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

Binyamini, Gal; Masser, David

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

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 $\widehat{\mathrm{e}}$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 $\widehat{\mathrm{e}}$tre effectivisée en utilisant les Q-fonctions de Novikov, Yakovenko et le premier auteur.

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.

Classification : 11G10, 11G15, 11G18, 11G50

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