The place of exceptional covers among all diophantine relations

Abstract

Let F q be the order q finite field. An F q cover φ : X → Y of absolutely irreducible normal varieties has a nonsingular locus. Then, φ is exceptional if it maps one–one on F q t points for ∞ -ly many t over this locus. Lenstra suggested a curve Y may have an Exceptional ( cover) Tower over F q Lenstra Jr. [Talk at Glasgow Conference, Finite Fields III, 1995]. We construct it, and its canonical limit group and permutation representation, in general. We know all one-variable tamely ramified rational function exceptional covers, and much on wildly ramified one variable polynomial exceptional covers, from Fried et al. [Schur covers and Carlitz's conjecture, Israel J. Math. 82 (1993) 157–225], Guralnick et al. [The rational function analogue of a question of Schur and exceptionality of permutations representations, Mem. Amer. Math. Soc. 162 (2003) 773, ISBN 0065-9266] and Lidl et al. [Dickson Polynomials, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 65, Longman Scientific, New York, 1993]. We use exceptional towers to form subtowers from any exceptional cover collections. This gives us a language for separating known results from unsolved problems. We generalize exceptionality to p(ossibly)r(educible)-exceptional covers by dropping irreducibility of X. Davenport pairs (DPs) are significantly different covers of Y with the same ranges (where maps are nonsingular) on F q t points for ∞ -ly many t. If the range values have the same multiplicities, we have an iDP. We show how a pr-exceptional correspondence on F q covers characterizes a DP. You recognize exceptional covers and iDPs from their extension of constants series. Our topics include some of their dramatic effects • How they produce universal relations between Poincaré series. • How they relate to the Guralnick–Thompson genus 0 problem and to Serre's open image theorem. Historical sections capture Davenport's late 1960s desire to deepen ties between exceptional covers, their related cryptology, and the Weil conjectures.