Integral representations of p-class groups in \({\mathbb{Z}}_ p\)-extensions, semisimple differentials and Jacobians
type: Item
instance of: scholarly article
title: Integral representations of p-class groups in \({\mathbb{Z}}_ p\)-extensions, semisimple differentials and Jacobians
zbMATH Open document ID: 0689.12008
DOI: 10.1007/BF01190213
author: Manohar L. Madan
author: Gabriel D. Villa Salvador
published in: Archiv der Mathematik
publication date: 1991-01-01T00:00:00Z
review text: For a ramified p-extension of cyclotomic \({\mathbb{Z}}_ p\)-number fields with Galois group G, the \({\mathbb{Z}}_ p[G]\)-module structure of the minus part of the p-primary ideal class group is obtained when the p-th roots of unity are not present. When the p-th roots of unitary are present, the integral representation is determined if all the decomposition groups of the ramified primes are contained in one of them and this one is normal. Analogous results are obtained for a wildly ramified p-extension of algebraic function fields of one variable over an algebraically closed field k of characteristic \(p>0\). Assuming the existence of a unique maximal decomposition group, the k[G]-module structure of the holomorphic semisimple differentials is obtained. For this family is also obtained the \({\mathbb{F}}_ p[G]\)-module structure of the elements of order dividing p in the null class group, \({\mathbb{F}}_ p\) denoting the field with p elements. Finally, the structure of the p-art of the Jacobian variety associated with the overfield, is determined as \({\mathbb{Z}}_ p[G]\)-module for the same class.
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11R20
Mathematics Subject Classification ID: 11R33
zbMATH DE Number: 4128941
zbMATH Keywords: cyclotomic \({\mathbb Z}_ p\)-extensions
zbMATH Keywords: integral representations
zbMATH Keywords: cyclotomic \({\mathbb Z}_ p\)-number fields
zbMATH Keywords: algebraic function fields
zbMATH Keywords: holomorphic semisimple differentials
zbMATH Keywords: null class group
zbMATH Keywords: Jacobian variety
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01190213
OpenAlex ID: W2051057996
description: scientific article; zbMATH DE number 4128941