Umberto Zannier

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.

On integral points on surfaces

Let us now consider two-dimensional problems, i.e. problems reducing to the distribution of integral points on surfaces.

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.

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.

A proof of Pisot's d th root conjecture

Let {b(n) : n E N} be the sequence of coefficients in the Taylor expansion of a rational function R(X) E Q(X) and suppose that b(n) is a perfect dth power for all large n. A conjecture of Pisot states that one can choose a dth root a(n) of b(n) such that E a(n)Xn is also a rational function. Actually, this is the fundamental case of an analogous statement formulated for fields more general than Q. A number of papers have been devoted to various special cases. In this note we shall completely settle the general case.

A note on Thue's equation over function fields

By a refinement of Masons method we improve upon the estimate of the height of solutions of Thues equation over function fields. We also give an application to the diophantine approximation of algebraic functions.

Bounded height in pencils of finitely generated subgroups

In this paper we prove a general bounded height result for specializations in finitely generated subgroups varying in families which complements and sharpens the toric Mordell-Lang Theorem by replacing finiteness by emptyness, for the intersection of varieties and subgroups, all moving in a pencil, except for bounded height values of the parameters (and excluding identical relations). More precisely, an instance of the results is as follows. Consider the torus scheme G r m/C over a curve C defined over Q, and let Γ be a subgroup-scheme generated by finitely many sections (satisfying some necessary conditions). Further, let V be any subscheme. Then there is a bound for the height of the points P ∈ C(Q) such that, for some γ ∈ Γ which does not generically lie in V , γ(P) lies in the fiber VP. We further offer some direct diophantine applications, to illustrate once again that the results implicitly contain information absent from the previous bounds in this context.

Fractional parts of Linear polynomials and an application to hypergeometric functions

Using a result on arithmetic progressions, we describe a method for finding the rational h –tuples ρ = (ρ l ,…,ρ h ) such that all the multiples m ρ (for m coprime to a denominator of ρ) lie in a linear variety modulo Z. We give an application to hypergeometric functions.