Davenport pairs over finite fields

Aitken, Wayne; Fried, Michael D.; Holt, Linda M.

doi:10.2140/pjm.2004.216.1

Cited by 4 publications

(8 citation statements)

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“…This is exactly pr-exceptionality for X 1 × Y X 2 → X i . It is also exactly the DP condition as in [AFH03,Theorem 3.8]. So, this is equivalent to both conditions of (3.4).…”

Section: Dps and Pr-exceptionalitymentioning

confidence: 78%

“…Let Y 0 be Y minus the discriminant locus of , and X 0 the pullback by of Y 0 . Aitken et al [AFH03,Remark 3.9] extends in generality, with only notational change, the short proof of Fried and Jarden [FrJ86,Lemma 19.27] for DPs of polynomials. This proof shows the equivalence of : X → Y pr-exceptional over F q t (without assuming X is absolutely irreducible) with the following Galois theoretic statement.…”

Section: Lifting Rational Pointsmentioning

confidence: 99%

“…Investigating DPs started with proving Schur's conjecture. [AFH03] used DP to mean a pair of polynomials over F q as we do in Definition 3.5: Equal ranges on F q t for ∞-ly many t. We usually include the not-linearly related assumption §8.2.1 to exclude such exceptionality situations as a degree one cover together with any exceptional cover. We do not expect covers in an isovalent DP to have the same degree.…”

Section: Some History Of Dpsmentioning

confidence: 99%

“…Now we call that a strong Davenport pair (of polynomials) over K. An SDP over (Y, K) is a pair of covers i : X i → Y , i = 1, 2, over K satisfying Range equality: (8.5) 1 (X 1 (O/p)) = 2 (X 2 (O/p)) for almost all prime ideals p of O K . Aitken et al [AFH03] reserves the acronym DP over (Y, K) to mean equality on ranges holds for infinitely many p. An iDP is then an isovalent DP ( §8.2.1 and Proposition 3.9), iSDP means isovalent SDP, etc. ( 1 , 2 , K) is an iSDP for ∞-ly many p, then it is an iSDP for almost all p.…”

Section: History Of Davenport Pairsmentioning

confidence: 99%

“…Aitken et al [AFH03,Definition 1.3] calls this possibility an example of masking. Müller [Mu98a, §4] found a version of it, motivating our name.…”

Section: C12 Maskingmentioning

confidence: 99%

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The place of exceptional covers among all diophantine relations

Fried

2005

Finite Fields and Their Applications

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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.

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“…This is exactly pr-exceptionality for X 1 × Y X 2 → X i . It is also exactly the DP condition as in [AFH03,Theorem 3.8]. So, this is equivalent to both conditions of (3.4).…”

Section: Dps and Pr-exceptionalitymentioning

confidence: 78%

Section: Lifting Rational Pointsmentioning

confidence: 99%

Section: Some History Of Dpsmentioning

confidence: 99%

Section: History Of Davenport Pairsmentioning

confidence: 99%

“…Aitken et al [AFH03,Definition 1.3] calls this possibility an example of masking. Müller [Mu98a, §4] found a version of it, motivating our name.…”

Section: C12 Maskingmentioning

confidence: 99%

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The place of exceptional covers among all diophantine relations

Fried

2005

Finite Fields and Their Applications

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Relating Two Genus 0 Problems of John Thompson

Fried¹

Developments in Mathematics

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Abstract. Excluding a precise list of groups like alternating, symmetric, cyclic and dihedral, from 1st year algebra ( §7.2.3), we expect there are only finitely many monodromy groups of primitive genus 0 covers. Denote this nearly proven genus 0 problem as Problem g=0 2 . We call the exceptional groups 0-sporadic. Example: Finitely many Chevalley groups are 0-sporadic. A proven result: Among polynomial 0-sporadic groups, precisely three produce covers falling in nontrivial reduced families. Each (miraculously) defines one natural genus 0 Q cover of the j-line. The latest Nielsen class techniques apply to these dessins d'enfant to see their subtle arithmetic and interesting cusps.John Thompson earlier considered another genus 0 problem: To find θ-functions uniformizing certain genus 0 (near) modular curves. We call this Problem g=01 . We pose uniformization problems for j-line covers in two cases. First: From the three 0-sporadic examples of Problem g=0 2 . Second: From finite collections of genus 0 curves with aspects of Problem g=01 .

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Cryptography

2013

Handbook of Finite Fields

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Davenport pairs over finite fields

Cited by 4 publications

References 20 publications

The place of exceptional covers among all diophantine relations

The place of exceptional covers among all diophantine relations

Relating Two Genus 0 Problems of John Thompson

Cryptography

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