Some remarks on Leopoldt's conjecture

1992 ◽  
Vol 77 (1) ◽  
pp. 405-414
Author(s):  
Tsutomu Shimada
Keyword(s):  
Leopoldt's Conjecture

scholarly journals K-groups with finite coefficients and arithmetic

2003 ◽  
Vol 93 (1) ◽  
pp. 41
Author(s):  
Paul Arne Østvær
Keyword(s):  
Leopoldt's Conjecture ◽  
K Theory

In this paper we prove rank formulas for the even K-groups of number rings and relate Leopoldt's conjecture to K-theory. These results follow from a computation of the higher K-groups with finite coefficients.

scholarly journals Stark units and the main conjectures for totally real fields

2009 ◽  
Vol 145 (5) ◽  
pp. 1163-1195 ◽  
Author(s):  
Kâzım Büyükboduk
Keyword(s):  
System Theory ◽  
Iwasawa Theory ◽  
The Core ◽  
Kolyvagin Systems ◽  
Leopoldt's Conjecture ◽  
Stark Units ◽  
Stark Conjecture ◽  
Totally Real ◽  
Totally Real Fields ◽  
Iwasawa Algebra

AbstractThe main theorem of the author’s thesis suggests that it should be possible to lift the Kolyvagin systems of Stark units, constructed by the author in an earlier paper, to a Kolyvagin system over the cyclotomic Iwasawa algebra. In this paper, we verify that this is indeed the case. This construction of Kolyvagin systems over the cyclotomic Iwasawa algebra from Stark units provides the first example towards a more systematic study of Kolyvagin system theory over an Iwasawa algebra when the core Selmer rank (in the sense of Mazur and Rubin) is greater than one. As a result of this construction, we reduce the main conjectures of Iwasawa theory for totally real fields to a statement in the context of local Iwasawa theory, assuming the truth of the Rubin–Stark conjecture and Leopoldt’s conjecture. This statement in the local Iwasawa theory context turns out to be interesting in its own right, as it suggests a relation between the solutions to p-adic and complex Stark conjectures.

Kummer's and Iwasawa's Version of Leopoldt's Conjecture

1988 ◽  
Vol 31 (3) ◽  
pp. 338-346 ◽  
Author(s):  
Jonathan W. Sands
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Class Numbers ◽  
Root Of Unity ◽  
Leopoldt's Conjecture ◽  
Solvable Extension ◽  
Cyclotomic Units

AbstractWe present a refinement of Iwasawa's approach to Leopoldt's conjecture on the non-vanishing of the p-adic regulator of an algebraic number field K. As an application, the conjecture for K implies the conjecture for a solvable extension L of degree g over K if g is relatively prime to p — 1 and p does not divide g, the discriminant of K, and the quotient of class numbers where is a primitive pth root of unity. This can be viewed as generalizing a theorem of Kummer on cyclotomic units.

A simple remark on Leopoldt’s conjecture

2001 ◽  
pp. 862-870 ◽  
Author(s):  
Ichiro Satake ◽  
Genjiro Fujisaki ◽  
Kazuya Kato ◽  
Masato Kurihara ◽  
Shoichi Nakajima
Keyword(s):  
Leopoldt's Conjecture

scholarly journals Leopoldt's conjecture and Reiner's theorem

1984 ◽  
Vol 36 (1) ◽  
pp. 47-52 ◽  
Author(s):  
Hiroo MIKI ◽  
Hirotaka SATO
Keyword(s):  
Leopoldt's Conjecture
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