• Corpus ID: 117040368

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}
}
  • D. Bertrand
  • Published 27 April 2011
  • Mathematics
  • arXiv: Number Theory
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. 

Relative Manin-Mumford for abelian varieties

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

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)

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

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

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

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

  • Y. Andr'e
  • Mathematics
    International 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

  • M. Stoll
  • Mathematics
    Journal 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

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

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

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

Cohomological realization of a family of 1-motives

A Common Generalization of the Conjectures of André-Oort

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

© 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

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

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