Rational functions over finite fields with Galois closure of genus zero

Abstract

Let $k={\mathbb{F}}_q$. We classify, up to pre- and post-composition by $k$-Möbius transformations, all separable $k$-indecomposable rational functions $f\in k(X)$ of degree greater than one whose Galois closure has genus zero. The classification is valid in arbitrary characteristic and includes exact arithmetic conditions and class counts over the prescribed field. Semilinear Frobenius descent determines the finite-field forms and which geometric decompositions descend to $k$. For every separable $f\in k(X)$ of degree greater than one with Galois closure of genus zero and every $m⩾1$, we prove that $f$ permutes ${\mathbf{P}}^1({\mathbb{F}}_{q^m})$ if and only if it is exceptional over ${\mathbb{F}}_{q^m}$, meaning that it permutes ${\mathbf{P}}^1(L)$ for infinitely many finite extensions $L/{\mathbb{F}}_{q^m}$. No indecomposability assumption or lower bound on $q$ is needed. The argument combines fixed-point averaging with ramification on the Galois-closure curve. If the full constant field is ${\mathbb{F}}_{q^d}$, these properties depend only on $\gcd (m,d)$. For each classified family we determine the permutation extension degrees explicitly and characterize polynomial representatives.

arXiv comment81 pages