Colloquium Mathematics/ Prof. dr. G. Cornelissen

Title:

Dynamics in positive characteristic

Abstract:

This will be an expository talk in advanced algebra. We study periodic points of endomorphisms of abelian varieties over fields of positive characteristic p. Periodic points are counted by the
dynamical zeta function introduced by Artin and Mazur.

In the particular case of an abelian variety over a finite field and the Frobenius endomorphism, the dynamical zeta function coincides with the usual zeta function of an abelian variety and is rational.

We prove that in general the dynamical zeta function is rational or transcendental, and the former holds exactly if the action of the endomorphism on the local p-torsion group scheme is nilpotent. We also study the questions of analytic continuation, asymptotic behaviour of the orbit length distribution, and the analogues of the Prime Number Theorem.

No preliminary specialized knowledge is expected from the audience.
(Joint work with Jakub Byszewski.)

Colloquium coordinators are Prof.dr. A.J. van der Schaft ( a.j.van.der.schaft rug.nl ),

Dr. A.V. Kiselev (e-mail: a.v.kiselev rug.nl )

http://www.rug.nl/research/jbi/news/colloquia/mathematics-colloquia/