Eisenstein series and fields of definition

M. Karel

Compositio Mathematica (1978)

  • Volume: 37, Issue: 2, page 121-169
  • ISSN: 0010-437X

How to cite


Karel, M.. "Eisenstein series and fields of definition." Compositio Mathematica 37.2 (1978): 121-169. <http://eudml.org/doc/89378>.

author = {Karel, M.},
journal = {Compositio Mathematica},
keywords = {Eisenstein Series; Arithmetic Quotients of Bounded Symmetric Domains; Tube Domains; Fourier Coefficients; Cyclotomic Number Field; Congruence Subgroups},
language = {eng},
number = {2},
pages = {121-169},
publisher = {Sijthoff et Noordhoff International Publishers},
title = {Eisenstein series and fields of definition},
url = {http://eudml.org/doc/89378},
volume = {37},
year = {1978},

AU - Karel, M.
TI - Eisenstein series and fields of definition
JO - Compositio Mathematica
PY - 1978
PB - Sijthoff et Noordhoff International Publishers
VL - 37
IS - 2
SP - 121
EP - 169
LA - eng
KW - Eisenstein Series; Arithmetic Quotients of Bounded Symmetric Domains; Tube Domains; Fourier Coefficients; Cyclotomic Number Field; Congruence Subgroups
UR - http://eudml.org/doc/89378
ER -


