scholarly journals Belyi’s theorem in characteristic two

2019 ◽  
Vol 156 (2) ◽  
pp. 325-339 ◽  
Author(s):  
Yusuke Sugiyama ◽  
Seidai Yasuda
Keyword(s):  
Rational Function ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Characteristic Two ◽  
Obstruction Class

We prove an analogue of Belyi’s theorem in characteristic two. Our proof consists of the following three steps. We first introduce a new notion called pseudo-tameness for morphisms between curves over an algebraically closed field of characteristic two. Secondly, we prove the existence of a ‘pseudo-tame’ rational function by showing the vanishing of an obstruction class. Finally, we construct a tamely ramified rational function from the ‘pseudo-tame’ rational function.

scholarly journals Geometric Weil representation in characteristic two

2011 ◽  
Vol 11 (2) ◽  
pp. 221-271 ◽  
Author(s):  
Alain Genestier ◽  
Sergey Lysenko
Keyword(s):  
Weil Representation ◽  
Closed Field ◽  
Triangulated Category ◽  
Algebraically Closed Field ◽  
Witt Vectors ◽  
Suitable Sense ◽  
Characteristic Two ◽  
Greenberg Realization ◽  
Geometric Analogue

AbstractLet k be an algebraically closed field of characteristic two. Let R be the ring of Witt vectors of length two over k. We construct a group stack Ĝ over k, the metaplectic extension of the Greenberg realization of $\operatorname{\mathbb{S}p}_{2n}(R)$. We also construct a geometric analogue of the Weil representation of Ĝ, this is a triangulated category on which Ĝ acts by functors. This triangulated category and the action are geometric in a suitable sense.

scholarly journals THE 2-RANKS OF HYPERELLIPTIC CURVES WITH EXTRA AUTOMORPHISMS

2009 ◽  
Vol 05 (05) ◽  
pp. 897-910 ◽  
Author(s):  
DARREN GLASS
Keyword(s):  
Automorphism Group ◽  
Hyperelliptic Curve ◽  
Hyperelliptic Curves ◽  
Closed Field ◽  
Base Field ◽  
Characteristic P ◽  
Algebraically Closed Field ◽  
Characteristic Two ◽  
Carry Over ◽  
The Relationship

This paper examines the relationship between the automorphism group of a hyperelliptic curve defined over an algebraically closed field of characteristic two and the 2-rank of the curve. In particular, we exploit the wild ramification to use the Deuring–Shafarevich formula in order to analyze the ramification of hyperelliptic curves that admit extra automorphisms and use this data to impose restrictions on the genera and 2-ranks of such curves. We also show how some of the techniques and results carry over to the case where our base field is of characteristic p > 2.

scholarly journals Invariants of the dihedral group D2p in characteristic two

2011 ◽  
Vol 152 (1) ◽  
pp. 1-7 ◽  
Author(s):  
MARTIN KOHLS ◽  
MÜFİT SEZER
Keyword(s):  
Upper Bound ◽  
Dihedral Group ◽  
Explicit Description ◽  
Closed Field ◽  
Generating Set ◽  
Algebraically Closed Field ◽  
Finite Dimensional ◽  
Characteristic Two ◽  
Minimal Generating Set

AbstractWe consider finite dimensional representations of the dihedral group D2p over an algebraically closed field of characteristic two where p is an odd prime and study the degrees of generating and separating polynomials in the corresponding ring of invariants. We give an upper bound for the degrees of the polynomials in a minimal generating set that does not depend on p when the dimension of the representation is sufficiently large. We also show that p + 1 is the minimal number such that the invariants up to that degree always form a separating set. We also give an explicit description of a separating set.

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 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.

Three-dimensional zeropotent algebras over an algebraically closed field of characteristic two

2019 ◽  
Vol 48 (4) ◽  
pp. 1613-1625
Author(s):  
Kiyoshi Shirayanagi ◽  
Yuji Kobayashi ◽  
Sin-Ei Takahasi ◽  
Makoto Tsukada
Keyword(s):  
Three Dimensional ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Characteristic Two

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 The existence of Zariski dense orbits for polynomial endomorphisms of the affine plane

2017 ◽  
Vol 153 (8) ◽  
pp. 1658-1672 ◽  
Author(s):  
Junyi Xie
Keyword(s):  
Rational Function ◽  
Affine Plane ◽  
Positive Answer ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Closed Point ◽  
Polynomial Endomorphisms ◽  
Dense Orbits ◽  
Polynomial Endomorphism

In this paper we prove the following theorem. Let $f$ be a dominant polynomial endomorphism of the affine plane defined over an algebraically closed field of characteristic $0$. If there is no nonconstant invariant rational function under $f$, then there exists a closed point in the plane whose orbit under $f$ is Zariski dense. This result gives us a positive answer to a conjecture proposed by Medvedev and Scanlon, by Amerik, Bogomolov and Rovinsky, and by Zhang for polynomial endomorphisms of the affine plane.

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.

scholarly journals Uniform perfectness of the Berkovich Julia sets in non-archimedean dynamics

2021 ◽  
pp. 1-18
Author(s):  
YÛSUKE OKUYAMA
Keyword(s):  
Rational Function ◽  
Julia Set ◽  
Julia Sets ◽  
Projective Line ◽  
Closed Field ◽  
Fatou Component ◽  
Algebraically Closed Field ◽  
Absolute Value ◽  
Uniformly Perfect ◽  
Uniform Perfectness

Abstract We show that a rational function f of degree $>1$ on the projective line over an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value has no potentially good reductions if and only if the Berkovich Julia set of f is uniformly perfect. As an application, a uniform regularity of the boundary of each Berkovich Fatou component of f is also established.

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