scholarly journals Cogalois theory and drinfeld modules

2019 ◽  
Vol 19 (01) ◽  
pp. 2050001
Author(s):  
Marco Antonio Sánchez–Mirafuentes ◽  
Julio Cesar Salas–Torres ◽  
Gabriel Villa–Salvador
Keyword(s):  
Class Number ◽  
Present Form ◽  
Drinfeld Modules ◽  
Carlitz Module ◽  
Rank One

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].

scholarly journals On Artin's Conjecture for Rank One Drinfeld Modules

2001 ◽  
Vol 88 (1) ◽  
pp. 157-174 ◽  
Author(s):  
Chih-Nung Hsu ◽  
Jing Yu
Keyword(s):  
Drinfeld Modules ◽  
Artin's Conjecture ◽  
Rank One

scholarly journals Rank one Drinfeld modules on hyperelliptic curves

1995 ◽  
Vol 52 (1) ◽  
pp. 85-90
Author(s):  
Sunghan Bae ◽  
Pyung-Lyun Kang
Keyword(s):  
Elliptic Curves ◽  
Recent Work ◽  
Hyperelliptic Curves ◽  
Drinfeld Modules ◽  
Rank One

We extend the recent work of Dummit and Hayes on rank one Drinfeld modules on elliptic curves and hyperelliptic curves in the case that the infinite place is ramified to the case that the infinite place is inert.

scholarly journals On an Erdős–Pomerance conjecture for rank one Drinfeld modules

2015 ◽  
Vol 157 ◽  
pp. 1-36
Author(s):  
Yen-Liang Kuan ◽  
Wentang Kuo ◽  
Wei-Chen Yao
Keyword(s):  
Drinfeld Modules ◽  
Rank One

scholarly journals On primitive roots for rank one Drinfeld modules

2010 ◽  
Vol 130 (2) ◽  
pp. 370-385 ◽  
Author(s):  
Wei-Chen Yao ◽  
Jing Yu
Keyword(s):  
Drinfeld Modules ◽  
Rank One ◽  
Primitive Roots

scholarly journals Non-openness of v-adic Galois representation for A-motives

2020 ◽  
pp. 1-21
Author(s):  
Maike Ella Elisabeth Frantzen
Keyword(s):  
Elliptic Curves ◽  
Function Field ◽  
Galois Representations ◽  
Abelian Varieties ◽  
Galois Representation ◽  
Drinfeld Modules ◽  
Ground Field ◽  
Rank One

Drinfeld modules and [Formula: see text]-motives are the function field analogs of elliptic curves and abelian varieties. For both Drinfeld modules and [Formula: see text]-motives, one can construct their [Formula: see text]-adic Galois representations and ask whether the images are open. For Drinfeld modules, this question has been answered by Richard Pink and his co-authors; however, this question has not been addressed for [Formula: see text]-motives. Here, we clarify the rank-one case for [Formula: see text]-motives and show that the image of Galois is open if and only if the virtual dimension is prime to the characteristic of the ground field.

scholarly journals Refined class number formulas and Kolyvagin systems

2010 ◽  
Vol 147 (1) ◽  
pp. 56-74 ◽  
Author(s):  
Barry Mazur ◽  
Karl Rubin
Keyword(s):  
Class Number ◽  
Class Number Formula ◽  
Kolyvagin Systems ◽  
Rank One ◽  
Number Formula ◽  
Positive Integers

AbstractWe use the theory of Kolyvagin systems to prove (most of) a refined class number formula conjectured by Darmon. We show that, for every odd prime p, each side of Darmon’s conjectured formula (indexed by positive integers n) is ‘almost’ a p-adic Kolyvagin system as n varies. Using the fact that the space of Kolyvagin systems is free of rank one over ℤp, we show that Darmon’s formula for arbitrary n follows from the case n=1, which in turn follows from classical formulas.

scholarly journals PERIODS OF DRINFELD MODULES AND LOCAL SHTUKAS WITH COMPLEX MULTIPLICATION

2018 ◽  
Vol 19 (1) ◽  
pp. 175-208 ◽  
Author(s):  
Urs Hartl ◽  
Rajneesh Kumar Singh
Keyword(s):  
Present Article ◽  
Abelian Varieties ◽  
Complex Multiplication ◽  
Number Fields ◽  
Product Formula ◽  
Drinfeld Modules ◽  
Higher Dimensional ◽  
Carlitz Module ◽  
Logarithmic Derivatives ◽  
Faltings Height

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.

On the Reduction of Rank-One Drinfeld Modules

1991 ◽  
Vol 57 (195) ◽  
pp. 339
Author(s):  
David R. Hayes
Keyword(s):  
Drinfeld Modules ◽  
Rank One

Rank-One Drinfeld Modules on Elliptic Curves

1994 ◽  
Vol 62 (206) ◽  
pp. 875 ◽  
Author(s):  
D. S. Dummit ◽  
David Hayes
Keyword(s):  
Elliptic Curves ◽  
Drinfeld Modules ◽  
Rank One
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