Equidistribution de sous-variétés spéciales et o-minimalité: André-Oort géométrique

Shimura 簇上的典范高度与 André-Oort 猜想

Canonical Heights on Shimura Varieties and the André-Oort Conjecture

本文旨在完全一般性地证明 André-Oort 猜想，该猜想涉及 Shimura 簇上特殊子簇的分布。

Towards the André-Oort conjecture for mixed Shimura varieties: the Ax-Lindemann theorem and lower bounds for Galois orbits of special points

We prove in this paper the Ax-Lindemann-Weierstrass theorem for all mixed Shimura varieties and discuss the lower bounds for Galois orbits of special points of mixed Shimura varieties. In particular we reprove a result of Silverberg in a different approach. Then combining these results we prove the André-Oort conjecture for any mixed Shimura variety whose pure part is a subvariety of A_6^n.

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

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 ${\bf C}^2$ 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 …

A proof of the Andre-Oort conjecture for A_g

We give a proof of the André-Oort conjecture for $\mathcal{A}_g$ - the moduli space of principally polarized abelian varieties. In particular, we show that a recently proven `averaged' version of the Colmez conjecture yields lower bounds for Galois orbits of CM points. The André-Oort conjecture then follows from previous work of Pila and the author.

Local André-Oort conjecture for the universal abelian variety

We prove a $p$-adic analogue of the André-Oort conjecture for subvarieties of the universal abelian varieties containing a dense set of special points. Let $g$ and $n$ be integers with $n \geq 3$ and $p$ a prime number not dividing $n$. Let $R$ be a finite extension of $W[{\mathbb F}_p^{\mathrm alg}]$, the ring of Witt vectors of the algebraic closure of the field of $p$ elements. The moduli space $\cA = \cA_{g,1,n}$ of $g$-dimensional principally polarized abelian varieties with full level $n$-structure as well as the universal abelian variety $π:\cX \to \cA$ over $\cA$ may be defined over $R…

混合 Shimura 变种中的混合猜想

Hybrid Conjecture in a Mixed Shimura variety

本文研究混合 Shimura 变种中的混合猜想，结合等分布与 o-最小性方法，证明了混合猜想包含多个经典猜想，如混合 André-Oort 猜想、Manin-Mumford 猜想及 Mordell-Lang 猜想。