Bounded Height in Pencils of Finitely Generated Subgroups

We prove height bounds concerning intersections of finitely generated subgroups in a torus with algebraic subvarieties, all varying in a pencil. This vastly extends the previously treated constant case and involves entirely different, and more delicate, techniques

Mixing and linear equations over groups in positive characteristic

We prove a result on linear equations over multiplicative groups in positive characteristic. This is applied to settle a conjecture about higher order mixing properties of algebraicZ d -action

UNLIKELY INTERSECTIONS FOR CURVES IN MULTIPLICATIVE GROUPS OVER POSITIVE CHARACTERISTIC

The conjectures associated with the names of Zilber-Pink greatly generalize results associated with the names of Manin-Mumford and Mordell-Lang, but unlike the latter they are at present restricted to zero characteristic. We make a start on removing this restriction by stating a conjecture for curves in multiplicative groups over positive characteristic, and we verify the conjecture in three dimensions as well as for some special lines in general dimension. We also give an example where the finite set in question can be explicitly determine

UNLIKELY INTERSECTIONS FOR CURVES IN MULTIPLICATIVE GROUPS OVER POSITIVE CHARACTERISTIC

The conjectures associated with the names of Zilber-Pink greatly generalize results associated with the names of Manin-Mumford and Mordell-Lang, but unlike the latter they are at present restricted to zero characteristic. We make a start on removing this restriction by stating a conjecture for curves in multiplicative groups over positive characteristic, and we verify the conjecture in three dimensions as well as for some special lines in general dimension. We also give an example where the finite set in question can be explicitly determine

How to Solve a Quadratic Equation In Rationals

The title alludes to a similar title of the paper [3] by Grunewald and Segal, in which it is shown how to solve a quadratic equation in integers. This latter procedure seems to be quite difficult, and the algorithm outlined in [3] is rather involved, although it is completely effective in the logical sense. 1991 Mathematics Subject Classification 11D0

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 elliptic curve defined by Y 2=X(X − 1)(X − λ). Here we give a generalization to any two points with coordinates algebraic over the field Q(λ) and even over C(λ). This implies a special case of a variant of Pink's Conjecture for a variety inside a semiabelian scheme: namely for any curve inside any scheme isogenous to a fibred product of two isogenous elliptic scheme

Linear equations over multiplicative groups, recurrences, and mixing I

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135647/1/plms1045.pd

Lang's Conjecture and Sharp Height Estimates for the elliptic curves y2=x3+axy^{2}=x^{3}+ax

For elliptic curves given by the equation $E_{a}: y^{2}=x^{3}+ax$, we establish the best-possible version of Lang's conjecture on the lower bound of the canonical height of non-torsion points along with best-possible upper and lower bounds for the difference between the canonical and logarithmic height.Comment: published version. Lemmas 5.1 and 6.1 now precise (with resultant refinement to Theorem 1.2). Small corrections to

Galois Properties of Division Fields of Elliptic Curves

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135234/1/blms0247.pd