On a functorial property of power residue symbols. Erratum : Solution of the congruence subgroup problem for $SL_n (n \ge 3)$ and $Sp_{2n} (n \ge 2)$

Jean-Pierre Serre

Displaying similar documents to “On a functorial property of power residue symbols. Erratum : Solution of the congruence subgroup problem for $S L_{n} (n \geq 3)$ and $S p_{2 n} (n \geq 2)$ ”

On $2$ -class field towers of imaginary quadratic number fields

Franz Lemmermeyer (1994)

Journal de théorie des nombres de Bordeaux

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For a number field $k$ , let $k^{1}$ denote its Hilbert $2$ -class field, and put $k^{2} = {(k^{1})}^{1}$ . We will determine all imaginary quadratic number fields $k$ such that $G = G a l (k^{2} / k)$ is abelian or metacyclic, and we will give $G$ in terms of generators and relations.

Pólya fields and Pólya numbers

Amandine Leriche (2010)

Actes des rencontres du CIRM

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A number field $K$ , with ring of integers $𝒪_{K}$ , is said to be a Pólya field if the $𝒪_{K}$ -algebra formed by the integer-valued polynomials on $𝒪_{K}$ admits a regular basis. In a first part, we focus on fields with degree less than six which are Pólya fields. It is known that a field $K$ is a Pólya field if certain characteristic ideals are principal. Analogously to the classical embedding problem, we consider the embedding of $K$ in a Pólya field. We give a positive answer to this embedding problem by...

Hilbert symbols, class groups and quaternion algebras

Ted Chinburg, Eduardo Friedman (2000)

Journal de théorie des nombres de Bordeaux

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Let $B$ be a quaternion algebra over a number field $k$ . To a pair of Hilbert symbols ${a, b}$ and ${c, d}$ for $B$ we associate an invariant $ρ = ρ_{R} ([𝒟 (a, b)], [𝒟 (c, d)])$ in a quotient of the narrow ideal class group of $k$ . This invariant arises from the study of finite subgroups of maximal arithmetic kleinian groups. It measures the distance between orders $𝒟 (a, b)$ and $𝒟 (c, d)$ in $B$ associated to ${a, b}$ and ${c, d} .$ If $a = c$ , we compute $ρ_{R} ([𝒟 (a, b)], [𝒟 (c, d)])$ by means of arithmetic in the field $k (\sqrt{a}) .$ The problem of extending this algorithm to the general case leads to studying a finite...

Representation of prime powers in arithmetical progressions by binary quadratic forms

Franz Halter-Koch (2003)

Journal de théorie des nombres de Bordeaux

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Let $Γ$ be a set of binary quadratic forms of the same discriminant, $Δ$ a set of arithmetical progressions and $m$ a positive integer. We investigate the representability of prime powers $p^{m}$ lying in some progression from $Δ$ by some form from $Γ$ .

Maximal unramified extensions of imaginary quadratic number fields of small conductors, II

Ken Yamamura (2001)

Journal de théorie des nombres de Bordeaux

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In the previous paper [15], we determined the structure of the Galois groups $Gal (K_{u r} / K)$ of the maximal unramified extensions $K_{u r}$ of imaginary quadratic number fields $K$ of conductors $≦ 1000$ under the Generalized Riemann Hypothesis (GRH) except for 23 fields (these are of conductors $≧ 723$ ) and give a table of $Gal (K_{u r} / K)$ . We update the table (under GRH). For 19 exceptional fields $K$ of them, we determine $Gal (K_{u r} / K)$ . In particular, for $K = 𝐐 (\sqrt{- 856})$ , we obtain $Gal (K_{u r} / K) ≅ \tilde{S_{4}} \times C_{5} and K_{u r} = K_{4}$ , the fourth Hilbert class field of $K$ . This is the first example of a number...

Artin's primitive root conjecture for quadratic fields

Hans Roskam (2002)

Journal de théorie des nombres de Bordeaux

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Fix an element $α$ in a quadratic field $K$ . Define $S$ as the set of rational primes $p$ , for which $α$ has maximal order modulo $p$ . Under the assumption of the generalized Riemann hypothesis, we show that $S$ has a density. Moreover, we give necessary and sufficient conditions for the density of $S$ to be positive.

