Integrating on \(p\)-adic Lie groups
type: Item
instance of: scholarly article
title: Integrating on \(p\)-adic Lie groups
zbMATH Open document ID: 0917.22007
DOI: 10.1007/BF02762274
author: Graham Everest
author: Marcus du Sautoy
published in: Israel Journal of Mathematics
publication date: 1999-08-02T00:00:00Z
review text: The authors consider integrals of the type \[ I(s,F,H)=\int_H| F(h)|^s d\mu(h) \quad\text{and}\quad m_H(F)=\int_H\text{ord}_K(F(h))d\mu(h), \] where \(K\) is a non-archimedean local field of characteristic 0, \(H\) is a compact \(K\)-Lie group with Haar measure \(\mu\), \(F\) is an analytic function on \(H\) and \(s\in \mathbb{C}\), \(\text{Re}(s)>0\). It is known that the Igusa integral \(I(s,F,H)\) is a rational function of \(q^{-s}\) \((q\) the modulus of \(K)\) and that \(m_H(F)/ \mu(H)\) is rational. The purpose of this paper is to give a structure theory for \(I(s,F,H)\) and \(m_H(F)\) in terms of simpler functions of the same type, namely when \(H\) is of the form \(\mathbb{Z}_p^m\) and \(F\) is a polynomial. This is done roughly under the assumption that \(H\) is a compact open subgroup of \(G(K)\), where \(G\) is a split, semisimple linear algebraic group and \(F\) is a polynomial on \(G\) defined over a suitable field. The authors give examples to show how their result can be used and how it relates to arithmetic problems, such as Leopoldt's conjecture. They also show how one can compute the integrals effectively.
Mathematics Subject Classification ID: 22E35
Mathematics Subject Classification ID: 11D88
Mathematics Subject Classification ID: 11E95
zbMATH DE Number: 1250646
zbMATH Keywords: Mahler measure
zbMATH Keywords: Chevalley groups
zbMATH Keywords: Igusa integral
zbMATH Keywords: semisimple linear algebraic group
zbMATH Keywords: Leopoldt's conjecture
Wikidata QID: Q115391675
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02762274
OpenAlex ID: W2328435434
description: scientific article; zbMATH DE number 1250646