scholarly journals On the rational K2 of a curve of GL2 type over a global field of positive characteristic

2014 ◽  
Vol 14 (2) ◽  
pp. 313-342
Author(s):  
Masataka Chida ◽  
Satoshi Kondo ◽  
Takuya Yamauchi
Keyword(s):  
Positive Characteristic ◽  
Smooth Curve ◽  
Global Field ◽  
Integral Model ◽  
Large Rank ◽  
Boundary Map ◽  
Localization Sequence ◽  
Drinfeld Modular Curves ◽  
Rank 2 ◽  
Field Of Positive Characteristic

AbstractIf is an integral model of a smooth curve X over a global field k, there is a localization sequence comparing the K-theory of and X. We show that K1 () injects into K1(X) rationally, by showing that the previous boundary map in the localization sequence is rationally a surjection, for X of “GL2 type” and k of positive characteristic not 2. Examples are given to show that the relative G1 term can have large rank. Examples of such curves include non-isotrivial elliptic curves, Drinfeld modular curves, and the moduli of -elliptic sheaves of rank 2.

scholarly journals On a purely inseparable analogue of the Abhyankar conjecture for affine curves

2018 ◽  
Vol 154 (8) ◽  
pp. 1633-1658
Author(s):  
Shusuke Otabe
Keyword(s):  
Fundamental Group ◽  
Positive Characteristic ◽  
Smooth Curve ◽  
Group Scheme ◽  
Partial Answer ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Affine Curves ◽  
Finite Quotients ◽  
Field Of Positive Characteristic

Let$U$be an affine smooth curve defined over an algebraically closed field of positive characteristic. The Abhyankar conjecture (proved by Raynaud and Harbater in 1994) describes the set of finite quotients of Grothendieck’s étale fundamental group$\unicode[STIX]{x1D70B}_{1}^{\acute{\text{e}}\text{t}}(U)$. In this paper, we consider a purely inseparable analogue of this problem, formulated in terms of Nori’s profinite fundamental group scheme$\unicode[STIX]{x1D70B}^{N}(U)$, and give a partial answer to it.

THE RAMIFICATION GROUPS AND DIFFERENT OF A COMPOSITUM OF ARTIN–SCHREIER EXTENSIONS

2010 ◽  
Vol 06 (07) ◽  
pp. 1541-1564 ◽  
Author(s):  
QINGQUAN WU ◽  
RENATE SCHEIDLER
Keyword(s):  
Class Number ◽  
Function Field ◽  
Positive Characteristic ◽  
Field Extension ◽  
Constant Field ◽  
Divisor Class ◽  
Characteristic P ◽  
Similar Nature ◽  
Field Of Positive Characteristic ◽  
The Ideal

Let K be a function field over a perfect constant field of positive characteristic p, and L the compositum of n (degree p) Artin–Schreier extensions of K. Then much of the behavior of the degree pn extension L/K is determined by the behavior of the degree p intermediate extensions M/K. For example, we prove that a place of K totally ramifies/is inert/splits completely in L if and only if it totally ramifies/is inert/splits completely in every M. Examples are provided to show that all possible decompositions are in fact possible; in particular, a place can be inert in a non-cyclic Galois function field extension, which is impossible in the case of a number field. Moreover, we give an explicit closed form description of all the different exponents in L/K in terms of those in all the M/K. Results of a similar nature are given for the genus, the regulator, the ideal class number and the divisor class number. In addition, for the case n = 2, we provide an explicit description of the ramification group filtration of L/K.

The Hilbert series of the irreducible quotient of the polynomial representation of the rational Cherednik algebra of type An−1 in characteristic p for p|n − 1

2021 ◽  
pp. 2250191
Author(s):  
Merrick Cai ◽  
Daniil Kalinov
Keyword(s):  
Hilbert Series ◽  
Positive Characteristic ◽  
Closed Field ◽  
Simple Criterion ◽  
Characteristic P ◽  
Polynomial Representation ◽  
Cherednik Algebra ◽  
Rational Cherednik Algebra ◽  
Polynomial Formula ◽  
Field Of Positive Characteristic

In this paper, we study the irreducible quotient [Formula: see text] of the polynomial representation of the rational Cherednik algebra [Formula: see text] of type [Formula: see text] over an algebraically closed field of positive characteristic [Formula: see text] where [Formula: see text]. In the [Formula: see text] case, for all [Formula: see text] we give a complete description of the polynomials in the maximal proper graded submodule [Formula: see text], the kernel of the contravariant form [Formula: see text], and subsequently find the Hilbert series of the irreducible quotient [Formula: see text]. In the [Formula: see text] case, we give a complete description of the polynomials in [Formula: see text] when the characteristic [Formula: see text] and [Formula: see text] is transcendental over [Formula: see text], and compute the Hilbert series of the irreducible quotient [Formula: see text]. In doing so, we prove a conjecture due to Etingof and Rains completely for [Formula: see text], and also for any [Formula: see text] and [Formula: see text]. Furthermore, for [Formula: see text], we prove a simple criterion to determine whether a given polynomial [Formula: see text] lies in [Formula: see text] for all [Formula: see text] with [Formula: see text] and [Formula: see text] fixed.

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