Class groups and L-series of function fields
type: Item
instance of: scholarly article
title: Class groups and L-series of function fields
zbMATH Open document ID: 0581.14025
DOI: 10.1016/0022-314X(86)90068-5
published in: Journal of Number Theory
publication date: 1986-01-01T00:00:00Z
review text: The author considers Galois coverings Y/X of projective non-singular algebraic curves defined over a perfect field K of prime characteristic p.. Throughout the paper, the group G of Y/X is assumed to be prime to p. First, in considering the decomposition (with respect to characters \(\chi\) of G) of the group \(C^ 0(Y)(p)\) of divisor classes of Y annihilated by p, the author refines the notion of Hasse-Witt invariant, and describes the relation with the Cartier operator C on Y. For being able to formulate his principal results, he assumes now Y/X to be defined over a finite field \({\mathbb{F}}_ q\), and he gives a short review of the relevant L-series. The main results are contained in section 4, namely the relation between ''reductions'' of L-series modulo p and the characteristic polynomial of C (theorem 4.1) and the p-class number formula (theorem 4.2). These results refine and extend analogous results of Manin and others in more special situations (e.g. G \(= trivial\), or abelian with certain ramification conditions).
reviewed by: Ernst-Ulrich Gekeler
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 3929168
zbMATH Keywords: Galois coverings of projective non-singular algebraic curves
zbMATH Keywords: characteristic p.
zbMATH Keywords: Hasse-Witt invariant
zbMATH Keywords: Cartier operator
zbMATH Keywords: L-series
author: Hans-Georg Rück
MaRDI profile type: Publication
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full work available at URL: 0022-314x(86)90068-5
OpenAlex ID: W2011517086
description: scientific article; zbMATH DE number 3929168