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

Abstract

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) under bar over Z(2) with generic fiber AutQ(2) (L,Q), which satisfies (G) under bar (Z(2)) = Aut(Z2) (L,Q). Our method works for any unramified finite extension of Q(2). Therefore, we give a long awaited proof for the local density formula of Conway and Sloane and discover its generalization to unramified finite extensions of Q(2). As an example, we give the mass formula for the integral quadratic form Q(n) (x(1),...,x(n)) = x(1)(2) +...+ x(n)(2) associated to a number field k which is totally real and such that the ideal (2) is unramified over k