Temperedness of measures defined by polynomial equations over local fields

David Taylor, V. S. Varadarajan, Jukka Virtanen and David Weisbart

Vol. 296 (2018), No. 1, 227–256

DOI: 10.2140/pjm.2018.296.227

Abstract

We investigate the asymptotic growth of the canonical measures on the fibers of morphisms between vector spaces over local fields of arbitrary characteristic. For a single polynomial over $ℝ$, this is due to Igusa and Raghavan. For nonarchimedean local fields we use a version of the Łojasiewicz inequality which follows from work of Greenberg, together with the theory of the Brauer group of local fields to construct definite forms of arbitrarily high degree, and to transfer questions at infinity to questions near the origin. We then use these to generalize results of Hörmander on estimating the growth of polynomials at infinity in terms of the distance to their zero loci. Specifically, when a fiber corresponds to a noncritical value which is stable, i.e., remains noncritical under small perturbations, we show that the canonical measure on the fiber is tempered, which generalizes results of Igusa and Raghavan, and Virtanen and Weisbart.

Dedicated to the memory of Professor Jun-Ichi Igusa

Keywords

algebraic geometry, local fields, tempered measures, invariant measures, semisimple Lie algebra

Mathematical Subject Classification 2010

Primary: 11G25, 14G20

Secondary: 11S31, 22E35

Milestones

Received: 18 April 2017

Accepted: 27 April 2017

Published: 1 May 2018