相关论文: Generalised Andr\'e-Pink-Zannier Conjecture for Sh…

The Andr\'e-Pink-Zannier conjecture concerns the intersection of subvarieties and the generalized Hecke orbit of a given point in mixed Shimura varieties. It is part of the Zilber-Pink conjecture. In this paper we focus on the universal…

The authors previously formulated the hybrid conjecture, unifying Andr\'e-Pink-Zannier and Andr\'e-Oort conjectures, and proved it in Shimura varieties of abelian type. We study its analogue for mixed Shimura varieties, and consider the…

We introduce a ``hybrid'' conjecture which is a common generalisation of the Andr\'e-Oort conjecture and the Andr\'e-Pink-Zannier conjecture and we prove that it is a consequence of the Zilber-Pink conjecture. We also show that our hybrid…

We prove the 'hybrid conjecture' which is a common generalisation of the Andre\'e-Oort conjecture and the Andr\'e-Pink-Zannier conjecture, in the case of Shimura varieties of abelian type.

The universal Poincar\'e torsor, or more generally the universal semi-abelian scheme, can be viewed as a mixed Shimura variety. We give a classification of special subvarieties of the universal semi-abelian scheme of arbitrary toric rank.…

We obtain results on the so-called Andre-Pink-Zannier conjecture which is a special case of a the Zilber-Pink conjecture on unlikely intersections in Shimura varieties. Our methods rely on an ergodic theorem of Richard-Zamojski and we are…

In this note, we prove the Zilber--Pink conjecture for subvarieties of mixed Shimura varieties, which are not defined over~$\overline{\mathbb Q}$ in a strong sense. We prove similar results for general variations of mixed Hodge structure of…

We investigate the analogue of the Andr\'e--Pink--Zannier conjecture in characteristic $p$. Precisely, we prove it for ordinary function field-valued points with big monodromy, in Shimura varieties of Hodge type. We also prove an algebraic…

In this article, we prove that the Zilber-Pink conjecture for abelian varieties over an arbitrary field of characteristic $0$ is implied by the same statement for abelian varieties over the algebraic numbers. More precisely, the conjecture…

In 2014, Pila and Tsimerman gave a proof of the Ax-Schanuel conjecture for the $j$-function and, with Mok, have recently announced a proof of its generalization to any (pure) Shimura variety. We refer to this generalization as the…

We discuss the relationships between the Andr\'e-Oort, Andr\'e-Pink-Zannier, and Mordell-Lang conjectures for Shimura varieties. We then combine the latter with the geometric Zilber-Pink conjecture to obtain some new results on unlikely…

We prove a general likely intersections theorem, a counterpart to the Zilber-Pink conjectures, under the assumption that the Ax-Schanuel property and some mild additional conditions are known to hold for a given category of complex quotient…

Following our work in \cite{papas2022height}, we extend the height bounds established by Y. Andr\'e in his seminal research monograph \cite{andre1989g} for $1$-parameter families of abelian varieties defined over number fields. In our…

We prove that the modular Zilber--Pink conjecture (in Pink's formulation in terms of unlikely intersections) holds for all subvarieties $V$ of $ \mathrm{Y}(1)^n$ for which no projection to any $\dim V + 2$ coordinates is defined over the…

We prove some cases of the Zilber-Pink conjecture on unlikely intersections in Shimura varieties. Firstly, we prove that the Zilber-Pink conjecture holds for intersections between a curve and the union of the Hecke translates of a fixed…

The Poincar\'e torsor of a Shimura family of abelian varieties can be viewed both as a family of semi-abelian varieties and as a mixed Shimura variety. We show that the special subvarieties of the latter cannot all be described in terms of…

The purpose of this article is to explain the Pila-Zannier strategy for proving the Andr\'e-Oort conjecture. First, however, we will provide a brief introduction to the theory of Shimura varieties.

We prove the Mumford-Tate conjecture for those abelian varieties over number fields, whose simple factors of their adjoint Mumford-Tate groups have over $\dbR$ certain (products of) non-compact factors. In particular, we prove this…

In this paper we prove the equidistribution of $\Cbf$-special subvarieties in certain Kuga varieties, which implies a special case of the general Andr\'e-Oort conjecture formulated for mixed Shimura varieties proposed by R.Pink. The main…

In this paper we prove a special case of the Andr\'e-Oort conjecture for Kuga varieties. If $M$ is a Kuga variety fibred over a pure Shimura variety $S$ as an abelian scheme, and $(M_n)$ is a sequence of special subvarieties in $M$ which…