An arithmetic proof of Pop`s Theorem concerning Galois groups of function fields over number fields.

1996 ◽  
Vol 1996 (478) ◽  
pp. 107-126
Keyword(s):  
Function Fields ◽  
Number Fields ◽  
Galois Groups

scholarly journals Non-abelian Cohen–Lenstra heuristics over function fields

2017 ◽  
Vol 153 (7) ◽  
pp. 1372-1390 ◽  
Author(s):  
Nigel Boston ◽  
Melanie Matchett Wood
Keyword(s):  
Function Fields ◽  
Quadratic Function ◽  
Number Fields ◽  
Galois Groups ◽  
Quadratic Number ◽  
Quadratic Extensions ◽  
Quadratic Number Fields ◽  
Imaginary Quadratic Number ◽  
Unique Distribution ◽  
Quadratic Function Fields

Boston, Bush and Hajir have developed heuristics, extending the Cohen–Lenstra heuristics, that conjecture the distribution of the Galois groups of the maximal unramified pro-$p$extensions of imaginary quadratic number fields for$p$an odd prime. In this paper, we find the moments of their proposed distribution, and further prove there is a unique distribution with those moments. Further, we show that in the function field analog, for imaginary quadratic extensions of$\mathbb{F}_{q}(t)$, the Galois groups of the maximal unramified pro-$p$extensions, as$q\rightarrow \infty$, have the moments predicted by the Boston, Bush and Hajir heuristics. In fact, we determine the moments of the Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic function fields, leading to a conjecture on Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic number fields.

MILD PRO-p-GROUPS AS GALOIS GROUPS OVER GLOBAL FIELDS

2009 ◽  
Vol 05 (05) ◽  
pp. 779-795 ◽  
Author(s):  
LANDRY SALLE
Keyword(s):  
Galois Group ◽  
Cohomology Group ◽  
Function Fields ◽  
Number Fields ◽  
Galois Groups ◽  
Global Field ◽  
Global Fields ◽  
Global Principle

This paper is devoted to finding new examples of mild pro-p-groups as Galois groups over global fields, following the work of Labute ([6]). We focus on the Galois group [Formula: see text] of the maximal T-split S-ramified pro-p-extension of a global field k. We first retrieve some facts on presentations of such a group, including a study of the local-global principle for the cohomology group [Formula: see text], then we show separately in the case of function fields and in the case of number fields how it can be used to find some mild pro-p-groups.

Reciprocity Law for Compatible Systems of Abelian mod p Galois Representations

2005 ◽  
Vol 57 (6) ◽  
pp. 1215-1223 ◽  
Author(s):  
Chandrashekhar Khare
Keyword(s):  
Galois Representations ◽  
Function Fields ◽  
Arbitrary Dimension ◽  
Number Fields ◽  
Galois Groups ◽  
Reciprocity Law ◽  
Mod P

AbstractThe main result of the paper is a reciprocity law which proves that compatible systems of semisimple, abelianmod p representations (of arbitrary dimension) of absolute Galois groups of number fields, arise from Hecke characters. In the last section analogs for Galois groups of function fields of these results are explored, and a question is raised whose answer seems to require developments in transcendence theory in characteristic p.

scholarly journals FINITE GROUPS AS GALOIS GROUPS OF FUNCTION FIELDS WITH INFINITE FIELD OF CONSTANTS

2010 ◽  
Vol 88 (3) ◽  
pp. 301-312
Author(s):  
C. ÁLVAREZ-GARCÍA ◽  
G. VILLA-SALVADOR
Keyword(s):  
Finite Groups ◽  
Function Field ◽  
Function Fields ◽  
Galois Groups ◽  
Infinite Field ◽  
Separable Extension ◽  
Separable Extensions

AbstractLetE/kbe a function field over an infinite field of constants. Assume thatE/k(x) is a separable extension of degree greater than one such that there exists a place of degree one ofk(x) ramified inE. LetK/kbe a function field. We prove that there exist infinitely many nonisomorphic separable extensionsL/Ksuch that [L:K]=[E:k(x)] andAutkL=AutKL≅Autk(x)E.

scholarly journals Prodihedral Groups as Galois Groups over Number Fields

1996 ◽  
Vol 60 (2) ◽  
pp. 332-372 ◽  
Author(s):  
W.-D. Geyer ◽  
C.U. Jensen
Keyword(s):  
Number Fields ◽  
Galois Groups

scholarly journals Ã5 and Ã7 are Galois groups over number fields

1986 ◽  
Vol 104 (2) ◽  
pp. 231-260 ◽  
Author(s):  
Walter Feit
Keyword(s):  
Number Fields ◽  
Galois Groups

Introduction

2020 ◽  
pp. 1-6
Author(s):  
Peter Scholze ◽  
Jared Weinstein
Keyword(s):  
Moduli Spaces ◽  
Function Fields ◽  
Number Fields ◽  
Important Special Case ◽  
Langlands Correspondence ◽  
Key Innovation ◽  
Perfectoid Spaces ◽  
Definition Of ◽  
Special Case

This introductory chapter provides an overview of Drinfeld's work on the global Langlands correspondence over function fields. Whereas the global Langlands correspondence is largely open in the case of number fields K, it is a theorem for function fields, due to Drinfeld and L. Lafforgue. The key innovation in this case is Drinfeld's notion of an X-shtuka (or simply shtuka). The Langlands correspondence for X is obtained by studying moduli spaces of shtukas. A large part of this course is about the definition of perfectoid spaces and diamonds. There is an important special case where the moduli spaces of shtukas are classical rigid-analytic spaces. This is the case of local Shimura varieties. Some examples of these are the Rapoport-Zink spaces.

Sign in / Sign up

Export Citation Format

Share Document