Gan, Wee Teck; Hanke, Jonathan P.; Yu, Jiu-Kang On an exact mass formula of Shimura. (English) Zbl 1023.11019 Duke Math. J. 107, No. 1, 103-133 (2001). 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 B. H. Gross and W. T. Gan [Trans. Am. Math. Soc. 351, 1691-1704 (1999; Zbl 0991.20033)]. This is an extension of a fundamental result of 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. Reviewer: Ranjeet Sehmi (Chandigarh) Cited in 22 Documents MSC: 11E57 Classical groups 11E41 Class numbers of quadratic and Hermitian forms 20G35 Linear algebraic groups over adèles and other rings and schemes Keywords:maximal lattice; Hermitian space; mass formula; reductive group; Shimura’s formula; Bruhat-Tits theory Citations:Zbl 0695.22005; Zbl 0991.20033; Zbl 0941.22019 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] F. Bruhat and J. Tits, Groupes réductifs sur un corps local, II: Schémas en groupes; existence d’une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math. 60 (1984), 197–376. · Zbl 0597.14041 · doi:10.1007/BF02700560 [2] –. –. –. –., Schémas en groupes et immeubles des groupes classiques sur un corps local, II: Groupes unitaires , Bull. Soc. Math. France 115 (1987), 141–195. · Zbl 0636.20027 [3] B. H. Gross, On the motive of a reductive group , Invent. Math. 130 (1997), 287–313. · Zbl 0904.11014 · doi:10.1007/s002220050186 [4] B. H. Gross and W. T. Gan, Haar measure and the Artin conductor , Trans. Amer. Math. Soc. 351 (1999), 1691–1704. JSTOR: · Zbl 0991.20033 · doi:10.1090/S0002-9947-99-02095-4 [5] R. Kottwitz, Tamagawa numbers , Ann. of Math. (2) 127 (1988), 629–.\hs646. JSTOR: · Zbl 0678.22012 · doi:10.2307/2007007 [6] A. Moy and G. Prasad, Unrefined minimal \(K\)-types for \(p\)-adic groups , Invent. Math. 116 (1994), 393–.\hs408. · Zbl 0804.22008 · doi:10.1007/BF01231566 [7] A. Moy and G. Prasad, Jacquet functors and unrefined minimal \(K\)-types , Comment. Math. Helv. 71 (1996), 98–121. · Zbl 0860.22006 · doi:10.1007/BF02566411 [8] T. Ono, “On Tamagawa numbers” in Algebraic Groups and Discontinuous Subgroups (Boulder, Colo., 1965) , Proc. Sympos. Pure Math. 9 , Amer. Math. Soc., Providence, 1966, 122–132. [9] G. Prasad, Volumes of \(S\)-arithmetic quotients of semisimple groups , Inst. Hautes Études Sci. Publ. Math. 69 (1989), 91–117. · Zbl 0695.22005 · doi:10.1007/BF02698841 [10] G. Shimura, Euler Products and Eisenstein Series , CBMS Reg. Conf. Ser. Math. 93 , Amer. Math. Soc., Providence, 1997. · Zbl 0906.11020 [11] –. –. –. –., An exact mass formula for orthogonal groups , Duke Math. J. 97 (1999), 1–.\hs66. · Zbl 1161.11325 · doi:10.1215/S0012-7094-99-09701-6 [12] –. –. –. –., Some exact formulas on quaternion unitary groups , J. Reine Angew. Math. 509 (1999), 67–102. · Zbl 0918.11027 · doi:10.1515/crll.1999.509.67 [13] J.-P. Serre, Local Fields , Grad. Texts in Math. 67 , Springer, New York, 1979. [14] J. Tits, “Reductive groups over local fields” in Automorphic Forms, Representations, and L-functions, I (Oregon State Univ., Corvallis, 1977) , Proc. Sympos. Pure Math. 33 , Amer. Math. Soc., Providence, 1979, 29–.\hs69 · Zbl 0415.20035 [15] J.-K. Yu, Construction of tame supercuspidal representations , to appear in J. Amer. Math. Soc. JSTOR: · Zbl 0971.22012 · doi:10.1090/S0894-0347-01-00363-0 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.