Shifted dihedral isogeny quotients and the classification of exceptional rational functions of degree five

Abstract

We give a complete classification, up to $k$-Möbius equivalence, of exceptional rational functions of degree five over every finite field. The classification gives explicit normal forms in every characteristic, exact parameter identifications, and the resulting class counts. The main structural input is an arithmetic theory of shifted dihedral isogeny quotients valid in every odd degree. For each odd $n⩾3$, the separable degree-$n$ rational maps whose geometric monodromy group is isomorphic to $D_n$ and whose nontrivial inertia groups are generated by reflections are precisely the maps induced by cyclic $n$-isogenies on shifted Kummer quotients. We determine the exact ambiguity of the isogeny data under two-sided $k$-Möbius equivalence by means of a signed Frobenius-descent invariant. The theory allows arbitrary odd $n$, reduced kernels when the characteristic divides $n$, and wild reflection inertia. If Frobenius acts on the cyclic kernel by $\lambda \in (\mathbb{Z}/n\mathbb{Z}{)}^{\times }$, the induced map is exceptional exactly when both $\lambda -1$ and $\lambda +1$ are units modulo $n$.

41 pages. In v3, we add exact $k$-Möbius equivalence criteria and class counts, and develop an arithmetic theory of shifted dihedral isogeny quotients in every odd degree