Mixed Shimura Varieties Research Articles

Degeneracy loci in the universal family of Abelian varieties

Recent developments on the uniformity of the number of rational points on curves and subvarieties in a moving abelian variety rely on the geometric concept of the degeneracy locus. The first-named author investigated the degeneracy locus in certain mixed Shimura varieties. In this expository note we revisit some of these results while minimizing the use of mixed Shimura varieties while working in a family of principally polarized abelian varieties. We also explain their relevance for applications in diophantine geometry.

An overview on problems of Unlikely Intersections in families of abelian varieties

This short survey is part of a minicourse I gave during the CMI-HIMR Summer School “Unlikely Intersections in Diophantine Geometry” on the Zilber–Pink conjecture, formulated independently by Zilber (2002), Bombieri, Masser and Zannier (1999) in the case of tori and by Pink (2005) in the more general setting of mixed Shimura varieties. This conjecture, which includes in its general formulation many important results in number theory, has been intensively studied by several mathematicians in the past 20 years. We will mainly focus on these problems in the special setting of semiabelian varieties and families of abelian varieties.

Generic rank of Betti map and unlikely intersections

Let $\mathcal {A} \rightarrow S$ be an abelian scheme over an irreducible variety over $\mathbb {C}$ of relative dimension $g$. For any simply-connected subset $\Delta$ of $S^{\mathrm {an}}$ one can define the Betti map from $\mathcal {A}_{\Delta }$ to $\mathbb {T}^{2g}$, the real torus of dimension $2g$, by identifying each closed fiber of $\mathcal {A}_{\Delta } \rightarrow \Delta$ with $\mathbb {T}^{2g}$ via the Betti homology. Computing the generic rank of the Betti map restricted to a subvariety $X$ of $\mathcal {A}$ is useful to study Diophantine problems, e.g. proving the geometric Bogomolov conjecture over char $0$ and studying the relative Manin–Mumford conjecture. In this paper we give a geometric criterion to detect this rank. As an application we show that it is maximal after taking a large fibered power (if $X$ satisfies some conditions); it is an important step to prove the bound for the number of rational points on curves (Dimitrov et al., Uniformity in Mordell–Lang for Curves, Preprint (2020), arXiv:2001.10276). Another application is to answer a question of André, Corvaja and Zannier and improve a result of Voisin. We also systematically study its link with the relative Manin–Mumford conjecture, reducing the latter to a simpler conjecture. Our tools are functional transcendence and unlikely intersections for mixed Shimura varieties.

Pink’s conjecture on unlikely intersections and families of semi-abelian varieties

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 group subschemes of the former. This provides a counter-example to the relative Manin-Mumford conjecture, but also some evidence in favour of Pink's conjecture on unlikely intersections in mixed Shimura varieties. The main part of the article concerns mixed Hodge structures and the uniformization of the Poincar\'e torsor, but other, more geometric, approaches are also discussed.

Functoriality of motivic lifts of the canonical construction

Let (G,{mathfrak {X}}) be a Shimura datum and K a neat open compact subgroup of G(mathbb {A}_f). Under mild hypothesis on (G,{mathfrak {X}}), the canonical construction associates a variation of Hodge structure on text {Sh}_K(G,{mathfrak {X}})(mathbb {C}) to a representation of G. It is conjectured that this should be of motivic origin. Specifically, there should be a lift of the canonical construction which takes values in relative Chow motives over text {Sh}_K(G,{mathfrak {X}}) and is functorial in (G,{mathfrak {X}}). Using the formalism of mixed Shimura varieties, we show that such a motivic lift exists on the full subcategory of representations of Hodge type {(-1,0),(0,-1)}. If (G,{mathfrak {X}}) is equipped with a choice of PEL-datum, Ancona has defined a motivic lift for all representations of G. We show that this is independent of the choice of PEL-datum and give criteria for it to be compatible with base change. Additionally, we provide a classification of Shimura data of PEL-type and demonstrate that the canonical construction is applicable in this context.

