Comparing orders of Selmer groups
type: Item
instance of: scholarly article
title: Comparing orders of Selmer groups
zbMATH Open document ID: 1098.11056
author: Sébastien Bosca
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2006-03-30T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/52737
review text: The author calculates the quotient of the orders of two Selmer groups, using two different methods based upon class field theory, Galois groups or Kummer radicals of some particular extensions, the ambiguous class number formula, and local criteria. In particular, let \(K^*\) be the multiplicative group of a number field \(K\), \(I_K\) the group of fractional ideals of \(K\), \((x)\) the principal fractional ideal generated by \(x\), with \(x\in K^*\), and \(P_K\subseteq I_K\) the group of principal fractional ideals of \(K\). If \(F\) is an algebraic extension of \(K\), let \(N= N_{F/K}\) denote the norm map from \(F\) to \(K\). For a given cyclic extension of number fields \(F/K\) with degree \(n\), the author defines the groups \(S(F/K)= \{x\in K^*\mid(x)\in \{I^n_K, x\in N(F^*)\}/K^*\), and \(G(F/K)= \}I\in I_K\mid\exists J\in I_F, N(J)= I\}/I^n_K N(P_F)\). Let \(U_{K,n}\) denote the set of \(n\)th roots of unity in \(K\), \(P1(K)\) the set of places of \(K\), \(r_1\) the number of real places of \(K\), and \(r_2\) the number of complex places of \(K\). For a given finite place \(v\) of \(K\), let \(e_v\) denote the ramification index of \(v\) in \(F/K\). When \(v\) is an infinite place of \(K\), define \(e_v= 1\) when \(v\) totally splits at \(v\), and \(e_v= 2\) in the other case; \(v\) is viewed as a ramified place if \(e_v= 2\). The author's main result establishes the formula that if \(F/K\) is a cyclic extension of number fields with degree \(n\), then \(|S(F/K)|/|G(F/K)= |U_{K,n}| n^{r_1+r_2}/\prod_{v\in P(K)} e_v\).
reviewed by: Elliot Benjamin
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11R20
zbMATH DE Number: 5016569
zbMATH Keywords: class field
zbMATH Keywords: radical
zbMATH Keywords: Kummer extension
zbMATH Keywords: ray
zbMATH Keywords: ambiguous classes formula
zbMATH Keywords: idele group
zbMATH Keywords: set of places
MaRDI profile type: Publication
OpenAlex ID: W2071663772
cites work: Counting cyclic quartic extensions of a number field
cites work: Q4779057
DOI: 10.5802/JTNB.502
description: scientific article; zbMATH DE number 5016569