Lattices and orders in quaternion algebras with involution

Abstract

In this paper maximal orders Λ in quaternion algebras having an involution over a quadratic field extension K / F are studied. A quadratic quaternary lattice in the algebra which parametrises the optimally embedded orders in F-subalgebras is constructed. It is shown that the Clifford algebra of the dual of this lattice can naturally be embedded in the order. A theory relating quaternion orders to hermitian planes is also developed. Using these techniques, the optimally embedded suborders of Λ are classified up to genus. Finally, it is shown that the units of norm 1 in Λ maps surjectively to the spinorial kernel of the orthogonal group of the lattice.