Integral orbits over function fields

2014 ◽  
Vol 10 (08) ◽  
pp. 2187-2204
Author(s):  
Hsiu-Lien Huang ◽  
Chia-Liang Sun ◽  
Julie Tzu-Yueh Wang
Keyword(s):  
Rational Function ◽  
Function Field ◽  
Function Fields ◽  
Closed Field ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Forward Orbit ◽  
Siegel Theorem

Over the function field of a smooth projective curve over an algebraically closed field, we investigate the set of S-integral elements in a forward orbit under a rational function by establishing some analogues of the classical Siegel theorem.

scholarly journals On Unramified Separable Abelian p-Extensions of Function Fields I

1959 ◽  
Vol 14 ◽  
pp. 223-234 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Galois Group ◽  
Function Field ◽  
Function Fields ◽  
Abelian Extension ◽  
Closed Field ◽  
Jacobian Variety ◽  
Algebraically Closed Field

Let k be an algebraically closed field of characteristic p>0. Let K/k be a function field of one variable and L/K be an unramified separable abelian extension of degree pr over K. The galois automorphisms ε1, …, εpr of L/K are naturally extended to automorphisms η(ε1), … , η(εpr) of the jacobian variety JL of L/k. If we take a svstem of p-adic coordinates on JL, we get a representation {Mp(η(εv))} of the galois group G(L/K) of L/K over p-adic integers.

scholarly journals Fibrations with moving cuspidal singularities

1991 ◽  
Vol 122 ◽  
pp. 161-179 ◽  
Author(s):  
Yoshifumi Takeda
Keyword(s):  
Algebraic Geometry ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Closed Field ◽  
Projective Surface ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Smooth Projective Surface ◽  
Almost All

Let f: V → C be a fibration from a smooth projective surface onto a smooth projective curve over an algebraically closed field k. In the case of characteristic zero, almost all fibres of f are nonsingular. In the case of positive characteristic, it is, however, known that there exist fibrations whose general fibres have singularities. Moreover, it seems that such fibrations often have pathological phenomena of algebraic geometry in positive characteristic (see M. Raynaud [7], W. Lang [4]).

ON MAXIMALLY FROBENIUS DESTABILISED VECTOR BUNDLES

2019 ◽  
Vol 99 (2) ◽  
pp. 195-202
Author(s):  
LINGGUANG LI
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Closed Field ◽  
Projective Curve ◽  
Stable Vector Bundle ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field

Let $X$ be a smooth projective curve of genus $g\geq 2$ over an algebraically closed field $k$ of characteristic $p>0$. We show that for any integers $r$ and $d$ with $0<r<p$, there exists a maximally Frobenius destabilised stable vector bundle of rank $r$ and degree $d$ on $X$ if and only if $r\mid d$.

scholarly journals On nilpotent automorphism groups of function fields

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nurdagül Anbar ◽  
Burçin Güneş
Keyword(s):  
Automorphism Group ◽  
Function Field ◽  
Automorphism Groups ◽  
Function Fields ◽  
Infinite Family ◽  
Nilpotent Subgroup ◽  
Closed Field ◽  
Algebraically Closed Field

Abstract We study the automorphisms of a function field of genus g ≥ 2 over an algebraically closed field of characteristic p > 0. More precisely, we show that the order of a nilpotent subgroup G of its automorphism group is bounded by 16 (g – 1) when G is not a p-group. We show that if |G| = 16(g – 1), then g – 1 is a power of 2. Furthermore, we provide an infinite family of function fields attaining the bound.

scholarly journals Harder–Narasimhan reduction of a principal bundle

2004 ◽  
Vol 174 ◽  
pp. 201-223 ◽  
Author(s):  
Indranil Biswas ◽  
Yogish I. Holla
Keyword(s):  
Algebraic Group ◽  
Principal Bundle ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Mild Assumption

AbstractLet E be a principal G–bundle over a smooth projective curve over an algebraically closed field k, where G is a reductive linear algebraic group over k. We construct a canonical reduction of E. The uniqueness of canonical reduction is proved under the assumption that the characteristic of k is zero. Under a mild assumption on the characteristic, the uniqueness is also proved when the characteristic of k is positive.

scholarly journals Formal orbifolds and orbifold bundles in positive characteristic

2019 ◽  
Vol 30 (12) ◽  
pp. 1950067
Author(s):  
Manish Kumar ◽  
A. J. Parameswaran
Keyword(s):  
Fundamental Group ◽  
Characteristic Zero ◽  
Vector Bundles ◽  
Positive Characteristic ◽  
Closed Field ◽  
Fundamental Groups ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Arbitrary Characteristic

