## Algebra & Number Theory

### Dynamics on abelian varieties in positive characteristic

#### 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.

#### Article information

Source
Algebra Number Theory, Volume 12, Number 9 (2018), 2185-2235.

Dates
Revised: 29 June 2018
Accepted: 29 July 2018
First available in Project Euclid: 5 January 2019

https://projecteuclid.org/euclid.ant/1546657279

Digital Object Identifier
doi:10.2140/ant.2018.12.2185

Mathematical Reviews number (MathSciNet)
MR3894433

Zentralblatt MATH identifier
06999507

#### Citation

Byszewski, Jakub; Cornelissen, Gunther. Dynamics on abelian varieties in positive characteristic. Algebra Number Theory 12 (2018), no. 9, 2185--2235. doi:10.2140/ant.2018.12.2185. https://projecteuclid.org/euclid.ant/1546657279

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