Circular units of real abelian fields with four ramified primes

Vladimír Sedláček

Displaying similar documents to “Circular units of real abelian fields with four ramified primes”

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...

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...

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...

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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Some new maps and ideals in classical Iwasawa theory with applications

David Solomon (2014)

Acta Arithmetica

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We introduce a new ideal of the p-adic Galois group-ring associated to a real abelian field and a related ideal for imaginary abelian fields, Both result from an equivariant, Kummer-type pairing applied to Stark units in a $ℤ_{p}$ -tower of abelian fields, and is linked by explicit reciprocity to a third ideal studied more generally in [D. Solomon, Acta Arith. 143 (2010)]. This leads to a new and unifying framework for the Iwasawa theory of such fields including a real analogue of Stickelberger’s...

Capitulation and transfer kernels

K. W. Gruenberg, A. Weiss (2000)

Journal de théorie des nombres de Bordeaux

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If $K / k$ is a finite Galois extension of number fields with Galois group $G$ , then the kernel of the capitulation map $C l_{k} \to C l_{K}$ of ideal class groups is isomorphic to the kernel $X (H)$ of the transfer map $H / H^{'} \to A,$ where $H = Gal (\tilde{K} / k), A = Gal (\tilde{K} / K)$ and $\tilde{K}$ is the Hilbert class field of $K$ . H. Suzuki proved that when $G$ is abelian, $| G |$ divides $| X (H) |$ . We call a finite abelian group $X$ a transfer kernel for $G$ if $X ≅ X (H)$ for some group extension $A ↪ H ↠ G$ . After characterizing transfer kernels in terms of integral representations of $G$ , we show that $X$ is a transfer...

On sequences over a finite abelian group with zero-sum subsequences of forbidden lengths

Weidong Gao, Yuanlin Li, Pingping Zhao, Jujuan Zhuang (2016)

Colloquium Mathematicae

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Let G be an additive finite abelian group. For every positive integer ℓ, let $d i s c_{ℓ} (G)$ be the smallest positive integer t such that each sequence S over G of length |S| ≥ t has a nonempty zero-sum subsequence of length not equal to ℓ. In this paper, we determine $d i s c_{ℓ} (G)$ for certain finite groups, including cyclic groups, the groups $G = C ₂ \oplus C_{2 m}$ and elementary abelian 2-groups. Following Girard, we define disc(G) as the smallest positive integer t such that every sequence S over G with |S| ≥ t has nonempty zero-sum...

Quaternion extensions with restricted ramification

Peter Schmid (2014)

Acta Arithmetica

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In any normal number field having Q₈, the quaternion group of order 8, as Galois group over the rationals, at least two finite primes must ramify. The classical example by Dedekind of such a field is extraordinary in that it is totally real and only the primes 2 and 3 are ramified. In this note we describe in detail all Q₈-fields over the rationals where only two (finite) primes are ramified. We also show that, for any integer n>3 and any prime $p \equiv 1 (m o d 2^{n - 1})$ , there exist unique real and complex...

On the structure of the 2-Iwasawa module of some number fields of degree 16

Idriss Jerrari, Abdelmalek Azizi (2022)

Czechoslovak Mathematical Journal

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Let $K$ be an imaginary cyclic quartic number field whose 2-class group is of type $(2, 2, 2)$ , i.e., isomorphic to $ℤ / 2 ℤ \times ℤ / 2 ℤ \times ℤ / 2 ℤ$ . The aim of this paper is to determine the structure of the Iwasawa module of the genus field $K^{(*)}$ of $K$ .

On the subfields of cyclotomic function fields

Zhengjun Zhao, Xia Wu (2013)

Czechoslovak Mathematical Journal

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Let $K = 𝔽_{q} (T)$ be the rational function field over a finite field of $q$ elements. For any polynomial $f (T) \in 𝔽_{q} [T]$ with positive degree, denote by $Λ_{f}$ the torsion points of the Carlitz module for the polynomial ring $𝔽_{q} [T]$ . In this short paper, we will determine an explicit formula for the analytic class number for the unique subfield $M$ of the cyclotomic function field $K (Λ_{P})$ of degree $k$ over $𝔽_{q} (T)$ , where $P \in 𝔽_{q} [T]$ is an irreducible polynomial of positive degree and $k > 1$ is a positive divisor of $q - 1$ . A formula for the analytic class...

Subsequence sums of zero-sum free sequences over finite abelian groups

Yongke Qu, Xingwu Xia, Lin Xue, Qinghai Zhong (2015)

Colloquium Mathematicae

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Let G be a finite abelian group of rank r and let X be a zero-sum free sequence over G whose support supp(X) generates G. In 2009, Pixton proved that $| Σ (X) | \geq 2^{r - 1} (| X | - r + 2) - 1$ for r ≤ 3. We show that this result also holds for abelian groups G of rank 4 if the smallest prime p dividing |G| satisfies p ≥ 13.