Representation zeta functions of some compact p-adic analytic groups

Representation zeta functions of some compact p-adic analytic groups

Using the Kirillov orbit method, novel methods from p-adic integration and Clifford theory, we study representation zeta functions associated to compact p-adic analytic groups. In particular, we give general estimates for the abscissae of convergence of such zeta functions. We compute explicit formulae for the representation zeta functions of some compact p-adic analytic groups, defined over a compact discrete valuation ring O of characteristic 0. These include principal congruence subgroups of SL_2(O), without any restrictions on the residue field characteristic of O, as well as the norm one group SL_1(D) of a non-split quaternion algebra D over the field of fractions of O and its principal congruence subgroups. We also determine the representation zeta functions of principal congruence subgroups of SL_3(O) in the case that O has residue field characteristic 3 and is unramified over Z_3

1-34

Avni, Nir

6f0719d5-4bce-4d71-b312-33a9cea6b36f

Klopsch, Benjamin

3556ffc9-0748-4eee-aecc-d1dc70a30638

Voll, Christopher

b7bd2890-38ac-4e05-adb7-3b376916ff79

Avni, Nir

6f0719d5-4bce-4d71-b312-33a9cea6b36f

Klopsch, Benjamin

3556ffc9-0748-4eee-aecc-d1dc70a30638

Voll, Christopher

b7bd2890-38ac-4e05-adb7-3b376916ff79

Avni, Nir, Klopsch, Benjamin and Voll, Christopher
(2010)
Representation zeta functions of some compact p-adic analytic groups.
*Preprint*, .

## Abstract

Using the Kirillov orbit method, novel methods from p-adic integration and Clifford theory, we study representation zeta functions associated to compact p-adic analytic groups. In particular, we give general estimates for the abscissae of convergence of such zeta functions. We compute explicit formulae for the representation zeta functions of some compact p-adic analytic groups, defined over a compact discrete valuation ring O of characteristic 0. These include principal congruence subgroups of SL_2(O), without any restrictions on the residue field characteristic of O, as well as the norm one group SL_1(D) of a non-split quaternion algebra D over the field of fractions of O and its principal congruence subgroups. We also determine the representation zeta functions of principal congruence subgroups of SL_3(O) in the case that O has residue field characteristic 3 and is unramified over Z_3

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## More information

e-pub ahead of print date: 2010

## Identifiers

Local EPrints ID: 180863

URI: http://eprints.soton.ac.uk/id/eprint/180863

PURE UUID: 7e314c7e-c8e3-47df-a8d9-9d124e5739de

## Catalogue record

Date deposited: 12 Apr 2011 07:37

Last modified: 18 Sep 2020 16:36

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## Contributors

Author:
Nir Avni

Author:
Benjamin Klopsch

Author:
Christopher Voll

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