author.

Dans les démonstrations de la plupart des cas de la conjecture de André–Oort, il y a deux étapes différentes dont l’effectivité n’est pas claire : l’utilisation de généralisations de Brauer–Siegel et l’utilisation de Pila–Wilkie. Seulement le cas des courbes dans $ℂ^{2}$ est couramment effectivement connu (par des autres méthodes).

Nous donnons une démonstration effective de la conjecture pour les courbes non-compactes dans chaque surface modulaire de Hilbert et chaque variété modulaire de Hilbert de genre impair (sous condition secondaire de simplicité générique). En particulier nous montrons que dans ces cas, la première étape peut $\hat{e}$ tre remplacée par les majorations d’endomorphismes de Wüstholz et le deuxième auteur combinées avec la méthode de spécialisation de André par les G-fonctions, et la deuxième étape peut $\hat{e}$ tre effectivisée en utilisant les Q-fonctions de Novikov, Yakovenko et le premier auteur.

Received: 2020-09-22
Accepted: 2021-01-08
Published online: 2021-04-20

DOI: https://doi.org/10.5802/crmath.177
Classification: 11G10, 11G15, 11G18, 11G50
Author's affiliations:

Gal Binyamini ¹; David Masser ²

1. Weizmann Institute of Science, Rehovot, Israel
2. Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, 4051 Basel, Switzerland

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Gal Binyamini; David Masser. Effective André–Oort for non-compact curves in Hilbert modular varieties. Comptes Rendus. Mathématique, Volume 359 (2021) no. 3, pp. 313-321. doi : 10.5802/crmath.177. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.177/

References
Cited by

[1] Yves André G-functions and geometry, Aspects of Mathematics, 13, Vieweg & Sohn, 1989 | Zbl 0688.10032

[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 | Article | Zbl 0918.14010

[3] Yuri Bilu; Lars Kühne Linear equations in singular moduli, Int. Math. Res. Not., Volume 2020 (2020) no. 21, pp. 7617-7643 | Article | MR 4176833 | Zbl 07319825

[4] Yuri Bilu; David Masser; Umberto Zannier An effective ‘Theorem of André’ for $C M$ -points on a plane curve, Math. Proc. Camb. Philos. Soc., Volume 154 (2013) no. 1, pp. 145-152 | Article | Zbl 1263.14028

[5] Gal Binyamini Density of algebraic points on Noetherian varieties, Geom. Funct. Anal., Volume 29 (2019) no. 1, pp. 72-118 | Article | MR 3925105 | Zbl 07040583

[6] Gal Binyamini Some effective estimates for André–Oort in $Y {(1)}^{n}$ , J. Reine Angew. Math., Volume 767 (2020), pp. 17-35 (with appendix by E. Kowalski) | Article | MR 4160301 | Zbl 07268721

[7] Gal Binyamini; Dmitry Novikov; Sergei Yakovenko On the number of zeros of Abelian integrals. A constructive solution of the infinitesimal Hilbert sixteenth problem, Invent. Math., Volume 181 (2010) no. 2, pp. 227-289 | Article | Zbl 1207.34039

[8] Gal Binyamini; Dmitry Novikov; Sergei Yakovenko Quasialgebraic functions, Algebraic methods in dynamical systems (Banach Center Publications), Volume 94, Polish Academy of Sciences, 2011, pp. 61-81 | MR 2882613 | Zbl 1241.34036

[9] Enrico Bombieri; Jonathan Pila The number of integral points on arcs and ovals, Duke Math. J., Volume 59 (1989) no. 2, pp. 337-357 | MR 1016893 | Zbl 0718.11048

[10] Christopher Daw; Martin Orr Quantitative reduction theory and unlikely intersections (2019) (https://arxiv.org/abs/1911.05618)

[11] Christopher Daw; Martin Orr Unlikely intersections with $E \times C M$ curves in $𝒜_{2}$ (2019) (https://arxiv.org/abs/1902.10483)

[12] Albrecht Fröhlich; John C. Shepherdson Effective procedures in field theory, Philos. Trans. Roy. Soc. London, Volume 248 (1956), pp. 407-432 | MR 74349 | Zbl 0070.03502

[13] Gareth Jones; Margaret E. M. Thomas Effective Pila–Wilkie bounds for unrestricted Pfaffian surfaces (2018) (https://arxiv.org/abs/1804.08232, to appear in Math. Ann.)

[14] Lars Kühne An effective result of André–Oort type, Ann. Math., Volume 176 (2012) no. 1, pp. 651-671 | Article | Zbl 1341.11035

[15] Lars Kühne Logarithms of algebraic numbers, J. Théor. Nombres Bordeaux, Volume 27 (2015) no. 2, pp. 499-535 | Article | Numdam | MR 3393165 | Zbl 1387.11055

[16] David Masser Specializations of some hyperelliptic Jacobians, Number theory in progress, Walter de Gruyter, 1999, pp. 293-307 | Zbl 0942.14015

[17] David Masser; Gisbert Wüstholz Endomorphism estimates for abelian varieties, Math. Z., Volume 215 (1994) no. 4, pp. 641-653 | Article | MR 1269495 | Zbl 0826.14025

[18] Jean-François Mestre Families of hyperelliptic curves with real multiplication, Arithmetic algebraic geometry (Progress in Mathematics), Volume 89, Birkhäuser, 1991, pp. 193-208 | Article

[19] Ya’acov Peterzil; Sergei Starchenko Definability of restricted theta functions and families of abelian varieties, Duke Math. J., Volume 162 (2013) no. 4, pp. 731-765 | MR 3039679 | Zbl 1284.03215

[20] Jonathan Pila On the algebraic points of a definable set, Sel. Math., New Ser., Volume 15 (2009) no. 1, pp. 151-170 | Article | MR 2511202 | Zbl 1218.11068

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

[22] Jonathan Pila; Jacob Tsimerman The André–Oort conjecture for the moduli space of abelian surfaces, Compos. Math., Volume 149 (2013) no. 2, pp. 204-216 | Article | Zbl 1304.11055

[23] Jonathan Pila; Alex Wilkie The rational points of a definable set, Duke Math. J., Volume 133 (2006) no. 3, pp. 591-616 | MR 2228464 | Zbl 1217.11066

[24] Jonathan Pila; Umberto Zannier Rational points in periodic analytic sets and the Manin–Mumford conjecture, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl., Volume 19 (2008) no. 2, pp. 149-162 | Article | MR 2411018 | Zbl 1164.11029

[25] Jacob Tsimerman Brauer–Siegel for arithmetic tori and lower bounds for Galois orbits of special points, J. Am. Math. Soc., Volume 25 (2012) no. 4, pp. 1091-1117 | Article | MR 2947946 | Zbl 1362.11057

[26] Jacob Tsimerman The André–Oort conjecture for $𝒜_{g}$ , Ann. Math., Volume 187 (2015) no. 2, pp. 379-390 | Article | MR 3744855 | Zbl 1415.11086

[27] John Wilson Explicit moduli for curves of genus 2 with real multiplication by $ℚ (\sqrt{5})$ , Acta Arith., Volume 93 (2000) no. 2, pp. 121-138 | Article | MR 1757185 | Zbl 0966.11027

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