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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.