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 THE p-TORSION OF CURVES WITH LARGE p-RANK

2009 ◽  
Vol 05 (06) ◽  
pp. 1103-1116 ◽  
Author(s):  
RACHEL PRIES
Keyword(s):  
Moduli Space ◽  
Open Problem ◽  
Hyperelliptic Curve ◽  
Closed Field ◽  
Generic Point ◽  
Characteristic P ◽  
Smooth Curves ◽  
Algebraically Closed Field ◽  
Group Schemes

Consider the moduli space of smooth curves of genus g and p-rank f defined over an algebraically closed field k of characteristic p. It is an open problem to classify which group schemes occur as the p-torsion of the Jacobians of these curves for f < g - 1. We prove that the generic point of every component of this moduli space has a-number 1 when f = g - 2 and f = g - 3. Likewise, we show that a generic hyperelliptic curve with p-rank g - 2 has a-number 1 when p ≥ 3. We also show that the locus of curves with p-rank g - 2 and a-number 2 is non-empty with codimension 3 in [Formula: see text] when p ≥ 5. We include some other results when f = g - 3. The proofs are by induction on g while fixing g - f. They use computations about certain components of the boundary of [Formula: see text].

scholarly journals ORBIT CLOSURES IN THE WITT ALGEBRA AND ITS DUAL SPACE

2014 ◽  
Vol 13 (05) ◽  
pp. 1350146 ◽  
Author(s):  
MARTIN MYGIND
Keyword(s):  
Automorphism Group ◽  
Dual Space ◽  
Closed Field ◽  
Witt Algebra ◽  
Conditional Statement ◽  
Characteristic P ◽  
Algebraically Closed Field ◽  
Orbit Closures

Working over an algebraically closed field of characteristic p > 3, we calculate the orbit closures in the Witt algebra W under the action of its automorphism group G. We also outline how the same techniques can be used to determine closures of orbits of all heights except p - 1 (in which case we only obtain a conditional statement) in the dual space W* under the induced action of G. As a corollary we prove that the algebra of invariants k[W*]G is trivial.

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 Resolutions With Conical Slices and Descent for the Brauer Group Classes of Certain Central Reductions of Differential Operators in Characteristic p

2019 ◽  
Author(s):  
Dmitry Kubrak ◽  
Roman Travkin
Keyword(s):  
Differential Operators ◽  
Closed Field ◽  
Brauer Group ◽  
Affine Variety ◽  
Smooth Variety ◽  
Characteristic P ◽  
Algebraically Closed Field ◽  
Quiver Varieties ◽  
Hypertoric Varieties ◽  
General Reduction

Abstract “Even more so is the word ‘crystalline’, a glacial and impersonal concept of his which disdains viewing existence from a single portion of time and space” Eileen Myles, “The Importance of Being Iceland” For a smooth variety $X$ over an algebraically closed field of characteristic $p$ to a differential 1-form $\alpha $ on the Frobenius twist $X^{\textrm{(1)}}$ one can associate an Azumaya algebra ${{\mathcal{D}}}_{X,\alpha }$, defined as a certain central reduction of the algebra ${{\mathcal{D}}}_X$ of “crystalline differential operators” on $X$. For a resolution of singularities $\pi :X\to Y$ of an affine variety $Y$, we study for which $\alpha $ the class $[{{\mathcal{D}}}_{X,\alpha }]$ in the Brauer group $\textrm{Br}(X^{\textrm{(1)}})$ descends to $Y^{\textrm{(1)}}$. In the case when $X$ is symplectic, this question is related to Fedosov quantizations in characteristic $p$ and the construction of noncommutative resolutions of $Y$. We prove that the classes $[{{\mathcal{D}}}_{X,\alpha }]$ descend étale locally for all $\alpha $ if ${{\mathcal{O}}}_Y\widetilde{\rightarrow }\pi _\ast{{\mathcal{O}}}_X$ and $R^{1}\pi _*\mathcal O_X = R^2\pi _*\mathcal O_X =0$. We also define a certain class of resolutions, which we call resolutions with conical slices, and prove that for a general reduction of a resolution with conical slices in characteristic $0$ to an algebraically closed field of characteristic $p$ classes $[{{\mathcal{D}}}_{X,\alpha }]$ descend to $Y^{\textrm{(1)}}$ globally for all $\alpha $. Finally we give some examples; in particular, we show that Slodowy slices, Nakajima quiver varieties, and hypertoric varieties are resolutions with conical slices.

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 Modular finite W-algebras

2018 ◽  
Vol 2019 (18) ◽  
pp. 5811-5853 ◽  
Author(s):  
Simon M Goodwin ◽  
Lewis W Topley
Keyword(s):  
Algebraic Group ◽  
Recent Work ◽  
Nilpotent Element ◽  
Direct Approach ◽  
Closed Field ◽  
Characteristic P ◽  
Reductive Algebraic Group ◽  
Algebraically Closed Field ◽  
Equivalence Of Categories ◽  
Pbw Theorem

Abstract Let ${\mathbb{k}}$ be an algebraically closed field of characteristic p > 0 and let G be a connected reductive algebraic group over ${\mathbb{k}}$. Under some standard hypothesis on G, we give a direct approach to the finite W-algebra $U(\mathfrak{g},e)$ associated to a nilpotent element $e \in \mathfrak{g} = \textrm{Lie}\ G$. We prove a PBW theorem and deduce a number of consequences, then move on to define and study the p-centre of $U(\mathfrak{g},e)$, which allows us to define reduced finite W-algebras $U_{\eta}(\mathfrak{g},e)$ and we verify that they coincide with those previously appearing in the work of Premet. Finally, we prove a modular version of Skryabin’s equivalence of categories, generalizing recent work of the second author.

