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

Daniel Bertrand,Bas Edixhoven

doi:10.5802/jep.126

Abstract

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.

Highlights

In the unpublished preprint [25] Pink formulated a very influential conjecture on so-called “unlikely intersections” in mixed Shimura varieties
We merely recall the statement of his Conjecture 1.3: if Y is a Hodge generic irreducible closed subvariety of a mixed Shimura variety S, the union of the intersections of Y with the special subvarieties of S of codimension at least dim(Y ) + 1 is not Zariski dense in Y
In the last section of [25], Pink states a relative version of the Manin-Mumford conjecture for families of semi-abelian varieties, Conjecture 6.1: if B → X is a family of semi-abelian varieties over C and Y is an irreducible closed subvariety in B that is not contained in any proper closed subgroup scheme of B → X, the union of the intersections of Y with algebraic subgroups of codimension at least dim(Y ) + 1 of the fibres of B → X is not Zariski dense in Y

Summary

Introduction

In the unpublished preprint [25] Pink formulated a very influential conjecture (the equivalent Conjectures 1.1–1.3) on so-called “unlikely intersections” in mixed Shimura varieties. The conclusion is that the context of mixed Hodge structures is the right one for a relative Manin-Mumford conjecture for families of semi-abelian varieties: the image of a Ribet section is a special subvariety that can in general not be interpreted as a subgroup scheme (see Remark 5.4.2 below). He showed that, with a small change, and a more detailed description of the special subvarieties of the mixed Shimura varieties involved, Pink’s argument gives that Conjecture 1.3 implies Conjecture 5.2 (unlikely intersection generalisation of Mordell-Lang), and by Theorem 5.5 of [25], implies Conjecture 5.1 The details of this will appear in an article in preparation by Edixhoven. We thank the referees of the paper for their comments and suggestions to improve our text

The example with elliptic curves

The example with abelian schemes

The Poincaré torsor as mixed Shimura variety

The universal Poincaré torsor as moduli space of mixed Hodge structures

Ribet varieties are special subvarieties