Automorphism groups of endomorphisms of $\mathbb{P}^1 (\bar{\mathbb{F}}_p)$
Glasgow mathematical journal, Tome 65 (2023) no. 1, pp. 222-255
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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.
Cai, Julia; Hutz, Benjamin; Mayer, Leo; Weinreich, Max. Automorphism groups of endomorphisms of $\mathbb{P}^1 (\bar{\mathbb{F}}_p)$. Glasgow mathematical journal, Tome 65 (2023) no. 1, pp. 222-255. doi: 10.1017/S0017089522000222
@article{10_1017_S0017089522000222,
author = {Cai, Julia and Hutz, Benjamin and Mayer, Leo and Weinreich, Max},
title = {Automorphism groups of endomorphisms of $\mathbb{P}^1 (\bar{\mathbb{F}}_p)$},
journal = {Glasgow mathematical journal},
pages = {222--255},
year = {2023},
volume = {65},
number = {1},
doi = {10.1017/S0017089522000222},
url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089522000222/}
}
TY - JOUR
AU - Cai, Julia
AU - Hutz, Benjamin
AU - Mayer, Leo
AU - Weinreich, Max
TI - Automorphism groups of endomorphisms of $\mathbb{P}^1 (\bar{\mathbb{F}}_p)$
JO - Glasgow mathematical journal
PY - 2023
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EP - 255
VL - 65
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%J Glasgow mathematical journal
%D 2023
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