On the generalized Carlitz module

In this paper, we generalize the results of [M. Sánchez-Mirafuentes and G. Villa–Salvador, Radical extensions for the Carlitz module, J. Algebra 398 (2014) 284–302] to rank one Drinfeld modules with class number one. We show that, in the present form, there does not exist a cogalois theory for Drinfeld modules of rank or class number larger than one. We also consider the torsion of the Carlitz module for the extension [Formula: see text].

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Lubin-Tate extensions and Carlitz module over a projective line: an explicit connection

Чебышевский сборник ◽

10.22405/2226-8383-2021-22-2-90-103 ◽

2021 ◽

Vol 22 (2) ◽

pp. 90-103

Author(s):

Nikita Vyacheslavovich Elizarov ◽

Sergei Vladimirovich Vostokov

Keyword(s):

Projective Line ◽

Carlitz Module

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Indépendance algébrique de logarithmes en caractéristique P

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700040508 ◽

2006 ◽

Vol 74 (3) ◽

pp. 461-470 ◽

Cited By ~ 1

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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