Dynamics on abelian varieties in positive characteristic

Byszewski, Jakub; Cornelissen, Gunther

doi:10.2140/ant.2018.12.2185

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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

DOI: 10.2140/ant.2018.12.2185

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

Vol. 12, No. 9, 2018