Exceptional Covers of Surfaces

Exceptional Covers of Surfaces

Consider a finite morphism f : X → Y of smooth, projective varieties over a finite field F. Suppose X is the vanishing locus in PN of r forms of degree at most d. We show that there is a constant C depending only on (N, r, d) and deg( f ) such that if |F| > C, then f (F) : X(F) → Y(F) is injective if and only if it is surjective.

Chebyshev Covers and Exceptional Number Fields

Among the simplest of the classical polynomials are the Chebyshev polynomials of the first and second kind, Tk(x) and Uk(x). In our normalization, the indices are allowed to be halfintegers as well as integers, and the “polynomials” actually live in Z[x, √ 2− x, √ 2 + x]. We will show that the rational functions Tm/2(x) Tn/2(x) and Um/2(x) Un/2(x) are very remarkable from the point of view of G...

Exceptional Covers and Bijections on Rational Points

We show that if f : X −→ Y is a finite, separable morphism of smooth curves defined over a finite field Fq, where q is larger than an explicit constant depending only on the degree of f and the genus of X , then f maps X(Fq) surjectively onto Y (Fq) if and only if f maps X(Fq) injectively into Y (Fq). Surprisingly, the bounds on q for these two implications have different orders of magnitude. T...

COVERS OF KLEIN SURFACES by

The place of exceptional covers among all diophantine relations

Let Fq be the order q finite field. An Fq cover : X → Y of absolutely irreducible normal varieties has a nonsingular locus. Then, is exceptional if it maps one–one on Fqt points for ∞-ly many t over this locus. Lenstra suggested a curve Y may have an Exceptional (cover) Tower over Fq Lenstra Jr. [Talk at Glasgow Conference, Finite Fields III, 1995]. We construct it, and its canonical limit grou...