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On the distribution of eigenspaces in classical groups over finite rings and the Cohen-Lenstra heuristic

  • In 2006 Jeffrey Achter proved that the distribution of divisor class groups of degree 0 of function fields with a fixed genus and the distribution of eigenspaces in symplectic similitude groups are closely related to each other. Gunter Malle proposed that there should be a similar correspondence between the distribution of class groups of number fields and the distribution of eigenspaces in ceratin matrix groups. Motivated by these results and suggestions we study the distribution of eigenspaces corresponding to the eigenvalue one in some special subgroups of the general linear group over factor rings of rings of integers of number fields and derive some conjectural statements about the distribution of \(p\)-parts of class groups of number fields over a base field \(K_{0}\). Where our main interest lies in the case that \(K_{0}\) contains the \(p\)th roots of unity, because in this situation the \(p\)-parts of class groups seem to behave in an other way like predicted by the popular conjectures of Henri Cohen and Jacques Martinet. In 2010 based on computational data Malle has succeeded in formulating a conjecture in the spirit of Cohen and Martinet for this case. Here using our investigations about the distribution in matrixgroups we generalize the conjecture of Malle to a more abstract level and establish a theoretical backup for these statements.

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Author:Michael Adam
URN (permanent link):urn:nbn:de:hbz:386-kluedo-37322
Advisor:Gunter Malle
Document Type:Doctoral Thesis
Language of publication:English
Publication Date:2014/02/17
Year of Publication:2014
Publishing Institute:Technische Universität Kaiserslautern
Granting Institute:Technische Universität Kaiserslautern
Acceptance Date of the Thesis:2014/02/13
Date of the Publication (Server):2014/02/18
Number of page:70
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
Licence (German):Standard gemäß KLUEDO-Leitlinien vom 10.09.2012