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- Title
Automorphism groups of endomorphisms of $\mathbb{P}^1 (\bar{\mathbb{F}}_p)$.
- Authors
Cai, Julia; Hutz, Benjamin; Mayer, Leo; Weinreich, Max
- Abstract
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.
- Subjects
AUTOMORPHISM groups; ENDOMORPHISMS; ENDOMORPHISM rings; FINITE fields; AUTOMORPHISMS; DYNAMICAL systems; GEOMETRY
- Publication
Glasgow Mathematical Journal, 2023, Vol 65, Issue 1, p222
- ISSN
0017-0895
- Publication type
Article
- DOI
10.1017/S0017089522000222