Spinor genera under Z(p)-extensions

Let L be a quadratic lattice over a number field F. We lift the lattice L a long a Z(p)-extension of F and investigate the growth of the number of spin or genera in the genus of L. Let L, be the lattice obtained from L by exten ding scalars to the n-th layer of the Z(p)-extension. We show that, under v arious conditions on L and F, the number of spinor genera in the genus of L -n is 2(eta pn+O(1)) where eta is some rational number depending on L and t he Z(p)-extension. The work involves Iwasawa's theory of Z(p)-extensions an d explicit calculation of spinor norm groups of local integral rotations.