On p-adic L-series, p-adic cohomology and class field theory

2017 ◽  
Vol 2017 (732) ◽  
pp. 55-83 ◽  
Author(s):  
David Burns ◽  
Daniel Macias Castillo
Keyword(s):  
Field Theory ◽  
Class Field Theory ◽  
Iwasawa Theory ◽  
Class Groups ◽  
Class Field ◽  
Weil Theorem ◽  
Positive Integers ◽  
Natural Families

Abstract We establish several close links between the Galois structures of a range of arithmetic modules including certain natural families of ray class groups, the values at strictly positive integers of p-adic Artin L-series, the Shafarevich–Weil Theorem and the conjectural surjectivity of certain norm maps in cyclotomic {\mathbb{Z}_{p}} -extensions. Non-commutative Iwasawa theory and the theory of organising matrices play a key role in our approach.

scholarly journals Capitulation in Class Field Extensions of Type (p, p)

1980 ◽  
Vol 32 (5) ◽  
pp. 1229-1243 ◽  
Author(s):  
S. M. Chang ◽  
R. Foote
Keyword(s):  
Field Theory ◽  
Abelian Group ◽  
Number Field ◽  
Class Field Theory ◽  
Class Groups ◽  
Class Field ◽  
Field Extensions ◽  
Class Fields ◽  
Image Position ◽  
Hilbert Class Field

Let K be a number field, K(1) its Hilbert class field, that is, the maximal abelian unramified extension of K, let K(2) be the Hilbert class field of K(1), and let G = Gal(K(2)/K) (alternatively, for p a prime the first and second p class fields enjoy properties analogous to those of the respective class fields discussed in this introduction; the particulars may be found surrounding Lemma 2). Since G/G’ is the largest abelian quotient of G, G/G′ = Gal (K(1))/K) and so G’ is the abelian group Gal(K(2)K(1)); moreover, class field theory provides (Artin) maps φK, φK(1) which are isomorphisms of the class groups Ck, Ck(1) onto G/G′, G′ respectively. In the remarkable paper [1] E. Artin computed the composition VG′where e is the homomorphism induced on the class groups by extending ideals of K to ideals of K(1), and he gave a formula for computing VG′, the now familiar transfer (Verlagerung) homomorphism, in terms of the group G alone (see Lemma 1).

scholarly journals Cohomological Approach to Class Field Theory in Arithmetic Topology

2019 ◽  
Vol 71 (4) ◽  
pp. 891-935 ◽  
Author(s):  
Tomoki Mihara
Keyword(s):  
Field Theory ◽  
Class Group ◽  
Existence Theorems ◽  
Multiplicative Group ◽  
Three Dimensional ◽  
Class Field Theory ◽  
Class Groups ◽  
Class Field ◽  
Abstract Class

AbstractWe establish class field theory for three-dimensional manifolds and knots. For this purpose, we formulate analogues of the multiplicative group, the idèle class group, and ray class groups in a cocycle-theoretic way. Following the arguments in abstract class field theory, we construct reciprocity maps and verify the existence theorems.

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