The Place of Exceptionality
نویسنده
چکیده
Let Fq be the order q finite field. An Fq cover φ : X → Y of absolutely irreducible normal varieties has a nonsingular locus. Then, φ is exceptional if it maps one-one on F qt points for ∞-ly many t over this locus. Lenstra suggested a curve Y may have an Exceptional (cover) Tower over Fq [Le95]. 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 [FGS93], [GMS03] and [LMT93]. 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 non-singular) on F qt 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 Davenport pair. 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 ’60s desire to deepen ties between exceptional covers, their related cryptology, and the Weil conjectures.
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تاریخ انتشار 2009