Spinor genera under Zp-extensions

Abstract

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