On an exact mass formula of Shimura
In a recent series of 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 \(k\). The mass formula was obtained by Shimura as a consequence of his theory of Euler products and Eisenstein series for the corresponding orthogonal or unitary group. On the other hand, if \(G\) is a connected reductive group over \(k\) which is anisotropic at all Archimedean places and quasi split at all finite places, an explicit mass formula was obtained by \textit{B. H. Gross} and \textit{W. T. Gan} [Trans. Am. Math. Soc. 351, 1691-1704 (1999; Zbl 0991.20033)]. This is an extension of a fundamental result of \textit{G. Prasad} [Publ. Math., Inst. Hautes Étud. Sci. 69, 91-117 (1989; Zbl 0695.22005)]. The aim of this paper is to rederive Shimura's formula from the general mass formula using Bruhat-Tits theory at least in the cases where the quadratic or Hermitian form is totally definite. This derivation is based on the observation that the stabilizer of a maximal lattice at a finite place is always a maximal parahoric subgroup of the orthogonal or unitary group.
scientific article; zbMATH DE number 1820796