Schur covers and Carlitz’s conjecture

Abstract

We use the classification of finite simple groups and covering theory in positive characteristic to solve Carlitz’s conjecture (1966). An exceptional polynomialf over a finite field\({\mathbb{F}}_q \) is a polynomial that is a permutation polynomial on infinitely many finite extensions of\({\mathbb{F}}_q \). Carlitz’s conjecture saysf must be of odd degree (ifq is odd). Indeed, excluding characteristic 2 and 3, arithmetic monodromy groups of exceptional polynomials must be affine groups.We don’t, however, know which affine groups appear as the geometric metric monodromy group of exceptional polynomials. Thus, there remain unsolved problems. Riemann’s existence theorem in positive characteristic will surely play a role in their solution. We have, however, completely classified the exceptional polynomials of degree equal to the characteristic. This solves a problem from Dickson’s thesis (1896). Further, we generalize Dickson’s problem to include a description of all known exceptional polynomials.Finally: The methods allow us to consider coversX→\({\mathbb{P}}^1 \) that generalize the notion of exceptional polynomials. These covers have this property: Over each\({\mathbb{F}}_{q^t } \) point of\({\mathbb{P}}^1 \) there is exactly one\({\mathbb{F}}_{q^t } \) point ofX for infinitely manyt. ThusX has a rare diophantine property whenX has genus greater than 0. It has exactlyq t+1 points in\({\mathbb{F}}_{q^t } \) for infinitely manyt. This gives exceptional covers a special place in the theory of counting rational points on curves over finite fields explicitly. Corollary 14.2 holds also for a primitive exceptional cover having (at least) one totally ramified place over a rational point of the base. Its arithmetic monodromy group is an affine group.