Class numbers of quadratic forms over real quadratic fields

Abstract

Let k be an algebraic number field, let K be a Galois extension of k of finite degree, and let OK, Ok be the maximal orders of K, k, respectively. We consider the conjugate operation: for a given quadratic lattice M over OK equipped with a bilinear form B and for an automorphism σ ∈ G(K/k), we define a new quadratic lattice Mσ over OK. Here Mσ has the same underlying abelian group as M, but a new OK-action ; the new bilinear form Bσ on Mσ is defined by