Group schemes and local densities

Abstract

WEE TECK GAN and JIU-KANG YU 1. Introduction. The subject matter of this paper is an old one with a rich history, beginning with the work of Gauss and Eisenstein, maturing at the hands of Smith and Minkowski, and culminating in the fundamental results of Siegel. More precisely, if L is a lattice over Z (for simplicity), 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 ω ld over Aut Z p (L, Q), which is an open compact subgroup of Aut Q p (L, Q). The question that remains is whether one can find an explicit formula for the local density. Through the work of Pall (for p = 2) and Watson (for p = 2), such an explicit formula for the local density is in fact known for an arbitrary lattice over Z p (see [P] and [Wa]). The formula is obviously structured, although [CS] seems to be the first to comment on this. Unfortunately, the known proof (as given in [P] and [K]) does not explain this structure and involves complicated recursions. On the other hand, Conway and Sloane [CS, Section 13] have given a heuristic explanation of the formula. In this paper, we give a simple and conceptual proof of the local density formula for p = 2. The viewpoint taken here is similar to that of our earlier work [GHY], and the proof is based on the observation that there exists a smooth affine group scheme G over Z p with generic fiber Aut Q p (L, Q), which satisfies G(Z p) = Aut Z p (L, Q). This follows from general results of smoothening [BLR], as we explain in Section 3. To obtain an explicit formula, it is necessary to have an explicit construction of G. The main contribution of this paper is to give such an explicit construction of G (in Section 5), and to determine its special fiber (in Section 6). Finally, by comparing ω ld and the canonical volume form ω can of G, we obtain the explicit formula for the local density in Section 7. The smooth group schemes constructed in this paper should also be of independent interest. Our method works over any non-Archimedean local field of residue characteristic p = 2 and also works for all types of classical groups. Therefore, we obtain new explicit formulas for local densities of lattices in symplectic spaces, Hermitian spaces, and quaternionic Hermitian and anti-Hermitian spaces. For lattices in a symplectic

Untitled

“…Then the problem essentially reduces to constructing G explicitly, so that one can compute the cardinality of the group G(κ) of κ-points of its special fiber. This tells us what the analog of (1-1) for G is, and further, it so turns out that one can deduce the expression (1-1) from its analog for G. For a detailed explanation about this, see Section 3 of [Gan and Yu 2000].…”

“…A. Sloane [1988] further developed the formula for any p and gave a heuristic explanation for it. Later, W. T. Gan and J.-K. Yu [2000] (for p = 2) and S. Cho [2015a] (for p = 2) provided a simple and conceptual proof of Conway and Sloane's formula by explicitly constructing a smooth affine group scheme G over ‫ޚ‬ 2 with generic fiber Aut ‫ޑ‬ 2 (L , H ), which satisfies G(‫ޚ‬ 2 ) = Aut ‫ޚ‬ 2 (L , H ).…”

“…In addition, M. Mischler [2000] computed the formula for a ramified hermitian lattice ( p = 2) under restricted conditions. Later, Gan and Yu [2000] found a conceptual and elegant proof of the local density formula for an unramified hermitian lattice without any restriction on p, and for a ramified hermitian lattice with the restriction p = 2, by explicitly constructing certain smooth affine group schemes (called smooth integral models) of a unitary group.…”

“…In conclusion, this paper, combined with [Gan and Yu 2000] and [Cho 2015a], allows the computation of the mass formula for a hermitian R -lattice (L , H ) when k v ‫ޑ/‬ 2 is unramified, and k v /k v satisfies Case 1 or is unramified. Here, k v (resp.…”

“…Now let us describe some of the ideas involved in the computation of the special fiber G of G. Since the quotients of some pairs of lattices of the form L alluded to in the previous paragraph naturally support symplectic or quadratic forms, it is not hard to construct a map ϕ from G to a suitable product of symplectic and orthogonal groups. This step occurs in [Gan and Yu 2000], too. However, p being even for us poses at least two new difficulties.…”