Counting periodic points over finite fields

2017/05/25 by Laura Walton, Walton, Laura
Computer Science · Mathematics · #11G25 #14G15 #37P25 #37P35 #37P55 #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1705.09034

openalex publication_date 2017/05/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let V be a quasiprojective variety defined over \mathbbF_q, and let ϕ:V→ V be an endomorphism of V that is also defined over \mathbbF_q. Let G be a finite subgroup of Aut_\mathbbF_q(V) with the property that ϕ commutes with every element of G. We show that idempotent relations in the group ring ℚ[G] give relations between the periodic point counts for the maps induced by ϕ on the quotients of V by the various subgroups of G. We also show that if G is abelian, periodic point counts for the endomorphism on V/G induced by ϕ are related to periodic point counts on V and all of its twists by G.

Related

• An alternate proof of idempotent relations among periodic points and\n quotients
• A remark on periodic points on varieties over a field of finite type over Q
• Uniform Bounds for Periods of Endomorphisms of Varieties
• Rational periodic points for quadratic maps
• Boundedness results for periodic points on algebraic varieties