[Paper Review] Automorphism Groups and Invariant Theory on PN

João Alberto de Faria, Benjamin Hutz|arXiv (Cornell University)|Sep 22, 2015

Advanced Differential Equations and Dynamical Systems21 references3 citations

TL;DR

This paper develops computational methods in invariant theory to study automorphism groups of endomorphisms on projective space $\mathbb{P}^N$, extending the Faber-Manes-Viray algorithm from $\mathbb{P}^1$ to $\mathbb{P}^2$. It proves that every finite subgroup of $\mathrm{PGL}_{N+1}$ occurs infinitely often as a subgroup of automorphism groups of morphisms and provides an explicit bound on automorphism group size depending on degree, enabling algorithmic determination of automorphism groups via fixed-point analysis.

ABSTRACT

Let $K$ be a field and $f:\mathbb{P}^N o \mathbb{P}^N$ a morphism. There is a natural conjugation action on the space of such morphisms by elements of the projective linear group $ ext{PGL}_{N+1}$. The group of automorphisms, or stabilizer group, of a given $f$ for this action is known to be a finite group. In this article, we address two mainly computational problems concerning automorphism groups. Given a finite subgroup of $ ext{PGL}_{N+1}$ determine endomorphisms of $\mathbb{P}^N$ with that group as subgroup of its automorphism group. In particular, we show that every finite subgroup occurs infinitely often and discuss some associated rationality problems. Inversely, given an endomorphism determine its automorphism group. In particular, we extended the Faber-Manes-Viray fixed-point algorithm for $\mathbb{P}^1$ to endomorphisms of $\mathbb{P}^2$. A key component is an explicit bound on the size of the automorphism group depending on the degree of the endomorphism.

Motivation & Objective

• To address the inverse problem of constructing endomorphisms on $\mathbb{P}^N$ with a given finite subgroup of $\mathrm{PGL}_{N+1}$ as an automorphism subgroup.
• To extend the Faber-Manes-Viray fixed-point algorithm from $\mathbb{P}^1$ to $\mathbb{P}^2$ for computing automorphism groups of morphisms.
• To establish an explicit upper bound on the size of the automorphism group of a morphism in terms of its degree.
• To investigate rationality and moduli problems related to the field of definition versus field of moduli in dynamical systems.

Proposed method

• Use classical invariant theory to construct morphisms whose automorphism groups contain a prescribed finite subgroup of $\mathrm{PGL}_{N+1}$.
• Apply conjugation action of $\mathrm{PGL}_{N+1}$ on morphisms to define the moduli space $M_d^N$ and stabilize automorphism groups.
• Implement a fixed-point algorithm in Sage to compute rational periodic points and test conjugation invariance under candidate automorphisms.
• Use reduction modulo primes to verify absence of periodic points (e.g., 3-periodic points) when direct computation is infeasible.
• Leverage the equivalence between nontrivial automorphisms and nontrivial rational twists to analyze field of moduli issues.
• Utilize computational algebra systems (e.g., Magma, Sage) to verify automorphism group membership via matrix conjugation checks on homogeneous polynomials.

Experimental results

Research questions

• RQ1 Can every finite subgroup of $\mathrm{PGL}_{N+1}$ be realized as a subgroup of the automorphism group of some morphism on $\mathbb{P}^N$?
• RQ2 How can the automorphism group of a morphism on $\mathbb{P}^2$ be algorithmically determined, generalizing the $\mathbb{P}^1$ case?
• RQ3 What is an explicit upper bound on the size of the automorphism group of a morphism of degree $d$ on $\mathbb{P}^N$?
• RQ4 How do rational periodic points and their cycles over finite fields help in verifying the absence of certain periodic orbits in number fields?
• RQ5 What is the relationship between the field of moduli and the field of definition for morphisms with nontrivial automorphisms?

Key findings

• Every finite subgroup of $\mathrm{PGL}_{N+1}$ occurs infinitely often as a subgroup of the automorphism group of some morphism on $\mathbb{P}^N$.
• An explicit bound on the size of the automorphism group of a morphism on $\mathbb{P}^N$ is established, depending only on the degree of the morphism.
• The Faber-Manes-Viray fixed-point algorithm is successfully extended to $\mathbb{P}^2$, with runtime examples showing feasibility on morphisms of degree 3 and 4.
• For morphisms with trivial automorphism group, the algorithm completes in under 2.5 seconds, while those with nontrivial symmetries (e.g., octahedral, tetrahedral) take up to 54 seconds.
• The absence of 3-periodic points over required number fields was verified using reduction modulo primes, enabling completion of otherwise infeasible computations.
• Explicit morphisms with automorphism groups isomorphic to the octahedral group (size 24), tetrahedral group (size 12), and dihedral groups (size 6) were constructed and verified computationally.