Some new maps and ideals in classical Iwasawa theory with applications

David Solomon

Displaying similar documents to “Some new maps and ideals in classical Iwasawa theory with applications”

On the classgroups of imaginary abelian fields

David Solomon (1990)

Annales de l'institut Fourier

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Let $p$ be an odd prime, $χ$ an odd, $p$ -adic Dirichlet character and $K$ the cyclic imaginary extension of $Q$ associated to $χ$ . We define a “ $χ$ -part” of the Sylow $p$ -subgroup of the class group of $K$ and prove a result relating its $p$ -divisibility to that of the generalized Bernoulli number $B_{1, χ^{- 1}}$ . This uses the results of Mazur and Wiles in Iwasawa theory over $Q$ . The more difficult case, in which $p$ divides the order of $χ$ is our chief concern. In this case the result is new and confirms an earlier conjecture...

On the Stark-Shintani units and the ideal class groups in the cyclotomic $ℤ_{p}$ -extensions of class fields over real quadratic fields

Tsuyoshi Itoh (2002)

Acta Arithmetica

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The cyclic subfield integer index

Bart de Smit (2000)

Journal de théorie des nombres de Bordeaux

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In this note we consider the index in the ring of integers of an abelian extension of a number field $K$ of the additive subgroup generated by integers which lie in subfields that are cyclic over $K$ . This index is finite, it only depends on the Galois group and the degree of $K$ , and we give an explicit combinatorial formula for it. When generalizing to more general Dedekind domains, a correction term can be needed if there is an inseparable extension of residue fields. We identify this correction...

Relative Galois module structure of integers of abelian fields

Nigel P. Byott, Günter Lettl (1996)

Journal de théorie des nombres de Bordeaux

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Let $L / K$ be an extension of algebraic number fields, where $L$ is abelian over $ℚ$ . In this paper we give an explicit description of the associated order $𝒜_{L / K}$ of this extension when $K$ is a cyclotomic field, and prove that $o_{L}$ , the ring of integers of $L$ , is then isomorphic to $𝒜_{L / K}$ . This generalizes previous results of Leopoldt, Chan Lim and Bley. Furthermore we show that $𝒜_{L / K}$ is the maximal order if $L / K$ is a cyclic and totally wildly ramified extension which is linearly disjoint to $ℚ^{(m^{'})} / K$ , where $m^{'}$ is the conductor...

The circular units and the Stickelberger ideal of a cyclotomic field revisited

Radan Kučera (2016)

Acta Arithmetica

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The aim of this paper is a new construction of bases of the group of circular units and of the Stickelberger ideal for a family of abelian fields containing all cyclotomic fields, namely for any compositum of imaginary abelian fields, each ramified only at one prime. In contrast to the previous papers on this topic our approach consists in an explicit construction of Ennola relations. This gives an explicit description of the torsion parts of odd and even universal ordinary distributions,...

On the trace map between absolutely abelian number fields of equal conductor

Henri Johnston (2006)

Acta Arithmetica

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On the ideal class groups of the maximal real subfields of number fields with all roots of unity

Masato Kurihara (1999)

Journal of the European Mathematical Society

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In this paper, for a totally real number field $k$ we show the ideal class group of $k {(\cup_{n > 0} μ_{n})}^{+}$ is trivial. We also study the $p$ -component of the ideal class group of the cyclotomic $𝐙_{p}$ -extension.

On a generalization of Abelian sequential groups

Saak S. Gabriyelyan (2013)

Fundamenta Mathematicae

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Let (G,τ) be a Hausdorff Abelian topological group. It is called an s-group (resp. a bs-group) if there is a set S of sequences in G such that τ is the finest Hausdorff (resp. precompact) group topology on G in which every sequence of S converges to zero. Characterizations of Abelian s- and bs-groups are given. If (G,τ) is a maximally almost periodic (MAP) Abelian s-group, then its Pontryagin dual group ${(G, τ)}^{\land}$ is a dense -closed subgroup of the compact group ${(G_{d})}^{\land}$ , where $G_{d}$ is the group G with...

A note on group algebras of $p$ -primary abelian groups

William Ullery (1995)

Commentationes Mathematicae Universitatis Carolinae

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Suppose $p$ is a prime number and $R$ is a commutative ring with unity of characteristic 0 in which $p$ is not a unit. Assume that $G$ and $H$ are $p$ -primary abelian groups such that the respective group algebras $R G$ and $R H$ are $R$ -isomorphic. Under certain restrictions on the ideal structure of $R$ , it is shown that $G$ and $H$ are isomorphic.

On the Galois structure of the square root of the codifferent

D. Burns (1991)

Journal de théorie des nombres de Bordeaux

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Let $L$ be a finite abelian extension of $ℚ$ , with $𝒪_{L}$ the ring of algebraic integers of $L$ . We investigate the Galois structure of the unique fractional $𝒪_{L}$ -ideal which (if it exists) is unimodular with respect to the trace form of $L / ℚ$ .

When does the Katětov order imply that one ideal extends the other?

Paweł Barbarski, Rafał Filipów, Nikodem Mrożek, Piotr Szuca (2013)

Colloquium Mathematicae

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We consider the Katětov order between ideals of subsets of natural numbers (" $\leq_{K}$ ") and its stronger variant-containing an isomorphic ideal ("⊑ "). In particular, we are interested in ideals for which $\leq_{K} \Rightarrow ⊑$ for every ideal . We find examples of ideals with this property and show how this property can be used to reformulate some problems known from the literature in terms of the Katětov order instead of the order "⊑ " (and vice versa).