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 group and permutation representation, in general. We know all onevariable 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 Fqt 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 Fq covers characterizes a DP. E-mail addresses: [email protected], [email protected]. 1071-5797/$ see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.ffa.2005.06.005 368 M.D. Fried / Finite Fields and Their Applications 11 (2005) 367–433 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. © 2005 Elsevier Inc. All rights reserved.