ICM 1998, Berlin, Aug. 18–27

Abstracts of Plenary and Invited Lectures

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 11 J 13, 11 J 68, 11 J 61, 11 D 61, 11 D 72 Schlickewei, Hans Peter, Philipps-Universität Marburg, Germany The Absolute Subspace Theorem and Applications

1. The Subspace Theorem The Subspace Theorem in Diophantine Approximation by W. M. Schmidt (1972) reads as follows: Let L_{1}, …, L_{n} be linearly independent linear forms, L_{i}(X) = λ_{i1}X_{1} + ⋯ + λ_{in}X_{n} with algebraic coefficients λ_{ij}. Suppose ε > 0. Consider the inequality

|L_{1}(x) … L_{n}(x)| < |x|^{−ε}  (1)

in x ∈ ℤ^{n}, where |x| = (x_{1}^{2} + ⋯ + x_{n}^{2})^{1/2}. Then there exist finitely many proper linear subspaces T_{1}, …, T_{t} of ℚ^{n} such that the set of solutions of (1) is contained in the union

T_{1} ∪ … ∪ T_{t}.  (2)

Later on the Subspace Theorem has been extended to include a finite set S of p-adic valuations and also to deal with the situation where the variables are taken from the ring of integers of a number field K. It is in fact this generalization that is most interesting for applications. In the last decade quite some activity was going on to give uniform upper bounds for the number t of subspaces T_{1}, … T_{t} needed in (2), to cover the set of solutions of (1). We will discuss in detail the most recent result in that context due to Evertse and Schlickewei, the Quantitative Absolute Subspace Theorem (to appear). Here the variables are allowed to vary inside \overline{ℚ}^{n} (the field of all algebraic numbers) instead of ℚ^{n} or K^{n}. Roughly speaking, if we disregard the “very small“ solutions of (1), we obtain in this general situation an upper bound for the number of subspaces in (2) that depends only upon the dimension n and upon ε.

2. Applications We will give several applications to Diophantine equations. Let us mention here one of them: Let {u_{n}}_{n∈ℤ} be a linear recurrence sequence of order k with elements in a field of characteristic 0, i. e., a sequence satisfying a recurrence relation

u_{n+k} = a_{k−1}u_{n+k−1} + ⋯ + a_{0}u_{n} (n ∈ ℤ).

A classical conjecture has been as follows: If the sequence {u_{n}} is “nondegenerate” then the number of solutions n ∈ ℤ of the equation

u_{n} = 0

is bounded in terms of the order k of the sequence only. This conjecture now is a Theorem. It has been settled in the case when {u_{n}} has only simple characteristic roots by Evertse, Schlickewei and W. M. Schmidt (to appear) and shortly afterwards by W. M. Schmidt (to appear) in general.