There are many almost strongly minimal generalized n-gons that do not interpret an infinite group

1998 ◽  
Vol 63 (2) ◽  
pp. 485-508 ◽  
Author(s):  
Mark J. Debonis ◽  
Ali Nesin
Keyword(s):  
Amalgamation Property ◽  
Projective Planes ◽  
Infinite Group ◽  
Closed Field ◽  
Morley Rank ◽  
Algebraically Closed Field ◽  
Finite Graphs ◽  
Incidence Geometries ◽  
Finite Morley Rank ◽  
Carry Over

Generalized n-gons are certain geometric structures (incidence geometries) that generalize the concept of projective planes (the nontrivial generalized 3-gons are exactly the projective planes).In a simplified world, every generalized n-gon of finite Morley rank would be an algebraic one, i.e., one of the three families described in [9] for example. To our horror, John Baldwin [2], using methods discovered by Hrushovski [7], constructed ℵ1-categorical projective planes which are not algebraic. The projective planes that Baldwin constructed fail to be algebraic in a dramatic way.Indeed, every algebraic projective plane over an algebraically closed field is Desarguesian [12]. In particular, an algebraically closed field (isomorphic to the base field) can be interpreted in every one of them. However, in the projective planes that Baldwin constructed, one cannot even interpret an infinite group.In this article we show that the same phenomenon occurs for the generalized n-gons if n ≥ 3 is an odd integer. For each such n we construct many nonisomorphic generalized n-gons of finite Morley rank that do not interpret an infinite group. As one may expect, our method is inspired by Hrushovski and Baldwin, and we follow Baldwin's line of approach. Quite often our proofs are a verification of the fact that the proofs of Baldwin [2] for n = 3 carry over to an arbitrary positive odd integer n (which is sometimes far from being obvious). As in [2], we begin by defining a certain collection of finite graphs K* and a binary relation ≤ on these graphs. We show that (K*, ≤) satisfies the amalgamation property.

scholarly journals Weight Parameterization of Simple Modules for p-Solvable Groups

2013 ◽  
Vol 20 (01) ◽  
pp. 1-46
Author(s):  
Lluis Puig
Keyword(s):  
Solvable Groups ◽  
Conjugacy Classes ◽  
Closed Field ◽  
Characteristic P ◽  
Isomorphism Classes ◽  
Outer Automorphisms ◽  
Odd Order ◽  
The One ◽  
A Finite Group ◽  
The Relationship

The weights for a finite group G with respect to a prime number p were introduced by Jon Alperin, in order to formulate his celebrated conjecture affirming that the number of G-conjugacy classes of weights of G coincides with the number of isomorphism classes of simple kG-modules, where k is an algebraically closed field of characteristic p. Thirty years ago, Tetsuro Okuyama already proved that in the class of p-solvable groups this conjecture holds. In this paper, for the p-solvable groups, on the one hand we exhibit a natural bijection — namely compatible with the action of the group of outer automorphisms of G — between the sets of isomorphism classes of simple kG-modules M and of G-conjugacy classes of weights (R,Y), up to the choice of a polarization. On the other hand, we determine the relationship between a multiplicity module of M and Y. In an Appendix, we show that the bijection defined by Gabriel Navarro for the groups of odd order coincides with our bijection for a particular choice of the polarization.

scholarly journals The Representations of Quantum Double of Dihedral Groups

2013 ◽  
Vol 20 (01) ◽  
pp. 95-108 ◽  
Author(s):  
Jingcheng Dong ◽  
Huixiang Chen
Keyword(s):  
Group Algebra ◽  
Dihedral Group ◽  
Closed Field ◽  
Quantum Double ◽  
Dihedral Groups ◽  
Characteristic P ◽  
Algebraically Closed Field ◽  
Finite Dimensional

Let k be an algebraically closed field of odd characteristic p, and let Dn be the dihedral group of order 2n such that p|2n. Let D(kDn) denote the quantum double of the group algebra kDn. In this paper, we describe the structures of all finite-dimensional indecomposable left D(kDn)-modules, equivalently, of all finite-dimensional indecomposable Yetter-Drinfeld kDn-modules, and classify them.

An 𝔽 p2 -maximal Wiman sextic and its automorphisms

2021 ◽  
Vol 21 (4) ◽  
pp. 451-461
Author(s):  
Massimo Giulietti ◽  
Motoko Kawakita ◽  
Stefano Lia ◽  
Maria Montanucci
Keyword(s):  
Riemann Surface ◽  
Automorphism Group ◽  
Finite Field ◽  
Symmetric Group ◽  
Closed Field ◽  
Complex Field ◽  
Hermitian Curve ◽  
Full Automorphism Group ◽  
Algebraically Closed Field

Abstract In 1895 Wiman introduced the Riemann surface 𝒲 of genus 6 over the complex field ℂ defined by the equation X 6+Y 6+ℨ 6+(X 2+Y 2+ℨ 2)(X 4+Y 4+ℨ 4)−12X 2 Y 2 ℨ 2 = 0, and showed that its full automorphism group is isomorphic to the symmetric group S 5. We show that this holds also over every algebraically closed field 𝕂 of characteristic p ≥ 7. For p = 2, 3 the above polynomial is reducible over 𝕂, and for p = 5 the curve 𝒲 is rational and Aut(𝒲) ≅ PGL(2,𝕂). We also show that Wiman’s 𝔽192 -maximal sextic 𝒲 is not Galois covered by the Hermitian curve H19 over the finite field 𝔽192 .

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