The Galois module structure of certain arithmetic principal homogeneous spaces
type: Item
instance of: scholarly article
title: The Galois module structure of certain arithmetic principal homogeneous spaces
zbMATH Open document ID: 0776.11065
DOI: 10.1016/0021-8693(92)90153-D
published in: Journal of Algebra
publication date: 1993-03-28T00:00:00Z
review text: This paper describes a new connection between Galois module theory and values of \(p\)-adic \(L\)-functions. The author demonstrates such a connection both for unramified Kummer extensions of order \(p\) of the cyclotomic field \(\mathbb{Q}(\zeta_ p)\), and for principal homogeneous spaces arising from an elliptic curve with complex multiplication defined over an imaginary quadratic field \(K\). For the cyclotomic case, let \(N=\mathbb{Q}(\zeta_ p)\), \(p\) an odd prime number, let \(G\) be cyclic of order \(p\), and let \({\mathfrak B}=\text{Map}(G,{\mathcal O}_ N)\). The group \(\text{PH}({\mathfrak B})\) of principal homogeneous spaces for \({\mathfrak B}\) is the same as the group of isomorphism classes of Galois extensions of \({\mathcal O}_ N\) with group \(G\). The map \(\psi: \text{PH}({\mathfrak B})\to\text{Cl}({\mathfrak A})\) defined by viewing a principal homogeneous space as a (locally free) module over \({\mathfrak A}={\mathcal O}_ N G\) is a \(\Delta\)-homomorphism with the natural action of \(\Delta=\text{Gal}(N/\mathbb{Q})\) on \(\text{PH}({\mathfrak B})\) and \(\text{Cl}({\mathfrak A})\). So \(\text{ker}(\psi)\) is a \(\mathbb{Z}_ p\Delta\)- module, hence decomposes into eigenspaces \(\text{ker}(\psi)_ i\) corresponding to the \(i\)-th power of the character \(\kappa: \Delta\to\mathbb{Z}^ \times_ p\) given by \(\zeta^ \delta=\xi^{\kappa(\delta)}\) for \(\delta\) in \(\Delta\). The author shows that \(\text{ker}(\psi)_ i=(0)\) if \(i=0\) or \(i\) is odd, while if \(i>0\) is even then \(\text{ker}(\psi)_ i=(0)\) iff \(L_ p(1,\kappa^ i)h_{F,i}^{-1}\not\equiv 0\pmod p\). Here \(L_ p(1,\kappa^ i)\) is the \(p\)-adic \(L\)-function associated to \(\kappa^ i\), and \(h_{F,i}\) is the cardinality of the \(i\)-part of the \(p\)-Sylow subgroup of the class group of \(F\). In an explicitly analogous way, the author obtains a similar result for principal homogeneous spaces arising from an elliptic curve \(E\) defined over \(K\) as above with complex multiplica…
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11S40
zbMATH DE Number: 130471
zbMATH Keywords: Galois module theory
zbMATH Keywords: values of \(p\)-adic \(L\)-functions
zbMATH Keywords: unramified Kummer extensions
zbMATH Keywords: cyclotomic field
zbMATH Keywords: homogeneous spaces
zbMATH Keywords: elliptic curve with complex multiplication
author: Martin J. Taylor
reviewed by: Lindsay N. Childs
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 130471