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

TL;DR

This paper explicitly derives the local density formula for quadratic lattices by constructing smooth integral group schemes, generalizing previous results to unramified extensions of Q_2, and applies it to specific quadratic forms.

Contribution

It provides a new explicit local density formula for quadratic lattices over unramified extensions of Q_2 using smooth integral group schemes, confirming and extending prior results.

Findings

Explicit local density formula for unramified extensions of Q_2

Construction of smooth integral group schemes for orthogonal groups

Application to mass formulas of quadratic forms over number fields

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 constructing a smooth integral group scheme model for an appropriate orthogonal group. 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.