We introduce a “hybrid” conjecture which is a common generalisation of the André-Oort conjecture and the André-Pink-Zannier conjecture and we prove that it is a consequence of the Zilber-Pink conjecture. We also show that our hybrid conjecture implies the Zilber-Pink conjecture for hypersurfaces contained in weakly special subvarieties.

In this paper, we introduce the notion of a bi-$\overline{\mathbb{Q}}$-structure on the tangent space at a CM point on a locally Hermitian symmetric domain. We prove that this bi-$\overline{\mathbb{Q}}$-structure decomposes into the direct sum of $1$-dimensional bi-$\overline{\mathbb{Q}}$-subspaces, and make this decomposition explicit for the moduli space of abelian varieties $\mathbb{A}_g$. We propose an Analytic Subspace Conjecture, which is the analogue of the Wüstholz's Analytic Subgroup Theorem in this context. We show that this conjecture, applied to $\mathbb{A}_g$, implies that all quadratic $\overline{\mathbb{Q}}$-relations among the holomorphic periods of CM abelian varieties arise from elementary ones.

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

In this paper, we introduce the notion of a bi-Q(-)-structure on the tangent space at a CM point on a locally Hermitian symmetric domain. We prove that this bi-Q(-)-structure decomposes into the direct sum of 1-dimensional bi-Q(-)-subspaces, and make this decomposition explicit for the moduli space of abelian varieties A(g). We propose an analytic subspace conjecture, which is the analogue of the W & uuml;stholz's analytic subgroup theorem in this context. We show that this conjecture, applied to A(g), implies that all quadratic Q(-)-relations among the holomorphic periods of CM abelian varieties arise from elementary ones.

The authors previously formulated the hybrid conjecture, unifying André-Pink-Zannier and André-Oort conjectures, and proved it in Shimura varieties of abelian type. We study its analogue for mixed Shimura varieties, and consider the prime example, the universal abelian scheme 𝒜_g→𝔸_g. In a radical departure from the Pila-Zannier strategy, typically applied to such questions, we employ instead a combination of equidistribution and o-minimality Our main result strictly includes the following: the Hybrid Conjecture, in particular the André-Pink-Zannier and André-Oort conjectures, for 𝔸_g; the mixed André-Oort conjecture for 𝒜_g; and Manin-Mumford conjecture for arbitrary abelian varieties. It also yields an analogue of the “Manin-Mumford in arithmetic pencil", a result of Baldi-Richard-Ullmo, for abelian schemes over a variety. The mixed hybrid conjecture in 𝒜_g also encompasses the Mordell-Lang conjecture. We actually reduce the mixed hybrid conjecture for 𝔸_g to its "mordellic" part. We also prove, Galois-theoretic results: uniform variants on the Ribet's Kummer theory of Abelian varieties, and Serre's theorem on Lang's conjecture.

In this paper, we prove the generalised Andr\'e-Pink-Zannier conjecture (an important case of the Zilber-Pink conjecture) for all Shimura varieties of abelian type. Questions of this type were first asked by Y. Andr\'e in 1989. We actually prove a general statement for all Shimura varieties, subject to certain assumptions that are satisfied for Shimura varieties of abelian type and are expected to hold in general. We also prove another result, a p-adic Kempf-Ness theorem, on the relation between good reduction of homogeneous spaces over p-adic integers with Mumford stability property in p-adic geometric invariant theory.

This note describes the results of [6]. The main result is the proof of the Generalised André–Pink–Zannier conjecture in Shimura varieties of abelian type. The core result is a lower bound, in terms of height functions defined in [7], for the sizes of Galois orbits of points in generalised Hecke orbits, which is unconditional for Shimura varieties of abelian type.

We introduce and study the notion of a generalised Hecke orbit in a Shimura variety. We define a height function on such an orbit and study its properties. We obtain a lower bounds for the size of Galois orbits of points in a generalised Hecke orbit in terms of these height, assuming a version of the Mumford-Tate conjecture. We then use it to prove the generalised Andr\'e-Pink-Zannier conjecture under this assumption by implementing the Pila-Zannier strategy.

In this paper we study the following nonlinear Hamiltonian elliptic system with gradient term Under a local super-quadratic condition on the nonlinearity, we obtain a new existence result of nontrivial solutions by using variational method. Here we do not need the global super-quadratic condition, and in this case our result allows the nonlinearity to be super-quadratic at some domains and asymptotically quadratic at other domains. In the proofs we apply some special techniques to demonstrate the link geometry of the energy functional.

We introduce and study the notion of a generalised Hecke orbit in a Shimura variety. We define a height function on such an orbit and study its properties. We obtain lower bounds for the sizes of Galois orbits of points in a generalised Hecke orbit in terms of this height function, assuming the “weakly adelic Mumford–Tate hypothesis” and prove the generalised André–Pink–Zannier conjecture (a special case of the Zilber-Pink conjecture) under this assumption using the Pila–Zannier strategy.

Let S be a Shimura variety. We conjecture that the heights of special points in S(ℚ) are discriminant negligible with respect to some Weil height function h:S(ℚ)→ℝ . Assuming this conjecture to be true, we prove that the sizes of the Galois orbits of special points grow as a fixed power of their discriminant (an invariant we will define in the text). In particular, we give a new proof of a theorem of Tsimerman on lower bounds for Galois degrees of special points in Shimura varieties of abelian type. This gives a new proof of the André–Oort conjecture for such varieties that avoids the use of Masser–Wüstholz isogeny estimates, replacing them by a point-counting argument.

In this note we derive a non-commutative version of the Wüstholz analytic subgroup theorem in transcendence theory from the original (commutative) theorem and provide an application.

We present some applications of recent results in homogeneous dynamics to an unlikely intersections problem in Shimura varieties (the Andre-Pink-Zannier conjecture) and its refinements. (C) 2019 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

We prove a hyperbolic analogue of the Bloch-Ochiai theorem about the Zariski closure of holomorphic curves in abelian varieties.

In this paper we prove two results on algebraic flows on Shimura varieties. One is the so-called ‘logarithmic’ Ax-Lindemann theorem. The other concerns the closure of the image of a totally geodesic sub-variety of a symmetric space by the uniformisation map.

This is a preliminary version of a monograph on homogeneous dynamics and application to some problems of unlikely intersections in Shimura varieties. It consists of four articles, which can be read independently. The first one, by the two first named authors, discuss the main application, to some refinement of the André-Pink-Zannier conjecture about closures of subsets of restricted Hecke orbits in Shimura varieties. The second article, by the first and last named author, establishes results about the dynamics of sequences translates of some measures in spaces of $S$-arithmetic lattices. These results are the cornerstone of the methods in the first article. The last two articles, by the first named author, establish results which are crucial for the second article. The third article gives an ultrametric analogue of archimedean results of Richard and Shah, and contains methods of independent interest about stability, Berkovich spaces and Bruhat-Tits buildings. The last article is set in the same context as the second article, and discusses non-divergence of the studied sequences of translates.

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 able to obtain stronger conclusions that those of the Andre-Pink-Zannier conjecture in the special case we consider. We work under the assumption of the S-Shafarevich conjecture and S-semisimplicity conjecture which hold for Shimura varieties of abelian type.

Let $A$ be an abelian variety over ${\bf C}$ of dimension $n$ and $\pi\colon {\bf C}^n \rightarrow A$ be the complex uniformisation. Let $X$ be an unbounded subset of ${\bf C}^n$ definable in a suitable o-minimal structure. We give a description of the Zariski closure of $\pi(X)$.