Vol. 12, No. 9, 2018

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Dynamics on abelian varieties in positive characteristic

Jakub Byszewski and Gunther Cornelissen

Appendix: Robert Royals and Thomas Ward

Vol. 12 (2018), No. 9, 2185–2235
Abstract

We study periodic points and orbit length distribution for endomorphisms of abelian varieties in characteristic p > 0. We study rationality, algebraicity and the natural boundary property for the dynamical zeta function (the latter using a general result on power series proven by Royals and Ward in the appendix), as well as analogues of the prime number theorem, also for tame dynamics, ignoring orbits whose order is divisible by p. The behavior is governed by whether or not the action on the local p-torsion group scheme is nilpotent.

Keywords
abelian variety, inseparability, fixed points, Artin–Mazur zeta function, recurrence sequence, natural boundary
Mathematical Subject Classification 2010
Primary: 37P55
Secondary: 11N45, 14G17, 14K02, 37C25, 37C30
Milestones
Received: 30 March 2018
Revised: 29 June 2018
Accepted: 29 July 2018
Published: 21 December 2018
Authors
Jakub Byszewski
Wydział Matematyki i Informatyki
Uniwersytet Jagielloński
Kraków
Poland
Gunther Cornelissen
Mathematisch Instituut
University of Utrecht
The Netherlands
Robert Royals
School of Mathematics
University of East Anglia
Norwich
United Kingdom
Thomas Ward
Ziff Building
University of Leeds
United Kingdom