Group schemes and local densities of quadratic lattices in residue characteristic 2

The celebrated Smith-Minkowski-Siegel mass formula expresses the mass of a quadratic lattice (L, Q) as a product of local factors, called the local densities of (L, Q). This mass formula is an essential tool for the classification of integral quadratic lattices. In this paper, we will describe the local density formula explicitly by observing the existence of a smooth affine group scheme G&barbelow; over Z2 with the generic fiber AutQ2 (L, Q), which satisfies G&barbelow;( Z2&parr0; =AutZ2 (L, Q). Our method works for any unramified finite extension of Q2 . Therefore, we give the long awaited proof for the local density formula of Conway and Sloane and discover its generalization to unrami fied finite extensions of Q2 . As an example, we give the mass formula for the integral quadratic form Qn(x1, ···, xn) = x21+&cdots;+x2n associated to a number field k which is totally real and such that the ideal (2) is unramified over k..