On the 2-primary part of K₂ of rings of integers in certain quadratic number fields

A. Vazzana (1997)

Acta Arithmetica

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1. Introduction. For quadratic fields whose discriminant has few prime divisors, there are explicit formulas for the 4-rank of $K ₂_{E}$ . For quadratic fields whose discriminant has arbitrarily many prime divisors, the formulas are less explicit. In this paper we will study fields of the form $ℚ (\sqrt (p ₁ . . . p_{k}))$ , where the primes $p_{i}$ are all congruent to 1 mod 8. We will prove a theorem conjectured by Conner and Hurrelbrink which examines under what conditions the 4-rank of $K ₂_{E}$ is zero for such fields. In the course...

Weber's class invariants revisited

Reinhard Schertz (2002)

Journal de théorie des nombres de Bordeaux

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Let $K$ be a quadratic imaginary number field of discriminant $d$ . For $t \in ℕ$ let $𝔒_{t}$ denote the order of conductor $t$ in $K$ and $j (𝔒_{t})$ its modular invariant which is known to generate the ring class field modulo $t$ over $K$ . The coefficients of the minimal equation of $j (𝔒_{t})$ being quite large Weber considered in [We] the functions $f, f_{1}, f_{2}, γ_{2}, γ_{3}$ defined below and thereby obtained simpler generators of the ring class fields. Later on the singular values of these functions played a crucial role in Heegner’s solution [He] of...

The class number one problem for some non-abelian normal CM-fields of degree $24$

F. Lemmermeyer, S. Louboutin, R. Okazaki (1999)

Journal de théorie des nombres de Bordeaux

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We determine all the non-abelian normal CM-fields of degree 24 with class number one, provided that the Galois group of their maximal real subfields is isomorphic to $𝒜_{4}$ , the alternating group of degree $4$ and order $12$ . There are two such fields with Galois group $𝒜_{4} \times 𝒞_{2}$ (see Theorem 14) and at most one with Galois group SL $_{2} (𝔽_{3})$ (see Theorem 18); if the generalized Riemann hypothesis is true, then this last field has class number $1$ .

Initial layers of $Z_{l}$ -extensions of complex quadratic fields

J. E. Carroll, H. Kisilevsky (1976)

Compositio Mathematica

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Note on the Hilbert 2-class field tower

Abdelmalek Azizi, Mohamed Mahmoud Chems-Eddin, Abdelkader Zekhnini (2022)

Mathematica Bohemica

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Let $k$ be a number field with a 2-class group isomorphic to the Klein four-group. The aim of this paper is to give a characterization of capitulation types using group properties. Furthermore, as applications, we determine the structure of the second 2-class groups of some special Dirichlet fields $𝕜 = ℚ (\sqrt{d}, \sqrt{- 1})$ , which leads to a correction of some parts in the main results of A. Azizi and A. Zekhini (2020).

Stark's conjecture in multi-quadratic extensions, revisited

David S. Dummit, Jonathan W. Sands, Brett Tangedal (2003)

Journal de théorie des nombres de Bordeaux

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Stark’s conjectures connect special units in number fields with special values of $L$ -functions attached to these fields. We consider the fundamental equality of Stark’s refined conjecture for the case of an abelian Galois group, and prove it when this group has exponent $2$ . For biquadratic extensions and most others, we prove more, establishing the conjecture in full.

Maximal unramified extensions of imaginary quadratic number fields of small conductors

Ken Yamamura (1997)

Journal de théorie des nombres de Bordeaux

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We determine the structures of the Galois groups Gal $(K_{u r} / K)$ of the maximal unramified extensions $K_{u r}$ of imaginary quadratic number fields $K$ of conductors $≦ 420 (≦ 719$ under the Generalized Riemann Hypothesis). For all such $K$ , $K_{u r}$ is $K$ , the Hilbert class field of $K$ , the second Hilbert class field of $K$ , or the third Hilbert class field of $K$ . The use of Odlyzko’s discriminant bounds and information on the structure of class groups obtained by using the action of Galois groups on class groups is essential. We...

Displaying similar documents to “On a functorial property of power residue symbols. Erratum : Solution of the congruence subgroup problem for S L n ( n ≥ 3 ) and S p 2 n ( n ≥ 2 ) ”

Displaying similar documents to “On a functorial property of power residue symbols. Erratum : Solution of the congruence subgroup problem for $S L_{n} (n \geq 3)$ and $S p_{2 n} (n \geq 2)$ ”