School Seminars and Colloquia

On some integral representations of finite groups

Geometry/Topology Seminar

by Dr Dmitry Malinin

Institution: University of the South Pacific
Date: Tue 9th January 2007
Time: 3:15 PM
Location: Room 213, Richard Berry Building, The University of Melbourne

Abstract: We consider the natural action of Galois groups on finite arithmetic groups and finite subgroups of GL(n,R) for orders R in number fields.

Let K/Q be a finite Galois extension with the ring of integers O_K and Galois group \Gamma. We consider finite \Gamma-stable subgroups G< GL_n(O_K) and prove that they are generated by matrices with coefficients in O_{K_ab}, K_ab the maximal abelian subextension of K over Q. This implies in particular a positive answer to a conjecture of J. Tate on the classification of p-divisible groups over Z and answers also a longstanding question of Y. Kitaoka on totally real scalar extensions of positive definite integral quadratic lattices.

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