Exceptional Covers and Bijections on Rational Points

Abstract

We show that if f : X −→ Y is a finite, separable morphism of smooth curves defined over a finite field F q , where q is larger than an explicit constant depending only on the degree of f and the genus of X, then f maps. Surprisingly, the bounds on q for these two implications have different orders of magnitude. The main tools used in our proof are the Chebotarev density theorem for covers of curves over finite fields, the Castelnuovo genus inequality, and ideas from Galois theory.

Full classification of permutation rational functions and complete rational functions of degree three over finite fields

“…For example, one can use the method described in [1,9]: if a ∈ F q is such that ϕ(a) = ∞ and ϕ(∞) = b, the fractional jump construction provides a new function ϕ such that ϕ(x) = ϕ(x) for any x = a, and ϕ(a) = b. The methods we use to list permutation rational functions are mostly number theoretical: the strategy is to characterize such rational functions in terms of Galois group properties of function field extensions associated to them (similarily to [11]); from these properties we deduce equations which we are able to solve and whose solutions parametrise exactly permutation rational functions of degree 3. As a corollary of our results we also obtain a classification of complete permutation rational functions.…”

Iterates of generic polynomials and generic rational functions

“…Let M/K be a finite Galois extension. If p is a prime of K and q is any prime of M extending p, we define e(q|p) to be the inertia degree of q over p and f (q|p) to be the residue degree of q over p. The next result is useful in determining the structure of inertia groups, similar results can be found in [9], [24]. Lemma 2.4.…”

A Gap Principle for Subvarieties with Finitely Many Periodic Points

“…We may view G as a group of permutations of the set T i . The following lemma is part of Lemmas 3.1 and 3.2 of [GTZ07]. This lemma gives a unified approach of treating ramified and unramified primes.…”