UCalgary Algebra and Number Theory Seminar: Joseph Silverman

Let X be an algebraic variety defined over a number field K such that the set of K -rational points X ( K ) is Zariski dense in X . For example, we could take X = P N to be projective space. Let f : X → X be a surjective endomorphism of X defined over K . For points P ∈ X ( K ) , we consider the forward orbit O ( f , P ) = { P , f ( P ) , f ( f ( P ) ) , . . . } . Recently Hector Pasten and I conjectured that there are "lots and lots" of "widely spaced" orbits. In this talk I will give two interpretations to the phrases "lots and lots" and "widely spaced", leading to a weak conjecture and a strong conjecture. As time permits, I will sketch proofs of the conjectures in various settings, including the case of endomorphisms of P N . This talk is in collaboration with the Nosh team and is co-organized by Antoine Leudière.