We define formal orbifolds over an algebraically closed field of arbitrary characteristic as curves together with some branch data. Their étale coverings and their fundamental groups are also defined. These fundamental groups approximate the fundamental group of an appropriate affine curve. We also define vector bundles on these objects and the category of orbifold bundles on any smooth projective curve. Analogues of various statements about vector bundles which are true in characteristic zero are also proved. Some of these are positive characteristic avatars of notions which appear in the second author’s work [A. J. Parmeswaran, Parabolic coverings I: Case of curves, J. Ramanujam Math. Soc. 25(3) (2010) 233–251.] in characteristic zero.

scholarly journals Galois module structure of holomorphic differentials

2000 ◽  
Vol 62 (3) ◽  
pp. 493-509 ◽  
Author(s):  
Martha Rzedowski-Calderón ◽  
Gabriel Villa-Salvador ◽  
Manohar L. Madan
Keyword(s):  
Rational Function ◽  
Prime Divisor ◽  
Function Field ◽  
Module Structure ◽  
Closed Field ◽  
Galois Module ◽  
Indecomposable Modules ◽  
Algebraically Closed Field ◽  
Rational Function Field ◽  
Galois Module Structure

For a finite cyclic P–extension L/K of a rational function field K = κ(x) over an algebraically closed field κ of characteristic P > 0 such that every ramified prime divisor is fully ramified, we find a basis of the κ[G]-module structure of ωL(0) in terms of indecomposable modules.

scholarly journals Oort groups and lifting problems

2008 ◽  
Vol 144 (4) ◽  
pp. 849-866 ◽  
Author(s):  
T. Chinburg ◽  
R. Guralnick ◽  
D. Harbater
Keyword(s):  
Finite Groups ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Closed Field ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Cyclic Groups ◽  
Characteristic Two ◽  
Field Of Positive Characteristic

AbstractLet k be an algebraically closed field of positive characteristic p. We consider which finite groups G have the property that every faithful action of G on a connected smooth projective curve over k lifts to characteristic zero. Oort conjectured that cyclic groups have this property. We show that if a cyclic-by-p group G has this property, then G must be either cyclic or dihedral, with the exception of A4 in characteristic two. This proves one direction of a strong form of the Oort conjecture.

scholarly journals Portraits of preperiodic points for rational maps

2015 ◽  
Vol 159 (1) ◽  
pp. 165-186 ◽  
Author(s):  
DRAGOS GHIOCA ◽  
KHOA NGUYEN ◽  
THOMAS J. TUCKER
Keyword(s):  
Normal Form ◽  
Rational Function ◽  
Function Field ◽  
Rational Maps ◽  
Closed Field ◽  
Minimum Period ◽  
Algebraically Closed Field ◽  
Almost All ◽  
Suitable Modification

AbstractLet K be a function field over an algebraically closed field k of characteristic 0, let ϕ ∈ K(z) be a rational function of degree at least equal to 2 for which there is no point at which ϕ is totally ramified and let α ∈ K. We show that for all but finitely many pairs (m, n) ∈ $\mathbb{Z}$⩾0 × $\mathbb{N}$ there exists a place $\mathfrak{p}$ of K such that the point α has preperiod m and minimum period n under the action of ϕ. This answers a conjecture made by Ingram–Silverman [13] and Faber–Granville [8]. We prove a similar result, under suitable modification, also when ϕ has points where it is totally ramified. We give several applications of our result, such as showing that for any tuple (c0, . . ., cd−2) ∈ kd−1 and for almost all pairs (mi, ni) ∈ $\mathbb{Z}$⩾0 × $\mathbb{N}$ for i = 0, . . ., d − 2, there exists a polynomial f ∈ k[z] of degree d in normal form such that for each i = 0, . . ., d − 2, the point ci has preperiod mi and minimum period ni under the action of f.

Multiplication complexe et équivalence élémentaire dans le langage des corps (Complex multiplication and elementary equivalence in the language of fields)

2002 ◽  
Vol 67 (2) ◽  
pp. 635-648
Author(s):  
Xavier Vidaux
Keyword(s):  
Function Fields ◽  
Complex Multiplication ◽  
Closed Field ◽  
Constant Symbol ◽  
Endomorphism Rings ◽  
Modular Invariant ◽  
Elementary Equivalence ◽  
Algebraically Closed Field

AbstractLet K and K′ be two elliptic fields with complex multiplication over an algebraically closed field k of characteristic 0. non k-isomorphic, and let C and C′ be two curves with respectively K and K′ as function fields. We prove that if the endomorphism rings of the curves are not isomorphic then K and K′ are not elementarily equivalent in the language of fields expanded with a constant symbol (the modular invariant). This theorem is an analogue of a theorem from David A. Pierce in the language of k-algebras.

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