Lattices and orders in quaternion algebras with involution

摘要

In this paper maximal orders A in quaternion algebras having an involution over a quadratic field extension KIF 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 A are classified up to genus. Finally, it is shown that the units of norm I in A maps surjectively to the spinorial kernel of the orthogonal group of the lattice. (c) 2006 Elsevier Inc. All rights reserved.