Integrating onp-adic Lie groups

Abstract

Inspired by the classical Mahler measure of a polynomial, we study the integral of the order of an arithmetic polynomial on a compactp-adic Lie group. A result of Denef and van den Dries guarantees this is always a rational number. Integrals of this kind arise naturally; for example, the local canonical height of a rational point on an elliptic curve is given by a Mahler measure. Also, the mean valuation of the normal integral generators in a finite Galois extension arises as a Mahler measure. There is interest in being able to calculate the value of this measure. We show that for some classical groups, it is possible to reduce the integral to a simpler form, one where explicit computations are feasible. The motivation comes from the calculus trick of integration by substitution, also from Weyl’s criterion. Applications are given to Galois Module Theory. Also, a close encounter with Leopoldt’s conjecture is recorded. We deduce our results on the Mahler measure from the more general setting of local zeta functions defined forp-adic Lie groups. Our techniques apply to certain zeta functions, so we state and prove our results at that level of generality in our main theorem.