Klopsch, B (2003) Zeta functions related to the pro-p group SL1(Delta(p)). Mathematical Proceedings of the Cambridge Philosophical Society, 135
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Let D-p be a central simple Q(p)-division algebra of index 2, with maximal Z(p)-order Delta(p). We give an explicit formula for the number of subalgebras of any given finite index in the Z(p) Lie algebra L := sl(1) (Delta(p)). From this we obtain a closed formula for the zeta function zeta(L)(s) := Sigma(M less than or equal to L) \ L: M\(-s). The results are extended to the p-power congruence subalgebras of L, and as an application we obtain the zeta functions of the corresponding congruence subgroups of the uniform pro-p group SL12 (Delta(p)).
This is a Submitted version This version's date is: 7/2003 This item is not peer reviewed
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