Abstract
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).
Highlights
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
In this note we discuss the known results on effective André-Oort, all of which are restricted to the context of modular curves, and announce a new effective result in the context of Hilbert modular varieties
In the André–Oort conjecture one considers a suitable ambient space X (in this note we will mention only X = Cn (n ≥ 2) or X = Ag (g ≥ 2) the moduli space of principally polarized abelian varieties of dimension g ) equipped with irreducible algebraic subvarieties that are known as special subvarieties
Summary
In this note we discuss the known results on effective André-Oort, all of which are restricted to the context of modular curves, and announce a new effective result in the context of Hilbert modular varieties. We provide a sketch of the key ideas, and will provide full details in a paper under preparation
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