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Intersecting a plane with algebraic subgroups of multiplicative groups
Consider an arbitrary algebraic curve defined over the field of all alge- braic numbers and sitting in a multiplicative commutative algebraic group. In an earlier article from 1999 bearing almost theExpand
Anomalous Subvarieties—Structure Theorems and Applications
When a fixed algebraic variety in a multiplicative group variety is intersected with the union of all algebraic subgroups of fixed dimension, a key role is played by what we call the anomalousExpand
Torsion anomalous points and families of elliptic curves
We prove that there are at most finitely many complex $\lambda \neq 0,1$ such that two points on the Legendre elliptic curve $Y^2 = X(X-1)(X-\lambda)$ with coordinates $X = 2,3$ both have finiteExpand
Counting algebraic numbers with large height II
Let ℚ denote the field of rational numbers, Open image in new window an algebraic closure of ℚ, and H : Open image in new window the absolute, multiplicative, Weil height. For each positive integer dExpand
Uniformly counting points of bounded height
1. Introduction. In this paper we give some new uniform estimates for the cardinalities of certain sets involving algebraic numbers of bounded height. The estimates are nearly optimal with respect toExpand
Rational values of the Riemann zeta function
Abstract We prove the existence of a constant C such that for any D ⩾ 3 there are at most C ( log D log log D ) 2 rational numbers s with 2 s 3 and denominator at most D such that ζ ( s ) is alsoExpand
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 theExpand
Linear equations over multiplicative groups, recurrences, and mixing I
Let u1,…,umu1,…,um be linear recurrences with values in a field KK of positive characteristic pp. We show that the set of integer vectors (k1,…,km)(k1,…,km) such thatExpand
Counting points of small height on elliptic curves
— Let k be a number field and let E be an elliptic curve defined over k. We prove a counting result which gives, among other things, the existence of a positive constant C, effectively computable inExpand