Automorphism Groups of Endomorphisms of ℙ¹ (\mathbbF_p)

2020/03/26 by Julia Cai, Benjamin Hutz, Cai, Julia +5
Mathematics · #(37P45) #37P05 #37P25 #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2003.12113

openalex publication_date 2020/03/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For any algebraically closed field K and any endomorphism f of ℙ¹(K) of degree at least 2, the automorphisms of f are the Möbius transformations that commute with f, and these form a finite subgroup of PGL₂(K). In the moduli space of complex dynamical systems, the locus of maps with nontrivial automorphisms has been studied in detail and there are techniques for constructing maps with prescribed automorphism groups that date back to Klein. We study the corresponding questions when K is the algebraic closure \mathbbF_p of a finite field. We use the classification of finite subgroups of PGL₂(\mathbbF_p) to show that every finite subgroup is realizable as an automorphism group. To construct examples, we use methods from modular invariant theory. Then, we calculate the locus of maps over \mathbbF_p of degree 2 with nontrivial automorphisms, showing how the geometry and possible automorphism groups depend on the prime p.

Citations

• Finite subgroups of PGL(2,K)
• The Moduli Space of Cubic Rational Maps
• Automorphism loci for degree 3 and degree 4 endomorphisms of the projective line

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• Automorphism loci for degree 3 and degree 4 endomorphisms of the projective line
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