Galois modular representation of associated Jacobians in the tamely ramified cyclic case (Q946501)

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 \[ 0\to\ker\rho\to P\to C_{0L}(\ell)\to 0 \] where \(P\) is \({\mathbb Z}_ \ell[G]\)-injective to find the structure of \(C_{0L}(\ell)\) when \(G\) is a cyclic group. The approach of the paper is in the spirit of the wild case \(\ell=p\) and the proof is much simpler than the one obtained using the cohomological Yakovlev's Theorem in the cyclic case [\textit{M. Rzedowski-Calderón}, the reviewer, and \textit{M. L. Madan}, Math. Z. 224, No. 1, 77--101 (1997; Zbl 0880.14014)]. First, the injective component of \(\ker\rho\) is found by means of the norm map. Once this is obtained, in order to use Heller's loop operator, the main step in the paper is to find the \({\mathbb Z}_ \ell[G]\)-structure of the non-injective component \(V\) of \(\ker \rho\). The main tool towards this end is the use of the cohomology of the various \(G\)-modules involved.