Integrating on \(p\)-adic Lie groups
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.
scientific article; zbMATH DE number 1250646
scientific article