# Special points and Poincar\'e bi-extensions, with an Appendix by Bas Edixhoven

@article{Bertrand2011SpecialPA, title={Special points and Poincar\'e bi-extensions, with an Appendix by Bas Edixhoven}, author={D. Bertrand}, journal={arXiv: Number Theory}, year={2011} }

i) in a joint project with D. Masser, A. Pillay and U. Zannier [7], we aim at extending to semi-abelian schemes the Masser-Zannier approach [11] to Conjecture 6.2 of R. Pink’s preprint [13]; this conjecture also goes under the name “Relative Manin-Mumford ”. Inspired by Anand Pillay’s suggestion that the semi-constant extensions of [6] may bring trouble, I found a counter-example, which is described in Section 1 below.

## 17 Citations

### Relative Manin-Mumford for abelian varieties

- Mathematics
- 2014

With an eye or two towards applications to Pell's equation and to Davenport's work on integration of algebraic functions, Umberto Zannier and I have recently characterised torsion points on a fixed…

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

- MathematicsJournal de l’École polytechnique — Mathématiques
- 2020

The Poincare 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…

### Torsion points on families of simple abelian surfaces and Pell's equation over polynomial rings (with an appendix by E. V. Flynn)

- Mathematics
- 2015

In recent papers we proved a special case of a variant of Pink's Conjecture for a variety inside a semiabelian scheme, namely for any curve inside anything isogenous to a product of two elliptic…

### Towards the Andr\'e-Oort conjecture for mixed Shimura varieties: the Ax-Lindemann theorem and lower bounds for Galois orbits of special points

- Mathematics
- 2013

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…

### Unlikely Intersections in Poincaré Biextensions over Elliptic Schemes

- MathematicsNotre Dame J. Formal Log.
- 2013

the 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…

### Relative Manin–Mumford for Semi-Abelian Surfaces

- MathematicsProceedings of the Edinburgh Mathematical Society
- 2016

Abstract We show that Ribet sections are the only obstruction to the validity of the relative Manin–Mumford conjecture for one-dimensional families of semi-abelian surfaces. Applications include…

### A Note on 1-Motives

- MathematicsInternational Mathematics Research Notices
- 2019

We prove that for $1$-motives defined over an algebraically closed subfield of $\mathbf{C}$, viewed as Nori motives, the motivic Galois group coincides with the Mumford–Tate group. In particular,…

### Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell–Weil rank

- MathematicsJournal of the European Mathematical Society
- 2018

We show that there is a bound depending only on g and [K:Q] for the number of K-rational points on a hyperelliptic curve C of genus g over a number field K such that the Mordell-Weil rank r of its…

### Unlikely intersections with isogeny orbits

- Mathematics
- 2019

This thesis consists of six chapters and two appendices. The first two chapters contain the introduction and some preliminaries.
Chapter 3 contains a characterization of curves in abelian schemes,…

### Linear relations in families of powers of elliptic curves

- Mathematics
- 2016

Motivated by recent work of Masser and Zannier on simultaneous torsion on the Legendre elliptic curve $E_\lambda$ of equation $Y^2=X(X-1)(X-\lambda)$, we prove that, given $n$ linearly independent…

## References

SHOWING 1-10 OF 13 REFERENCES

### A Lindemann-Weierstrass theorem for semi-abelian varieties over function fields

- Mathematics
- 2008

We prove an analogue of the Lindemann-Weierstrass theorem (that the exponentials of a Q-linearly independent set of algebraic numbers are algebraically independent), replacing Q alg by C(t) alg , and…

### A Common Generalization of the Conjectures of André-Oort

- Mathematics
- 2005

Consider any irreducible closed subvariety Z ⊂ S. Since any intersection of special subvarieties is a finite union of special subvarieties, there exists a unique smallest special subvariety…

### Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part

- Mathematics
- 1992

© Foundation Compositio Mathematica, 1992, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions…

### Conducteur, descente et pincement

- Mathematics
- 2003

Une somme amalgamee de schemas est decrite localement par un produit fibre d'anneaux. Ce texte donne un resultat global d'existence ( 5.4) de schemas definis comme certaines sommes amalgamees et un…

### Canonical Models of (Mixed) Shimura Varieties and Automorphic Vector Bundles

- Mathematics
- 2002

The article surveys what was known, or conjectured, about canonical models of Shimura varieties and related objects at the time it was written (1988). I Abelian varieties with complex multiplication…

### Key words : semi-abelian varieties, Manin-Mumford conjecture, one-motives, generalized jacobians, special points, mixed Shimura varieties

- Groupes algébriques et corps de classes; Hermann,
- 1959