×

zbMATH — the first resource for mathematics

Intersection cohomology and \(L\)-functions of some Shimura varieties. (Cohomologie d’intersection et fonctions \(L\) de certaines variétés de Shimura.) (French) Zbl 0553.12005
This paper presents the results of the authors which generalize the well known results of Eichler-Shimura, Deligne, and Langlands on modular curves to the case of Shimura varieties associated with certain rank-one \(\mathbb Q\)-groups (including the moduli of Hilbert-Blumenthal Abelian varieties). One replaces \(l\)-adic cohomology with the \(l\)-adic intersection cohomology of certain bundles and one uses the truth of the Zucker conjecture (for such groups) to interpret this cohomology as \(L_2\)-cohomology. Given a “good” prime one has two local Hasse-Weil factors: one coming from the intersection cohomology and the other by decomposing \(L_2\)-cohomology. The main result establishes that these two factors are equal and it is proved, á la Langlands, by comparing terms in the Selberg trace formula with terms in the Lefschetz trace formula. The purity theorem in intersection cohomology (established by O. Gabber) is then used to prove a version of the Ramanujan conjecture.

MSC:
11G18 Arithmetic aspects of modular and Shimura varieties
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11R39 Langlands-Weil conjectures, nonabelian class field theory
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] M. ARTIN , Some Numerical Criteria for Contractibility of Curves on Algebraic Surfaces (Americal Journal of Math., vol. 84, 1962 , p. 485-496). MR 26 #3704 | Zbl 0105.14404 · Zbl 0105.14404
[2] A. ASH , D. MUMFORD , M. RAPOPORT et Y. TAI , Smooth Compactifications of Locally Symmetric Varieties (Math. Science Press, Bookline, 1979 ). · Zbl 1209.14001
[3] W. L. BAILY et A. BOREL , Compactification of Arithmetic Quotients of Bounded Symmetric Domains (Annals of Math., vol. 84, 1966 ). MR 35 #6870 | Zbl 0154.08602 · Zbl 0154.08602
[4] A. BEILINSON , I. N. BERNSTEIN , P. DELIGNE et O. GABBER , Article à paraître dans les Comptes rendus de la Conférence de Luminy sur l”’Analyse et Topologie sur les espaces singuliers” , juillet 1981 .
[5] A. BOREL , Introduction aux groupes arithmétiques , Hermann, Paris, 1969 . MR 39 #5577 | Zbl 0186.33202 · Zbl 0186.33202
[6] A. BOREL , L\(^{2}\)-Cohomology and Intersection Cohomology of Certain Arithmetic Varieties , Preprint, I.A.S., 1982 .
[7] A. BOREL , Automorphic L-Functions , in Proc. Symp. Pure Math., vol. 33, part. 2, Amer. Math. Soc. R.I., 1979 , p. 27-62. MR 81m:10056 | Zbl 0412.10017 · Zbl 0412.10017
[8] A. BOREL et N. WALLACH , Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups (Annals of Math. Studies, n^\circ 94, Princeton, 1980 ). MR 83c:22018 | Zbl 0443.22010 · Zbl 0443.22010
[9] J. L. BRYLINSKI , ”1-motifs” et formes automorphes (Théorie arithmétique des domaines de Siegel), in Journées Automorphes, p. 43-106 (Public. math., Université Paris-VII, vol. 15, 1983 ). MR 85g:11047 | Zbl 0565.14001 · Zbl 0565.14001
[10] J. L. BRYLINSKI , (Co)-homologie d’intersection et faisceaux pervers , exposé au Séminaire Bourbaki, n^\circ 585, février 1982 . Numdam | MR 85i:32016a | Zbl 0574.14017 · Zbl 0574.14017
[11] P. DELIGNE , Formes modulaires et représentations l-adiques , exposé au séminaire Bourbaki, n^\circ 355, février 1969 . Numdam | Zbl 0206.49901 · Zbl 0206.49901
[12] P. DELIGNE , Travaux de Shimura , exposé au Séminaire Bourbaki, n^\circ 389. Numdam | Zbl 0225.14007 · Zbl 0225.14007
[13] P. DELIGNE , La conjecture de Weil II (Publ. Math. I.H.E.S., vol. 52, 1979 , p. 138-252). Numdam | Zbl 0456.14014 · Zbl 0456.14014
[14] M. DEMAZURE , Sous-groupes de rang maximum du groupe de Cremona (Ann. scient. Ec. norm. sup., 4e série, t. 3, fasc. 4, 1970 ). Numdam | MR 44 #1672 | Zbl 0223.14009 · Zbl 0223.14009
[15] M. DUFLO et J. P. LABESSE , Sur la formule des traces de Selberg (Ann. scient. Ec. norm. sup., 4e série, t. 4, 1971 , p. 193-284). Numdam | MR 55 #10392 | Zbl 0277.12011 · Zbl 0277.12011