  1. [1] A. Ash et al: Smooth Compactification of Locally Symmetric Varieties, Math. Sci. Press, Brookline, Massachusetts, 1976. 
  2. [2] W.L. Baily, Jr.: On the theory of θ-functions, the moduli of Abelian varieties, and the moduli of curves, Ann. of Math.75 (1962) 342-381. Zbl0147.39702
  3. [3] W.L. Baily, Jr.: On Hensel's lemma and exponential sums, Global Analysis: Papers in Honor of K. Kodaira (D.C. Spencer and S. Iyanga, ed.), Princeton University Press, Princeton, 1969, 85-100. Zbl0193.49201
  4. [4] W.L. Baily, Jr.: Eisenstein series on tube domains, Problems in Analysis: A symposium in Honor of Salomon Bochner, R.C. Gunning, ed., Princeton University Press, Princeton, 1970, 139-156. Zbl0211.10603MR361179
  5. [5] W.L. Baily, Jr.: Automorphic forms with integral Fourier coefficients, Several Complex Variables I—Maryland1970, Lecture Notes in Mathematics155, Springer-Verlag, Berlin - Heidelberg - New York, 1970, 1-8. Zbl0199.41501
  6. [6] W.L. Baily, Jr.: An exceptional arithmetic group and its Eisenstein series, Ann. of Math.91 (1970) 512-549. Zbl0202.07901MR269779
  7. [7] W.L. Baily, Jr.: On the Fourier coefficients of certain Eisenstein series on the adèle group, Number Theory, Algebraic Geometry and Commutative Algebra: in honor of Y. Akizuki, Kinokuniya, Tokyo, 1973. Zbl0283.10017MR367320
  8. [8] W.L. Baily, Jr.: Introductory Lectures on Automorphic Forms, Publications of the Mathematical Society of Japan12, Iwanami Shoten, Publishers, and Princeton University Press, Tokyo, 1973. Zbl0256.32001MR369750
  9. [9] W.L. Baily, Jr. and A. Borel: Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math.84 (1966) 442-528. Zbl0154.08602MR216035
  10. [10] A. Borel: Some finiteness properties of adèle groups over number fields, Publ. Math. I.H.E.S.16 (1963) 101-126. Zbl0135.08902MR202718
  11. [11] A. Borel: Introduction to Automorphic Forms, in Algebraic Groups and Discontinuous Subgroups (A. Borel and G.D. Mostow, ed.): Proc. Symp. Pure Math. IX, Amer. Math. Soc., Providence, R. I., 1966, 199-210. Zbl0191.09601MR207650
  12. [12] A. Borel: Linear Algebraic Groups (Notes by H. Bass), Benjamin Lecture Notes Series, New York, 1969. Zbl0186.33201MR251042
  13. [13] A. Borel: personal communication. 
  14. [14] A. Borel and J. Tits: Groupes réductifs, Publ. Math. I.H.E.S.27 (1965) 55-151. Zbl0145.17402MR207712
  15. [15] A. Borel, Compléments à l'article "Groupes réductifs,"Publ. Math. IHES.41 (1972) 253-276. Zbl0254.14018
  16. [16] F. Bruhat and J. Tits: Groupes réductifs sur un corps local: I. Données radicielles valuées, Publ. Math. I.H.E.S.41 (1972) 5-252. Zbl0254.14017MR327923
  17. [17] C. Chevalley: Séminaire C. Chevalley 1956-1958: Classification des groupes de Lie algébriques, mimeographed notes, École Normal Supérieure, Paris, 1958. Zbl0092.26301
  18. [18] U. Christian: Zur Theorie der Hilbert-Siegelschen Modulfunktionen, Math. Ann.152 (1963) 275-341. Zbl0143.30403MR159038
  19. [19] U. Christian, Zur Theorie der Hilbert-Siegelschen Modulfunktionen über den Eisensteinreihen, Abhandl. math. Sem. Hamburg Univ.27 (1964). Zbl0143.30403MR159038
  20. [20] P. Deligne: Travaux de Shimura, Séminaire Bourbaki, 1970/71, exposé 389, in Lecture Notes in Mathematics244, Springer-Verlag, Berlin-Heidelberg -New York, 1971. Zbl0225.14007MR498581
  21. [21] R. Godement: Introduction à la théorie de Langlands, Séminaire Bourbaki, 1966/67, exposé 321, W.A. Benjaminin, New York, and Amsterdam, 1968. Zbl0216.15001
  22. [22] R. Gunning: Lectures on Modular Forms, Princeton University Press, Princeton, 1962. Zbl0178.42901MR132828
  23. [23] E. Hecke: Analytische Arithmetik der positiven quadratischen Formen, Kgl. Danske Vid. Selsk. Math. - fys. Medd.XVII, 12 (1940), (Math. Werke; 789-918). Zbl0024.00902MR3665JFM66.0128.01
  24. [24] S. Helgason: Differential Geometry and Symmetric Spaces, Academic Press, New York, 1962. Zbl0111.18101MR145455
  25. [25] H. Klingen: Über den arithmetischen Charakter der Fourier koeffizienten von Modulformen, Math. Ann.147 (1962) 176-188. Zbl0104.26502MR138598
  26. [26] H.D. Kloosterman: Theorie der Eisensteinschen Reihen von mehreren Veränderlichen, Abhandl. math. Sem. Hamburg Univ.6 (1927) 163-188. Zbl54.0406.01JFM54.0406.01
  27. [27] M. Kneser: Starke Approximation in Algebraischen Gruppen. I, J. f. reine angew. Math.218 (1965) 190-203. Zbl0143.04701MR184945
  28. [28] A. Koranyi and J. Wolf: Realization of Hermitian Symmetric Spaces as Generalized Half-Planes, Ann. of Math.81 (1965) 265-288. Zbl0137.27402MR174787
  29. [29] A. Ogg: Modular Forms and Dirichlet Series, W.A. Benjamin, New York and Amsterdam, 1969. Zbl0191.38101MR256993
  30. [30] H. Petersson: Über den Körper der Fourierkoeffizienten der von Hecke untersuchten Eisensteinreihen, Abhandl. math. Sem. Hamburg Univ.16 (1949) 101-113. Zbl0034.34903MR33315
  31. [31] G. Prasad: Strong approximation for semi-simple groups over function fields. Ann. of Math.105 (1977) 553-572. Zbl0348.22006MR444571
  32. [32] G. Shimura: On canonical models of arithmetic quotients of bounded symmetric domains, I, II, Ann. of Math91 (1970) 144-222; 92 (1970) 528-549. Zbl0237.14010MR257031
  33. [33] G. Shimura: On the Fourier coefficients of modular forms of several variables, Nachr. Akad. Wiss. Göttingen, 1975, Nr. 17. Zbl0332.32024MR485706
  34. [34] C.L. Siegel: Über die analytische Theorie der quadratischen Formen, Ann. of Math36 (1935) 527-606 (Ges. Abhandl.I, 326-405). Zbl0012.19703MR1503238JFM61.0140.01
  35. [35] T.A. Springer: Jordan Algebras and Algebraic Groups, Springer-Verlag, NewYork-Heidelberg -Berlin, 1973. Zbl0259.17003
  36. [36] L.-C. Tsao: The rationality of the Fourier coefficients of certain Eisenstein series on tube domains, Compositio Math.32 (1976) 225-291. Zbl0346.10014MR419367
  37. [37] L.-C. Tsao: On saturated subgroups, preprint. 
  38. [38] L.-C. Tsao: Saturated normal congruence subgroups of the modular group, preprint. 
  39. [39] E.B. Vinberg: Homogeneous cones, Soviet Math.1 (1960) 787-793 (translated from Doklady Akad. Nauk SSSR133 (1960)). Zbl0143.05203MR141680
  40. [40] A. Weil: Adèles and algebraic groups (Notes by M. Demazure and T. Ono), Institute for Advanced Study, Princeton, 1961. MR670072
  41. [41] A. Weil: Basic Number Theory. Springer-Verlag, Berlin-Heidelberg-New York, 1967. Zbl0176.33601MR234930

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