( CEU)

## Subgroups of Mordell--Weil groups and reduction mod p

Let a_1, ..., a_n and b be nonzero rational numbers. A theorem of Schinzel states that if b can be written as a product of powers of the a_i modulo all but finitely many prime numbers, then b is a product of powers of the a_i. In my presentation I will show that the analogous statement for rational points on an elliptic curve holds as well.

Let E be an elliptic curve over the field of rational numbers k, given, say, by an equation     E: y^2 = x^3 + Ax + B     with A, B in k.
The Mordell--Weil theorem states that the rational points of E, i.e. the rational solutions of the equation E form a finitely generated, commutative group- the Mordell-Weil group. Let X be a subgroup of this group. If a rational point- i.e. solution P of E belongs to X modulo all but finitely many primes, does then P belong to X?

This is indeed the case. In the presentation I will try to explain the analogy between this and Schinzel's theorem, and how one can attack this kind of problems in general. The talk is aimed at a general mathematical audience.