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

Abstract

Abstract. 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 [57] in a different approach. Then combining these results we prove the André-Oort conjecture unconditionally for any mixed Shimura variety whose pure part is a subvariety of A n 6 and under GRH for all mixed Shimura varieties of abelian type.

Enlarged Mixed Shimura Varieties, Bi-Algebraic System and Some Ax Type Transcendental Results

“…We make some comparison of the proofs of Theorem 1.2 and Theorem 1.3 with the author's previous work on mixed Shimura varieties [16]. In both situations it is important to study the geometric bi-algebraic systems associated with the ambient varieties (we will elaborate on this in the next subsection).…”

“…Theorem 1.2 is proven for pure Shimura varieties by Moonen [27, 3.6, 3.7] and [2] It means that ( Z ♮ ) Zar is the complex analytic irreducible component of X ♮+ ∩(Zariski closure of Z ♮ in X ♮,∨ ) which contains Z ♮ . for mixed Shimura varieties by the author [16,Theorem 8.1]. Theorem 1.3 plays an essential role in the proof of the André-Oort conjecture.…”

“…Theorem 1.3 plays an essential role in the proof of the André-Oort conjecture. It is proven for Y (1) n by Pila [32], for projective pure Shimura varieties by Ullmo-Yafaev [46], for A g by Pila-Tsimerman [34], for any pure Shimura variety by Klingler-Ullmo-Yafaev [22], and for any mixed Shimura variety by the author [16,Theorem 1.2].…”

“…We should study the geometry of S ♮ more carefully and it is better to separate the proofs into two parts: first prove that geometrically bi-algebraic subvarieties of S ♮ are precisely quasi-linear subvarieties, then prove that the target objects in the conclusions are geometrically bi-algebraic. The first part is new compared to [16], but the second part is only a slight modification of [16].…”

The Ax–Schanuel conjecture for variations of Hodge structures

“…The Ax-Lindemann theorem was finally established in full generality for Shimura varieties in [KUY16] by Klingler, Ullmo, and Yafaev, and for mixed Shimura varieties by Gao [Gao17]. Motivated by an analogous (though much more difficult to carry out) approach to the more general Zilber-Pink conjectures, Mok, Pila, and the second author recently proved the full Ax-Schanuel conjecture for general Shimura varieties [MPT17].…”