Galois modular representation of associated Jacobians in the tamely ramified cyclic case
type: Item
instance of: scholarly article
title: Galois modular representation of associated Jacobians in the tamely ramified cyclic case
zbMATH Open document ID: 1151.11062
author: Martha Rzedowski-Calderón
author: V. Janitzio Mejía Huguet
published in: Manuscripta Mathematica
publication date: 2008-09-23T00:00:00Z
review text: Let \(k\) be an algebraically closed field of characteristic \(p\geq 0\) and let \(L/K\) be a finite Galois \(\ell\)-extension of function fields with Galois group \(G\). The group \(G\) acts naturally on the \(\ell\)-torsion of the Jacobian \({\mathbb J}_ L (\ell)\) associated with the field \(L\). We have that \({\mathbb J}_ L(\ell)\) is canonically isomorphic with the Sylow \(\ell\)--subgroup \(C_{0L}(\ell)\) of the group of divisors of degree \(0\) of \(L\). The \({\mathbb Z}_ \ell [G]\)-module \({\mathbb J}_ L(\ell)\) is an injective \({\mathbb Z}_ \ell\)--module of rank \(2g_ L\), where \({\mathbb Z} _ \ell\) denotes the ring of the \(\ell\)--adic integers and \(g_ L\) the genus of \(L\). That is, \({\mathbb J}_ L (\ell)\cong R^{2g_ K}\) as \({\mathbb Z}_\ell\)-modules where \(R\cong {\mathbb Q}_ \ell/{\mathbb Z}_ \ell\) and \({\mathbb Q}_ \ell\) is the field of \(\ell\)-adic numbers. If \({\mathcal N}\) is any nontrivial modulus in \(L\) containing all the ramified prime divisors in \(L/K\), there exists an exact \({\mathbb Z}_\ell[G]\)-sequence \[ 1\to{\mathcal R}\to C_{0{\mathcal N}}(\ell)\to C_{0L}(\ell)\to 1 \] where \(C_{0{\mathcal N}}(\ell)\) denotes the \(\ell\)-torsion of the generalized Jacobian associated to \(L\) and \({\mathcal N}\). When \(\ell=p\), \(C_{0{\mathcal N}}(\ell)\) is an injective \({\mathbb Z}_\ell[G]\)-module and the above sequence can be used to find the general structure of \(C_{0L} (\ell)\) as \({\mathbb Z}_ \ell [G]\)-module by means of the dual of the Heller's loop operator [\textit{P. R. López-Bautista} and \textit{G. D. Villa-Salvador}, Can. J. Math. 50, No. 6, 1253--1272 (1998; Zbl 0943.11050)]. When \(\ell \neq p\), \(C_{0{\mathcal N}}(\ell)\) is never \({\mathbb Z}_ \ell[G]\)-injective and the structure of \(C_{0L}(\ell)\) is known only in the cyclic case and in the general unramified case. In the paper under review, the authors modify \({\mathcal R}\) and \(C_{0{\mathcal N}}(\ell)\) and use an exact \(G\)-exact sequence \…
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 20C10
Mathematics Subject Classification ID: 12G05
zbMATH DE Number: 5346206
zbMATH Keywords: Integral representation
zbMATH Keywords: Jacobians
zbMATH Keywords: divisor class group
zbMATH Keywords: Galois modular representation
zbMATH Keywords: generalized Jacobians
reviewed by: Gabriel D. Villa Salvador
MaRDI profile type: Publication
full work available at URL: s00229-008-0196-5
OpenAlex ID: W2032683259
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DOI: 10.1007/S00229-008-0196-5
description: scientific article; zbMATH DE number 5346206