Wild sets in global function fields

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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On indecomposable quadratic lattices over global function fields

International Journal of Number Theory ◽

10.1142/s1793042117500919 ◽

2016 ◽

Vol 13 (06) ◽

pp. 1611-1616

Author(s):

Ruiqing Wang

Keyword(s):

Function Field ◽

Function Fields ◽

Global Function ◽

Global Function Fields ◽

Global Function Field ◽

Quadratic Lattices

In this paper, we prove that there exist indecomposable lattices of ranks 5 and 6 over a Hasse domain of any global function field in which [Formula: see text] is not a square, which solves a problem proposed by Gerstein.

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A NOTE ON THE GENUS OF GLOBAL FUNCTION FIELDS

Glasgow Mathematical Journal ◽

10.1017/s0017089512000742 ◽

2013 ◽

Vol 55 (3) ◽

pp. 559-565

Author(s):

ZHENGJUN ZHAO ◽

XIA WU

Keyword(s):

Algebraic Geometry ◽

Function Field ◽

Elementary Proof ◽

Function Fields ◽

Drinfeld Module ◽

Extension Field ◽

Global Function ◽

Global Function Fields ◽

Roch Theorem ◽

Stark Conjecture

AbstractTo give a relatively elementary proof of the Brumer–Stark conjecture in a function field context involving no algebraic geometry beyond the Riemann–Roch theorem for curves, Hayes Compos. Math., vol. 55, 1985, pp. 209–239) defined a normalizing field $H_\mathfrak{e}^*$ associated with a fixed sgn-normalized Drinfeld module and its extension field $K_\mathfrak{m}$, which is an analogue of cyclotomic function fields over a rational function field. We present explicitly in this note the formulae for the genus of the two fields and the maximal real subfield $H_\mathfrak{m}$ of $K_\mathfrak{m}$. In some sense, our results can be regarded as generalizations of formulae for the genus of classical cyclotomic function fields obtained by Hayes Trans. Amer. Math. Soc., vol. 189, 1974, pp. 77–91) and Kida and Murabayashi (Tokyo J. Math., vol. 14(1), 1991, pp. 45–56).

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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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Weak Approximation for Points with Coordinates in Rank-one Subgroups of Global Function Fields

Canadian Mathematical Bulletin ◽

10.4153/cmb-2018-008-3 ◽

2018 ◽

Vol 61 (4) ◽

pp. 878-890

Author(s):

Chia-Liang Sun

Keyword(s):

Function Field ◽

Function Fields ◽

Weak Approximation ◽

Affine Variety ◽

Finite Sets ◽

Global Function ◽

Arbitrary Rank ◽

Global Function Fields ◽

Rank One ◽

Multiplicative Subgroup

AbstractFor every affine variety over a global function field, we show that the set of its points with coordinates in an arbitrary rank-one multiplicative subgroup of this function field satisfies the required property of weak approximation for finite sets of places of this function field avoiding arbitrarily given finitely many places.

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