Algebra & Number Theory

Dynamics on abelian varieties in positive characteristic

Jakub Byszewski and Gunther Cornelissen

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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

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

Received: 30 March 2018
Revised: 29 June 2018
Accepted: 29 July 2018
First available in Project Euclid: 5 January 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37P55: Arithmetic dynamics on general algebraic varieties
Secondary: 11N45: Asymptotic results on counting functions for algebraic and topological structures 14G17: Positive characteristic ground fields 14K02: Isogeny 37C25: Fixed points, periodic points, fixed-point index theory 37C30: Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dynamical systems

abelian variety inseparability fixed points Artin–Mazur zeta function recurrence sequence natural boundary


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.

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