Exceptional extensions of local fields and the Carlitz--Wan conjecture

TL;DR

This paper extends the concept of exceptional polynomials over finite fields to local field extensions, characterizes these extensions, and provides new proofs for related theorems on ramification and exceptional maps.

Contribution

It introduces the notion of exceptional local field extensions, characterizes them, and links their properties to classical exceptional polynomials, offering new proofs of existing theorems.

Findings

Characterization of all exceptional local field extensions of degree coprime to the residue characteristic.

Relationship between exceptionality of a local extension and its subextensions.

Three new proofs of a theorem on ramification indices in exceptional maps between curves.

Abstract

For any prime power $q$, a polynomial $f(X)\in\F_q[X]$ is ``exceptional'' if it induces bijections of $\F_{q^k}$ for infinitely many $k$; this condition is known to be equivalent to $f(X)$ inducing a bijection of $\F_{q^k}$ for at least one $k$ with $q^k\ge \deg(f)^4$. In this paper, we introduce the notion of an ``exceptional'' extension of local fields of any characteristic, and show that if $f(X)\in\F_q[X]$ is exceptional in the classical sense then the field extension $\F_q(X)/\F_q(f(X))$ yields an exceptional local field extension upon passing to the completion at a degree-$1$ place. We describe all exceptional local field extensions of degree coprime to the residue characteristic, determine the relationship between exceptionality of a local field extension and exceptionality of a subextension, and give various Galois-theoretic characterizations of exceptional local field extensions. As a consequence, we obtain three new proofs, using quite different tools, of a theorem of Guralnick and M\"uller about ramification indices in exceptional maps between curves over $\F_q$. This theorem generalizes a result of Lenstra which subsumes earlier conjectures of Carlitz and Wan.