Arithmetic of Hermitian forms and related forms

Abstract

Let k be an algebraic number field of finite degree and K its quadratic extension. Let V be a finite dimensional vector space over K supplied with a non-degenerate Hermitian form H, which determines in a natural manner a symmetric bilinear form B (the real part of H) and an alternating form A (the imaginary part of H) on the k-vector space V. For an arbitrary prime ideal p in k, the Hasse symbol of the quadratic vector space V p is given in terms of the unitary invariants of the Hermitian vector space V p and a certain norm residue symbol. A necessary and sufficient condition for a lattice which is modular in the Hermitian vector space V to be modular in the quadratic vector space V is given in terms of the different of K k . Furthermore, it is shown that if A is an ideal in k, then an A -modular lattice in the Hermitian vector space V is a maximal lattice in the alternating vector space V. The elementary divisors of a lattice in the alternating vector space V, which is modular in the Hermitian vector space V, is determined under a certain condition (e.g., (2) is a prime ideal in k).