Zeta functions of three-dimensional p-adic Lie algebras

Abstract

We give an explicit formula for the subalgebra zeta function of a general three-dimensional Lie algebra over the p-adic integers \({\mathbb{Z}_{p}}\). To this end, we associate to such a Lie algebra a ternary quadratic form over \({\mathbb{Z}_{p}}\). The formula for the zeta function is given in terms of Igusa’s local zeta function associated to this form.

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Correspondence to Christopher Voll.

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We acknowledge support from the Mathematisches Forschungsinstitut Oberwolfach and the Nuffield Foundation.

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Klopsch, B., Voll, C. Zeta functions of three-dimensional p-adic Lie algebras. Math. Z. 263, 195–210 (2009). https://doi.org/10.1007/s00209-008-0416-4

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Keywords

  • Subgroup growth
  • Igusa’s local zeta function

Mathematics Subject Classification (2000)

  • 20E07
  • 11S40
  • 11E20