The Absolute Galois Group of a Rational Function Field in Characteristic Zero is a Semi-Direct Product

1984 ◽  
Vol 27 (3) ◽  
pp. 313-315 ◽  
Author(s):  
Lou Van Den Dries ◽  
Paulo Ribenboim
Keyword(s):  
Galois Group ◽  
Rational Function ◽  
Direct Product ◽  
Characteristic Zero ◽  
Function Field ◽  
Profinite Group ◽  
Absolute Galois Group ◽  
Rational Function Field ◽  
Free Profinite Group ◽  
The Absolute

AbstractLet K be a field of characteristic 0 and t an indeterminate. It is shown that the absolute Galois group of K(t) is the semi-direct product of a free profinite group with the absolute Galois group of K.

scholarly journals The differential Galois group of the rational function field

2021 ◽  
Vol 381 ◽  
pp. 107605
Author(s):  
Annette Bachmayr ◽  
David Harbater ◽  
Julia Hartmann ◽  
Michael Wibmer
Keyword(s):  
Galois Group ◽  
Rational Function ◽  
Function Field ◽  
Differential Galois Group ◽  
Rational Function Field

scholarly journals The Sylow subgroups of the absolute Galois group Gal(Q)

2015 ◽  
Vol 284 ◽  
pp. 186-212 ◽  
Author(s):  
Lior Bary-Soroker ◽  
Moshe Jarden ◽  
Danny Neftin
Keyword(s):  
Galois Group ◽  
Absolute Galois Group ◽  
Sylow Subgroups ◽  
The Absolute

Zeta functions of twisted modular curves

2006 ◽  
Vol 80 (1) ◽  
pp. 89-103 ◽  
Author(s):  
Cristian Virdol
Keyword(s):  
Zeta Function ◽  
Galois Group ◽  
Complex Plane ◽  
Zeta Functions ◽  
Modular Curves ◽  
Absolute Galois Group ◽  
Modular Curve ◽  
The Absolute

AbstractIn this paper we compute and continue meromorphically to the whole complex plane the zeta function for twisted modular curves. The twist of the modular curve is done by a modprepresentation of the absolute Galois group.

scholarly journals On L-Functions of Twisted 3-Dimensional Quaternionic Shimura Varieties

2008 ◽  
Vol 190 ◽  
pp. 87-104
Author(s):  
Cristian Virdol
Keyword(s):  
Galois Group ◽  
Complex Plane ◽  
Zeta Functions ◽  
Shimura Varieties ◽  
Absolute Galois Group ◽  
3 Dimensional ◽  
Entire Complex ◽  
The Absolute

In this paper we compute and continue meromorphically to the entire complex plane the zeta functions of twisted quaternionic Shimura varieties of dimension 3. The twist of the quaternionic Shimura varieties is done by a mod ℘ representation of the absolute Galois group.

scholarly journals Indépendance algébrique de logarithmes en caractéristique P

2006 ◽  
Vol 74 (3) ◽  
pp. 461-470 ◽  
Author(s):  
Laurent Denis
Keyword(s):  
Zeta Function ◽  
Rational Function ◽  
Riemann Zeta Function ◽  
Function Field ◽  
Fundamental Period ◽  
Rational Function Field ◽  
Linearly Independent ◽  
The Riemann Zeta Function ◽  
Carlitz Module ◽  
Riemann Zeta

Let k be the rational function field over the field with q elements with characteristic p. Since the work of Carlitz we know in this situation the function ζ analog of the Riemann zeta function and the function Logφ analog of the usual logarithm. We will show two main results. Firstly, if ξ denotes the fundamental period of Carlitz module, we prove that ξ, ζ(1),…, ζ(p – 2) are algebraically independent over k. Secondly if α1,…, αn are rational elements (of degree less than q/(q − 1) to ensure convergence of the logarithm) such that Logφ α1,…, Logφ αn are linearly independent over k then they are algebraically independent over k. The point is to find suitable functions taking these values and for which Mahler's method can be used.

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