[16] O. GABBER , Pureté de la cohomologie de la cohomologie de MacPherson-Goresky , rédigé par P. DELIGNE, prépublication I.H.E.S., 1981 .
[17] S. S. GELBART et H. JACQUET , Forms of GL(2) from the Analytic Point of View (Proc. of symp. in Pure Math., vol. 33, part 1, Amer. Math. Soc. R.I., 1979 , p. 213-251). MR 81e:10024 | Zbl 0409.22013 · Zbl 0409.22013
[18] M. GORESKY et R. MACPHERSON , Intersection Homology Theory II (à paraître). · Zbl 0529.55007
[19] G. HARDER , On the Cohomology of Discrete Arithmetically Defined Groups (Proc. of the Int. Coll. on Discrete Subgroups of Lie Groups and Applic. to Moduli, Bombay, 1973 , Oxford Univ. Press, 1975 ). Zbl 0317.57022 · Zbl 0317.57022
[20] M. HARRIS , Rationality Properties of Automorphic Forms (I. Preprint, 1980 , II. Preprint, 1981 ).
[21] H. JACQUET et R. P. LANGLANDS , Automorphic Forms on GL (2) , (Springer Lecture Notes, vol. 114, 1970 ). MR 53 #5481 | Zbl 0236.12010 · Zbl 0236.12010
[22] G. KEMPF , F. KNUDSEN , D. MUMFORD et D. SAINT-DONAT , Toroïdal Embeddings I (Springer Lecture Notes, vol. 339, 1972 ). Zbl 0271.14017 · Zbl 0271.14017
[23] B. KOSTANT , Lie Algebra Cohomology and the Generalized Borel-Weil Theorem (Annals of Math., vol. 74, 1961 ). MR 26 #265 | Zbl 0134.03501 · Zbl 0134.03501
[24] J. P. LABESSE et R. P. LANGLANDS , L-Indistinguishability for SL(2) (Canadian Journal of Math., vol. 31, 1979 , p. 726-785). MR 81b:22017 | Zbl 0421.12014 · Zbl 0421.12014
[25] R. P. LANGLANDS , Modular Forms and l-Adic Representations , Anvers II (Springer Lecture Notes, vol. 349, p. 362-500). MR 50 #7095 | Zbl 0279.14007 · Zbl 0279.14007
[26] R. P. LANGLANDS , On the Zeta-Function of Some Simple Shimura Varieties (Canadian Journal of Math., vol. 31, 1979 , p. 1121-1216). MR 82h:10042 | Zbl 0444.14016 · Zbl 0444.14016
[27] R. P. LANGLANDS , Lettre à M. Rapoport , 12 juin au 2 septembre 1974 .
[28] R. P. LANGLANDS , Stable Conjugacy (définitions and lemmas) (Canadian Journal of Math., vol. 39, 1979 , p. 700-725). MR 82j:10054 | Zbl 0421.12013 · Zbl 0421.12013
[29] R. P. LANGLANDS , Base Change for GL(2) (Ann. of Math. Studies, vol. 96, Princeton University Press, 1980 ). MR 82a:10032 | Zbl 0444.22007 · Zbl 0444.22007
[30] J. S. MILNE , Points on Shimura Varieties Mod p (Proc. Symp. in Pure Math., vol. 33, part 2, Amer. Math. Soc. R.I., 1979 , p. 165-184). MR 82e:10048 | Zbl 0418.14022 · Zbl 0418.14022
[31] K. NOMIZU , On the Cohomology of Compact Homogeneous Spaces of Nilpotent Lie Groups (Ann. of Math., vol. 59, 1954 , p. 531-538). MR 16,219c | Zbl 0058.02202 · Zbl 0058.02202
[32] M. RAPOPORT , Compactifications de l’espace des modules de Hilbert-Blumenthal (Comp. Math., vol. 36, 1978 , p. 255-335). Numdam | MR 80j:14009 | Zbl 0386.14006 · Zbl 0386.14006
[33] G. SHIMURA , On the Zeta-Functions of Algebraic Curves Uniformized by Certain Automorphic Functions (J. Math. Soc. Japan, vol. 13, 1961 , p. 275-331). Article | MR 26 #84 | Zbl 0218.14013 · Zbl 0218.14013
[34] G. SHIMURA , Construction of Class Fields and Zeta-Functions of Algebraic Curves (Annals of Math., vol. 85, 1967 , p. 58-159). MR 34 #4268 | Zbl 0204.07201 · Zbl 0204.07201
[35] W. VAN EST , A Generalization of the Cartan-Leray Spectral Sequence, II (Indag. Math., vol. XX, 1958 , p. 406-413). MR 21 #2236 | Zbl 0084.39202 · Zbl 0084.39202
[36] J. L. VERDIER , Classe d’homologie associée à un cycle (Séminaire de l’E.N.S., vol. 74-75, in Astérisque, vol. 36-37). Zbl 0346.14005 · Zbl 0346.14005
[37] S. ZUCKER , L2-Cohomology Warped Products and Arithmetic Groups (Invent. Math., vol. 70, fasc. 2, 1982 , p. 169-218). MR 86j:32063 | Zbl 0508.20020 · Zbl 0508.20020
[38] Variétés de Shimura et fonctions L , Publication mathématique de l’Université Paris-VII, n^\circ 6, 1979 . MR 84m:14025
[39] P. DELIGNE et M. RAPOPORT , Les schémas de modules de courbes elliptiques , Anvers II (Springer Lecture Notes, vol. 349, p. 143-316). MR 49 #2762 | Zbl 0281.14010 · Zbl 0281.14010
[40] D. MUMFORD , Hirzebruch’s Proportionality Principle in the Non-Compact Case (Inv. Math., vol. 42, 1971 , p. 239-272). MR 81a:32026 | Zbl 0365.14012 · Zbl 0365.14012
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.