ENLARGED MIXED SHIMURA VARIETIES, BI-ALGEBRAIC SYSTEM AND SOME AX TYPE TRANSCENDENTAL RESULTS

We develop a theory of enlarged mixed Shimura varieties, putting the universal vectorial bi-extension defined by Coleman into this framework to study some functional transcendental results of Ax type. We study their bi-algebraic systems, formulate the Ax-Schanuel conjecture and explain its relation with the logarithmic Ax theorem and the Ax-Lindemann theorem which we shall prove. All these bi-algebraic and transcendental results extend their counterparts for mixed Shimura varieties. In the end we briefly discuss the André–Oort and Zilber–Pink type problems for enlarged mixed Shimura varieties.

About the mixed André–Oort conjecture: Reduction to a lower bound for the pure case

We prove that the mixed André–Oort conjecture holds for any mixed Shimura variety if a lower bound for the size of Galois orbits of special points in pure Shimura varieties exists. This generalizes the current results for mixed Shimura varieties of Abelian type.

Bounded equidistribution of special subvarieties in mixed Shimura varieties

In this paper, we study the equidistribution of certain families of special subvarieties in general mixed Shimura varieties. We introduce the notion of bounded sequences of special subvarieties, and we prove that the Andre–Oort conjecture holds for such sequences. The proof follows the equidistribution approach used by Clozel, Ullmo, and Yafaev in the pure case. We then propose the notion of test invariant of a special subvariety, which is adapted from the lower bound formula of degrees of special subvarieties in the pure case studied by Ullmo and Yafaev, and we show that sequences of special subvarieties with bounded test invariants are bounded, hence the Andre–Oort conjecture holds in this case.

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

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 6 n {\mathcal{A}_{6}^{n}} and under the Generalized Riemann Hypothesis for all mixed Shimura varieties of abelian type.

Unlikely Intersections in Poincaré Biextensions over Elliptic Schemes

This paper concerns the relations between the relative Manin–Mumford conjecture and Pink’s conjecture on unlikely intersections in mixed Shimura varieties. The variety under study is the 4-dimensional Poincare biextension attached to a universal elliptic curve. A detailed list of its special subvarieties is drawn up, providing partial verifications of Pink’s conjecture in this case, and two open problems are stated in order to complete its proof.

On special subvarieties of Kuga varieties

In this paper we prove the equidistribution of certain families of special subvarieties in Kuga varieties, which is a special case of the general Andre–Oort conjecture formulated for mixed Shimura varieties. Our approach is parallel to the pure case treated in the works of L. Clozel, E. Ullmo, and A. Yafaev, which uses tools from ergodic theory.

Intersecting subvarieties of abelian varieties with algebraic subgroups of complementary dimension

In fact both authors stated stronger conjectures. Zilber conjectured a finiteness statement for subvarieties that are contained in an improper intersection of X with an algebraic subgroup of S. Pink was motivated by a more general conjecture in the context of mixed Shimura varieties. He also showed that Conjecture 1 implies the celebrated Mordell–Lang conjecture. An important precursor was set in 1999 by Bombieri, Masser, and Zannier [3] who proved the following result for an algebraic curve C embedded in the algebraic torus Gm , not contained in the translate of a proper algebraic subgroup, and defined over Q, an algebraic closure of Q: the set of algebraic points of C which are contained in the union of all algebraic subgroups of codimension at least 1 has bounded absolute Weil height. They then used their result to show that there are only finitely many points on C contained in the union of all algebraic subgroups of codimension at least 2.

Local André-Oort conjecture for the universal abelian variety

We prove a $p$-adic analogue of the Andr\'{e}-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 $\pi:\cX \to \cA$ over $\cA$ may be defined over $R$. We call a point $\xi \in \cX(R)$ \emph{$R$-special} if $\cX_{\pi(\xi)}$ is a canonical lift and $\xi$ is a torsion point of its fibre. We show that an irreducible subvariety of $\cX_R$ containing a dense set of $p$-special points must be a special subvariety in the sense of mixed Shimura varieties. Our proof employs the model theory of difference fields.