On an analogue of a conjecture of Gross
type: Item
instance of: scholarly article
title: On an analogue of a conjecture of Gross
zbMATH Open document ID: 0648.12012
DOI: 10.1007/BF01258443
author: Manohar L. Madan
author: Gabriel D. Villa Salvador
published in: Manuscripta Mathematica
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155319
review text: In this paper it is studied for some congruence function fields the validity of an analogue to a conjecture of Gross in Iwasawa theory. Let \(K_0\) be an algebraic function field of one variable with field of constants a finite field of characteristic \(p\), \(p\) an odd prime. Let \(K_{\infty}\) be a \(\mathbb{Z}_p\)-extension of \(K_0\), which is either one with no new constants (geometric extension) or a purely constant one (constant extension). Clearly \(G=\mathrm{Gal}(K_{\infty}/K_0)\) operates on \(A\), the \(p\)-part of the \(S\)-class group of \(K_{\infty}\), where \(S\) is a finite set of prime divisors of \(K_0\) (in the geometric case, \(S\) is the set of the ramified primes). The analogue of Gross' conjecture considered states that \(A^G\) is finite. The authors give, in the geometric case, necessary and sufficient conditions in order \(A^G\) to be finite. Using this, they may exhibit examples of fields for which the analogue of Gross' conjecture is true and also examples for which it fails. Moreover, for constant extensions, it is proved that the analogue of Gross' conjecture holds in this case.
Mathematics Subject Classification ID: 11R60
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R32
zbMATH DE Number: 4057635
zbMATH Keywords: invariants of Galois group
zbMATH Keywords: congruence function fields
zbMATH Keywords: Iwasawa theory
Wikidata QID: Q123094946
reviewed by: Nuria Vila
MaRDI profile type: Publication
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OpenAlex ID: W2035777527
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description: scientific article; zbMATH DE number 4057635