Incoherent definite spaces and Shimura varieties. (English) Zbl 1475.11124

Müller, Werner (ed.) et al., Relative trace formulas. Proceedings of the Simons symposium, Schloss Elmau, Germany, April 22–28, 2018. Cham: Springer. Simons Symp., 187-215 (2021).
Summary: In this paper, we define incoherent definite quadratic spaces over totally real number fields and incoherent definite Hermitian spaces over CM fields. We use the neighbors of these spaces to study the local points of orthogonal and unitary Shimura varieties.
For the entire collection see [Zbl 1469.11002].


11G18 Arithmetic aspects of modular and Shimura varieties
14G35 Modular and Shimura varieties
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14G15 Finite ground fields in algebraic geometry
Full Text: DOI arXiv


[1] I. Vollaard, T. Wedhorn, The supersingular locus of the Shimura variety of GU(n − 1,  1). Invent. Math · Zbl 1227.14027
[2] F. Oort, Which abelian surfaces are products of elliptic curves? Math. Ann. 214, 35-47 (1975) · Zbl 0283.14007
[3] K-Z. Li, F. Oort, Moduli of Supersingular Abelian Varieties. Springer Lecture Notes in Mathematics, vol. 1680 (Springer, Berlin, 1998) · Zbl 0920.14021
[4] S. Kudla, M. Rapoport, Cycles on Siegel threefolds and derivatives of Eisenstein series. Ann. Sci. École Norm. Sup. 33, 695-756 (2000) · Zbl 1045.11044
[5] M. Artin, Supersingular K3 surfaces. Ann. Sci. École Norm. Sup. 7, 543-567 (1974) · Zbl 0322.14014
[6] H. Carayol, Sur la mauvais réduction des courbes de Shimura. Compos. Math. 59, 151-230 (1986) · Zbl 0607.14021
[7] P. Deligne, Travaux de Shimura. Lecture Notes in Mathematics, vol. 244 (Springer, Berlin, 1971) · Zbl 0225.14007
[8] P. Deligne, Variétés de Shimura. Proc. Symp. Pure Math. 33, 247-290 (1979)
[9] K. Doi, H. Naganuma, On the algebraic curves uniformized by arithmetical automorphic functions. Ann. Math. 86, 449-460 (1967) · Zbl 0217.05002
[10] V.G. Drinfeld, S.G. Vladut, Number of points of an algebraic curve. Funct. Anal. Appl. 17, 53-54 (1983) · Zbl 0522.14011
[11] W.-T. Gan, B. Gross, D. Prasad, Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups. Sur les conjectures de Gross et Prasad Astérisque 346, 1-109 (2012) · Zbl 1280.22019
[12] B. Gross, On the motive of a reductive group. Invent. Math. 130, 287-313 (1997) · Zbl 0904.11014
[13] B. Gross, Heegner Points and Representation Theory. Heegner Points and Rankin L-Series, vol. 49 (MSRI Publication, Berkeley, 2004), pp. 37-65 · Zbl 1126.11032
[14] X. He, G. Pappas, M. Rapoport, Good and semi-stable reduction of Shimura varieties (2018). ArXiv: 1804.09615 · Zbl 1473.11133
[15] S. Kudla, M. Rapoport, Arithmetic Hirzebruch-Zagier cycles. J. Reine Angew. Math. 515, 155-244 (1999) · Zbl 1048.11048
[16] T. Lam, Algebraic Theory of Quadratic Forms(Addison-Wesley, Boston, 1980) · Zbl 0437.10006
[17] J. Milnor, D. Husemoller, Symmetric Bilinear Forms. Springer Ergebnisse, vol. 73 (Springer, Berlin, 1973) · Zbl 0292.10016
[18] M. Rapoport, B. Smithling, W. Zhang, On Shimura varieties for unitary groups. arXiv: 1906.12346, to appear in PAMQ · Zbl 1472.11195
[19] J.-P. Serre, Lie Algebras and Lie Groups. Springer Lecture Notes in Mathematics, vol. 1500 (Springer, Berlin, 2006)
[20] G. Shimura, Construction of class fields and zeta functions of algebraic curves. Ann. Math. 85, 58-159 (1967) · Zbl 0204.07201
[21] E. Viehmann, T. Wedhorn, Ekedahl-Oort and Newton strata for Shimura varieties of PEL type. Math. Ann. 356, 1493-1550 (2013) · Zbl 1314.14047
[22] G. Shimura, On the zeta functions of the algebraic curves uniformized by certain automorphic functions. J. Math. Soc. Japan 13, 275-331 (1961) · Zbl 0218.14013
[23] J.-P. Serre, A Course in Arithmetic. Springer GTM, vol. 7 (Springer, Berlin, 1973) · Zbl 0256.12001
[24] W.-T. Gan, J. Hanke, J.-K. Yu, On an exact mass formula of Shimura. Duke Math. J. 107, 103-133 (2001) · Zbl 1023.11019
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