The place of exceptional covers among all diophantine relations

Abstract

Let F q be the order q finite field. An F q cover : X → Y of absolutely irreducible normal varieties has a nonsingular locus. Then, 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 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 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.

Constructing Permutation Rational Functions from Isogenies

“…A map X → Y of (smooth, projective) algebraic curves over a finite field F q is called an exceptional cover when the induced map on F q t -points X(F q t ) → Y (F q t ) is a bijection for infinitely many values of t (necessarily including t = 1). The construction of exceptional covers is an important problem in arithmetic algebraic geometry [8], which also has applications to cryptography.…”

“…3.5], we describe an algorithm to generate permutation rational functions of any constant prime degree ℓ ≥ 5 (which are, in fact, exceptional covers) over large finite fields and show that it is efficient and practical. We also expand upon the RSA analogy alluded to in [8] and discuss how our algorithm might indeed be used to obtain new factoring-related trapdoor permutations that behave better than the RSA trapdoor permutation against certain classes of attacks.…”