Number Theory Seminar
Speaker: Emanuele Tron (Bordeaux).
Title: GCD in algebraic groups and the Ailon-Rudnick conjecture.
Abstract: The greatest common divisor of sequences of the form $a^n-1$ is an apparently simple subject, but quite sophisticated to study. While large values are well understood thanks to the Subspace theorem, small values are very mysterious, and the most basic problem, the Ailon-Rudnick conjecture, is unsolved. We will report on progress on some geometric versions of the conjecture, and we shall see how it all fits in a more general setting concerning orbits of points in algebraic groups. This is connected, through work of Silverman, to such topics as Vojta's conjecture and unlikely intersections.