David Masser

Intersecting a curve 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 the same title, we studied the intersection of the curve and the union of all algebraic subgroups of some fixed codimension. With codimension one the resulting set has bounded height properties, and with codimension two it has finiteness properties. The main aim of the present work is to make a start on such problems in higher dimension by proving the natural analogues for a linear surface (with codimensions two and three). These are in accordance with some general conjectures that we have recently proposed else- where.

Elliptic functions and transcendence

A transcendence measure.- Vanishing of linear forms without complex multiplication.- Vanishing of linear forms with complex multiplication.- An effective proof of a theorem of Coates.- A lower bound for non-vanishing linear forms.- Lemmas on elliptic functions with complex multiplication.- Linear forms in algebraic points.

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 d and real number \( \mathcal{H} \geqslant 1 \), it is well known that the number Open image in new window of points α in Open image in new window having degree d over ℚ and satisfying \( H\left( \alpha \right) \leqslant \mathcal{H} \) is finite. This is the one-dimensional case of Northcott’s Theorem [8] (see also [5, page 59]). The systematic study of the counting function Open image in new window , and that of related functions in higher dimensions, was begun by Schmidt [10]. It is relatively easy to prove the existence of a positive constant C = C(d) such that Open image in new window (1) and also the existence of positive constants c = c(d) and \( \mathcal{H}_0 = \mathcal{H}_0 \left( d \right) \) such that Open image in new window (2)

Estimating isogenies on elliptic curves

The object of this paper is to prove a new type of estimate for isogenies between elliptic curves. This has several diophantine applications (effective versions of Serres Galois irreducibility theorem and Shafarevichs theorem, for example) which are presented in another paper [MW3]. Later articles will deal with the corresponding problems for abelian varieties of arbitrary dimension. Right at the beginning we emphasize that we are identifying elliptic curves E with Weierstrass equations

Multiplicity estimates on group varieties

tions the field K will be the field Q of algebraic numbers. We remark at this point that the results remain true if we take instead of the field of complex numbers C its p-adic analogue C for some fixed prime p and for K a corresponding subfield Kp of CP. We restrict ourselves to the complex case in order to avoid some minor complications appearing in the p-adic domain. These complications always arise when analytic functions come in. Our functions are defined globally in the case of complex numbers but only locally in the case when we deal with the p-adic domain. These difficulties can be avoided by a purely algebraic approach. If we took this approach the only condition on the ground field would be that it should be algebraically closed and of characteristic zero. But we prefer to avoid such an approach in order to keep the text understandable, also for those who are mainly interested in the applications in transcendence.

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 algebraicZd-actions.

An effective "theorem of André" for CM-points on a plane curve

It is a well known result of Y. Andre (a basic special case of the Andre-Oort conjecture) that an irreducible algebraic plane curve containing infinitely many points whose coordinates are CM -invariants is either a horizontal or vertical line, or a modular curve Y 0 ( n ). Andres proof was partially ineffective, due to the use of (Siegels) class-number estimates. Here we observe that his arguments may be modified to yield an effective proof. For example, with the diagonal line X 1 + X 2 =1 or the hyperbola X 1 X 2 =1 it may be shown quite quickly that there are no imaginary quadratic τ 1 ,τ 2 with j (τ 1 )+ j (τ 2 )=1 or j (τ 1 ) j (τ 2 )=1, where j is the classical modular function.

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 that u1(k1)+⋯+um(km)=0u1(k1)+⋯+um(km)=0 is pp-normal in a natural sense generalizing that of the first author, who proved the result for m=1m=1. Furthermore the set is effectively computable if KK is. We illustrate this with an example for m=4m=4. We also show that the corresponding set for zero characteristic is not decidable for m=557844m=557844, thus verifying a conjecture of Cerlienco, Mignotte, and Piras.

Relative Manin-Mumford for semi-abelian surfaces

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 special cases of the Zilber-Pink conjecture for curves in a mixed Shimura variety of dimension four, as well as the study of polynomial Pell equations with non-separable discriminants.

Intersecting curves and algebraic subgroups: Conjectures and more results

This paper solves in the affirmative, up to dimension n = 5, a question raised in an earlier paper by the authors. The equivalence of the problem with a conjecture of Shou-Wu Zhang is proved in the Appendix.