Zur Frobeniusoperation auf der Homologie einiger arithmetischer Gruppen. (On the Frobenius operation on the homology of some arithmetic groups)
type: Item
instance of: scholarly article
title: Zur Frobeniusoperation auf der Homologie einiger arithmetischer Gruppen. (On the Frobenius operation on the homology of some arithmetic groups)
zbMATH Open document ID: 0643.20021
DOI: 10.1007/BF01313492
author: Ulrich Stuhler
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: Let \({\mathfrak a}\) be an ideal of the polynomial ring \({\mathbb{F}}_ q[t]\), \(\Gamma_{{\mathfrak a}}=\{\gamma \in GL_ 2({\mathbb{F}}_ q[t])|\gamma\equiv 1(mod {\mathfrak a})\}^ a \)congruence subgroup. The Frobenius endomorphism Frob: \(\Gamma\) \({}_{{\mathfrak a}}\to \Gamma_{{\mathfrak a}}\), \(\left( \begin{matrix} a\quad b\\ c\quad d\end{matrix} \right)\mapsto \left( \begin{matrix} a\quad p b p\\ c\quad p d p\end{matrix} \right)\), induces a linear transformation \((Frob)_*: H_ 1(\Gamma_{{\mathfrak a}},{\mathbb{Q}})\to H_ 1(\Gamma_{{\mathfrak a}},{\mathbb{Q}})\). In this paper it is shown that the eigenvalues of \((Frob)_*\) are roots of unity or zero. A formula for calculating the trace of \((Frob)^{{\mathfrak a}}_*\) is given as well.
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 20H05
Mathematics Subject Classification ID: 20J05
zbMATH DE Number: 4047945
zbMATH Keywords: eigenvalues of Frobenius map
zbMATH Keywords: trace of Frobenius map
zbMATH Keywords: homology of congruence group
zbMATH Keywords: congruence subgroup
zbMATH Keywords: Frobenius endomorphism
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4047945