Spinor genera under Z_p-extensions

Wai-kiu Chan

Vol. 185 (1998), No. 2, 237–267

DOI: 10.2140/pjm.1998.185.237

Abstract

Let L be a quadratic lattice over a number field F. We lift the lattice L along a Z_p-extension of F and investigate the growth of the number of spinor genera in the genus of L. Let L_n be the lattice obtained from L by extending scalars to the n-th layer of the Z_p-extension. We show that, under various conditions on L and F, the number of spinor genera in the genus of L_n is 2^ηpn+O(1) where η is some rational number depending on L and the Z_p-extension. The work involves Iwasawa’s theory of Z_p-extensions and explicit calculation of spinor norm groups of local integral rotations.

Milestones

Received: 11 March 1997

Revised: 20 May 1997

Published: 1 October 1998