Cogalois theory and drinfeld modules

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].

PERIODS OF DRINFELD MODULES AND LOCAL SHTUKAS WITH COMPLEX MULTIPLICATION

Colmez [Périodes des variétés abéliennes a multiplication complexe,Ann. of Math. (2)138(3) (1993), 625–683; available athttp://www.math.jussieu.fr/∼colmez] conjectured a product formula for periods of abelian varieties over number fields with complex multiplication and proved it in some cases. His conjecture is equivalent to a formula for the Faltings height of CM abelian varieties in terms of the logarithmic derivatives at$s=0$of certain Artin$L$-functions. In a series of articles we investigate the analog of Colmez’s theory in the arithmetic of function fields. There abelian varieties are replaced by Drinfeld modules and their higher-dimensional generalizations, so-called$A$-motives. In the present article we prove the product formula for the Carlitz module and we compute the valuations of the periods of a CM$A$-motive at all finite places in terms of Artin$L$-series. The latter is achieved by investigating the local shtukas associated with the$A$-motive.

Indépendance algébrique de logarithmes en caractéristique P

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.