Product of Local Points of Subvarieties of Almost Isotrivial Semi-Abelian Varieties Over a Global Function Field

In this paper, we formulate and prove a duality for cohomology of curves over perfect fields of positive characteristic with coefficients in Néron models of abelian varieties. This is a global function field version of the author’s previous work on local duality and Grothendieck’s duality conjecture. It generalizes the perfectness of the Cassels–Tate pairing in the finite base field case. The proof uses the local duality mentioned above, Artin–Milne’s global finite flat duality, the nondegeneracy of the height pairing and finiteness of crystalline cohomology. All these ingredients are organized under the formalism of the rational étale site developed earlier.

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Zsigmondy theorem for arithmetic dynamics induced by a drinfeld module

International Journal of Number Theory ◽

10.1142/s1793042119500611 ◽

2019 ◽

Vol 15 (06) ◽

pp. 1111-1125

Author(s):

Zhengjun Zhao ◽

Qingzhong Ji

Keyword(s):

Prime Ideal ◽

Function Field ◽

Drinfeld Module ◽

Torsion Point ◽

Arithmetic Dynamics ◽

Global Function ◽

Ideal Formula ◽

Global Function Field ◽

The Ideal

Let [Formula: see text] be a Drinfeld [Formula: see text]-module defined over a global function field [Formula: see text] Let [Formula: see text] be a non-torsion point of [Formula: see text] with infinite [Formula: see text]-orbit. For each [Formula: see text] write the ideal [Formula: see text] as a quotient of relatively prime integral ideals. We establish an analogue of the classical Zsigmondy theorem for the ideal sequence [Formula: see text] i.e. for all but finitely many [Formula: see text] there exists a prime ideal [Formula: see text] such that [Formula: see text] and [Formula: see text] for all [Formula: see text]

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Wild sets in global function fields

Mathematica Slovaca ◽

10.1515/ms-2017-0349 ◽

2020 ◽

Vol 70 (2) ◽

pp. 259-272

Author(s):

Alfred Czogała ◽

Przemysław Koprowski ◽

Beata Rothkegel

Keyword(s):

Function Field ◽

Function Fields ◽

The Self ◽

Global Function ◽

Global Function Fields ◽

Global Function Field ◽

Set Of Points

Abstract Given a self-equivalence of a global function field, its wild set is the set of points where the self-equivalence fails to preserve parity of valuation. In this paper we describe structure of finite wild sets.

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A Local-Global Criteria of Affine Varieties Admitting Points in Rank-One Subgroups of a Global Function Field

Journal de Théorie des Nombres de Bordeaux ◽

10.5802/jtnb.1016 ◽

2018 ◽

Vol 30 (1) ◽

pp. 59-79

Author(s):

Chia-Liang Sun

Keyword(s):

Function Field ◽

Global Function ◽

Global Function Field ◽

Rank One

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On the Computation of Integral Closures of Cyclic Extensions of Function Fields

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157000001340 ◽

2007 ◽

Vol 10 ◽

pp. 141-160

Author(s):

Robert Fraatz

Keyword(s):

Function Field ◽

Strong Approximation ◽

Approximation Theorem ◽

Function Fields ◽

Proper Subset ◽

Global Function ◽

Global Function Field ◽

Integral Closures ◽

Strong Approximation Theorem

AbstractLet S be a non-empty proper subset of the set of places of a global function field F and E a cyclic Kummer or Artin–Schreier–Witt extension of F. We present a method of efficiently computing the ring of elements of E which are integral at all places of S. As an important tool, we include an algorithmic version of the strong approximation theorem. We conclude with several examples.

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