Group schemes and local densities.
Let \(L\) be a lattice over \(\mathbb{Z}\) equipped with an integral quadratic form \(Q\). The celebrated Smith-Minkowski-Siegel mass formula expresses the total mass of \((L,Q)\), which is a weighted class number of the genus of \((L,Q)\), as a product of local factors. These local factors are known as the local densities of \((L,Q)\). Subsequent work of Kneser, Tamagawa and Weil resulted in an elegant formulation of the subject in terms of Tamagawa measures. In particular, the local density at a non-Archimedean place \(p\) can be expressed as the integral of a certain volume form \(w^{\text{Id}}\) over \(\Aut_{\mathbb{Z}_p}(L,Q)\), which is an open compact subgroup of \(\Aut_{\mathbb{Q}_p} (L,Q)\). An explicit formula for the local density for an arbitrary lattice over \(\mathbb{Z}_p\) is known through the work of Pall (for \(p\neq 2)\) and Watson (for \(p=2)\). Unfortunately, the known proof involves complicated recursions. On the other hand, Conway and Sloane have given a heuristic explanation of the formula. In this paper, a simple and conceptual proof of the local density fomula for \(p\neq 2\) is given. The proof is based on the observation that there exists a smooth affine group scheme \({\mathcal G}\) over \(\mathbb{Z}_p\) with generic fibre \(\Aut_{\mathbb{Q}_p} (L,Q)\) which satisfies \({\mathcal G}(\mathbb{Z}_p)= \Aut_{\mathbb{Z}_p}(L,Q)\). The main contribution of this paper is to give an explicit construction of \({\mathcal G}\) and to determine its special fibre. Finally, by comparing \(w^{\text{Id}}\) and the canonical volume form \(w^{\text{can}}\) of \({\mathcal G}\), the explicit formula for local density is obtained. As a consequence of the mass formula and the result on local densities, an explicit formula for the mass of an arbitrary lattice in a quaternionic Hermitian space is obtained which extends the result of Shimura for the mass of maximal lattice.
scientific article; zbMATH DE number 1820769
scientific article