Bicyclotomic polynomials and impossible intersections

@article{Masser2013BicyclotomicPA,
  title={Bicyclotomic polynomials and impossible intersections},
  author={David Masser and Umberto Zannier},
  journal={Journal de Theorie des Nombres de Bordeaux},
  year={2013},
  volume={25},
  pages={635-659}
}
In a recent paper we proved that there are at most finitely many complex numbers t 6= 0, 1 such that the points (2, √ 2(2− t)) and (3, √ 6(3− t)) are both torsion on the Legendre elliptic curve defined by y2 = x(x − 1)(x − t). In a sequel we gave a generalization to any two points with coordinates algebraic over the field Q(t) and even over C(t). Here we reconsider the special case (u, √ u(u− 1)(u− t)) and (v, √ v(v − 1)(v − t)) with complex numbers u and v. 
Simultaneous Torsion in the Legendre Family
TLDR
An explicit description of the set of parameters λ such that the points with x-coordinate α and β are simultaneously torsion in the case that α andβ are algebraic numbers that are not 2-adically close is obtained.
Torsion points on isogenous abelian varieties
Investigating a conjecture of Zannier, we study irreducible subvarieties of abelian schemes that dominate the base and contain a Zariski dense set of torsion points that lie on pairwise isogenous
Relative Manin–Mumford in additive extensions
  • Harry Schmidt
  • Mathematics
    Transactions of the American Mathematical Society
  • 2018
In recent papers Masser and Zannier have proved various results of “relative Manin–Mumford” type for various families of abelian varieties, some with field of definition restricted to the algebraic
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,
Finiteness theorems on elliptical billiards and a variant of the Dynamical Mordell-Lang Conjecture
We offer some theorems, mainly of finiteness, for certain patterns in elliptical billiards, related to periodic trajectories. For instance, if two players hit a ball at a given position and with

References

SHOWING 1-10 OF 13 REFERENCES
Torsion points on families of squares of elliptic curves
In a recent paper we proved that there are at most finitely many complex numbers λ ≠  0,1 such that the points $${(2,\sqrt{2(2-\lambda)})}$$ and $${(3, \sqrt{6(3-\lambda)})}$$ are both torsion on the
Torsion anomalous points and families of elliptic curves
Cyclotomic points on curves
We show that a plane algebraic curve f = 0over the complex numbers has on it either at most 22V (f) points whose coordinates are both roots of unity, or innitely many such points. Here V (f) is the
Some Problems of Unlikely Intersections in Arithmetic and Geometry
This book considers the so-called Unlikely Intersections, a topic that embraces well-known issues, such as Lang's and Manin-Mumford's, concerning torsion points in subvarieties of tori or abelian
Specializations of finitely generated subgroups of abelian varieties
Given a generic Mordell-Weil group over a function field, we can specialize it down to a number field. It has been known for some time that the resulting homomorphism of groups is injective
Integer points on the dilation of a subanalytic surface
Let � ⊂ R n be a compact subanalytic set of dimension 2 and t 1. This paper gives an upper bound as t →∞ for the number of integer points on the homothetic dilation tofthat do not reside on any
Foundations of Algebraic Geometry
ALGEBRAIC geometry, in spite of its beauty and importance, has long been reproached for lacking proper foundations. Great discoveries have been made, especially in Italy, by the intuition of a number
ELLIPTIC FUNCTIONS
The first systematic account of the theory of elliptic functions and the state of the art around the turn of the century. Preceding general class field theory and therefore incomplete. Contains a
Counting rational points on a certain exponential-algebraic surface
We study the distribution of rational points on a certain exponential-algebraic surface and we prove, for this surface, a conjecture of A. J. Wilkie. 2000 Mathematics Subject Classification: 11G99,
...
...