On an exact mass formula of Shimura

Abstract

In a series of recent papers, G. Shimura obtained an exact formula for the mass of a maximal lattice in a quadratic or hermitian space over a totally real number field. Using Bruhat-Tits theory, we obtain a quick and more conceptual proof of his formula when the form is totally definite.

Quaternary quadratic lattices over number fields

“…Now by [6] the mass of G ai (Q, n) does not depend on i, (as locally the lattices are just rescaled versions of each other) so for all i Mass(G ai (Q, n)) = Mass(M) 2 h −2 K 2 −1−s and the theorem follows from the computation of Mass(M) in Theorem 5.1.…”

Higher real K -theories and topological automorphic forms

“…The computation of the cardinality of GU A0 (Z (p) )\GU(A p,∞ )/K is a class number question for the group GU. Unfortunately, preliminary calculations of this cardinality by the first author, using the mass formulas of [8], seem to indicate that, even in the case of p = 5 and n = 4, this class number is quite large.…”

The mass of unimodular lattices

“…For a natural class of lattices L in (V , q), an explicit formula for Mass(L) was given in [GHY,Proposition 2.13], based on the fundamental work [P] and its extension [Gr]. These are the lattices for which the local stabilizers K L v are parahoric subgroups of G(F v ).…”

“…However, the derivation given in [GHY,§2] only relies on [GG,Proposition 9.3], and the latter holds without these restrictions, as long as the group G is semisimple.…”

“…(iii) The reader may notice that the formula in [GHY,Proposition 2.13] is cleaner than the one given above, and involves L-values at negative integers rather than positive integers. Not surprisingly, the two versions are related by the functional equation.…”