For any algebraically closed field K and any endomorphism f of $\mathbb{P}^1(K)$ of degree at least 2, the automorphisms of f are the Möbius transformations that commute with f, and these form a finite subgroup of $\operatorname{PGL}_2(K)$ . 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 $\bar{\mathbb{F}}_p$ of a finite field. We use the classification of finite subgroups of $\operatorname{PGL}_2(\bar{\mathbb{F}}_p)$ 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 $\bar{\mathbb{F}}_p$ of degree 2 with nontrivial automorphisms, showing how the geometry and possible automorphism groups depend on the prime p.
For any algebraically closed field K and any endomorphism f of P 1 (K) of degree at least 2, the automorphisms of f are the Möbius transformations which commute with f , and these form a finite subgroup of PGL 2 (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 which date back to Klein. We study the corresponding questions when K is the algebraic closure Fp of a finite field. We calculate the locus of maps over Fp of degree 2 with nontrivial automorphisms, showing how the geometry and possible automorphism groups depend on the prime p. Then, without restricting the degree to 2, we use the classification of finite subgroups of PGL 2 ( Fp ) to show that every subgroup is realizable as an automorphism group.To construct examples, we use methods from modular invariant theory.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.