Quelques applications du théorème de densité de Chebotarev

Jean-Pierre Serre

Publications Mathématiques de l'IHÉS (1981)

  • Volume: 54, page 123-201
  • ISSN: 0073-8301

How to cite

top

Serre, Jean-Pierre. "Quelques applications du théorème de densité de Chebotarev." Publications Mathématiques de l'IHÉS 54 (1981): 123-201. <http://eudml.org/doc/103977>.

@article{Serre1981,
author = {Serre, Jean-Pierre},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {Chebotarev density; infinite Galois extension; generalized Riemann hypothesis; modular form; elliptic curve without complex multiplication; l-adic representation},
language = {fre},
pages = {123-201},
publisher = {Institut des Hautes Études Scientifiques},
title = {Quelques applications du théorème de densité de Chebotarev},
url = {http://eudml.org/doc/103977},
volume = {54},
year = {1981},
}

TY - JOUR
AU - Serre, Jean-Pierre
TI - Quelques applications du théorème de densité de Chebotarev
JO - Publications Mathématiques de l'IHÉS
PY - 1981
PB - Institut des Hautes Études Scientifiques
VL - 54
SP - 123
EP - 201
LA - fre
KW - Chebotarev density; infinite Galois extension; generalized Riemann hypothesis; modular form; elliptic curve without complex multiplication; l-adic representation
UR - http://eudml.org/doc/103977
ER -

References

top
  1. [1] E. ARTIN, Über eine neue Art von L-Reihen, Hamb. Abh., 3 (1923), 89-108 (= Coll. Papers, 105-124). JFM49.0123.01
  2. [2] N. BOURBAKI, Variétés différentielles et analytiques. Fascicule de résultats, Paris, Hermann, 1971. Zbl0217.20401
  3. [3] N. BOURBAKI, Groupes et Algèbres de Lie, chapitre II : « Algèbres de Lie libres » ; chapitre III : « Groupes de Lie », Paris, Hermann, 1972. Zbl0244.22007
  4. [4] S. CHOWLA, On the least prime in an arithmetic progression, J. Indian Math. Soc., 1 (1934), 1-3. Zbl0009.00802JFM60.0144.01
  5. [5] R. DEDEKIND, Über die Diskriminanten endlicher Körper, Gött. Abh., 29 (1882), 1-56 (= Ges. Math. Werke, I, 351-397). JFM14.0059.02
  6. [6] R. DEDEKIND, Vorlesungen über Zahlentheorie von P. G. Lejeune-Dirichlet, 4e éd., Braunschweig, 1893 (réimpr. : New York, Chelsea, 1968). 
  7. [7] P. DELIGNE, Formes modulaires et représentations ℓ-adiques, Sém. Bourbaki 1968-1969, exposé 355, Lecture Notes in Math., 179, Springer-Verlag, 1971, 139-172. Zbl0206.49901
  8. [8] P. DELIGNE, La conjecture de Weil, I, Publ. Math. I.H.E.S., 43 (1974), 273-307. Zbl0287.14001MR49 #5013
  9. [9] P. DELIGNE et J.-P. SERRE, Formes modulaires de poids 1, Ann. Sci. E.N.S., 4e série, 7 (1974), 507-530. Zbl0321.10026MR52 #284
  10. [10] J. DIEUDONNÉ et A. GROTHENDIECK, Critères différentiels de régularité pour les localisés des algèbres analytiques, J. of Algebra, 5 (1967), 305-324. Zbl0163.03201MR34 #7560
  11. [11] H. FEDERER, Geometric Measure Theory, Berlin, Springer-Verlag, 1969. Zbl0176.00801MR41 #1976
  12. [12] G. F. FROBENIUS, Über Beziehungen zwischen den Primidealen eines algebraischen Körpers und den Substitutionen seiner Gruppe, Sitz. Akad. Berlin (1896), 689-703 (= Ges. Abh., II, 719-733). Zbl27.0091.04JFM27.0091.04
  13. [13] G. H. HARDY et J. E. LITTLEWOOD, Tauberian theorems concerning power series and Dirichlet series whose coefficients are positive, Proc. London Math. Soc., 2e série, 13 (1914), 174-191 (= G. H. HARDY, Coll. Papers, VI, 510-527). Zbl45.0389.02JFM45.0389.02
  14. [14] G. H. HARDY et J. E. LITTLEWOOD, Some theorems concerning Dirichlet series, Messenger of Math., 43 (1914), 134-147 (= G. H. HARDY, Coll. Papers, VI, 542-555). JFM45.0390.01
  15. [15] K. HENSEL, Über die Entwicklung der algebraischen Zahlen in Potenzreihen, Math. Ann., 55 (1902), 301-336. Zbl32.0208.01JFM32.0208.01
  16. [16] H. HIRONAKA, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math., 79 (1964), 109-326. Zbl0122.38603MR33 #7333
  17. [17] J. IGUSA, Complex Powers and Asymptotic Expansions II, J. Crelle, 278-279 (1975), 307-321. Zbl0315.41029MR53 #8018
  18. [18] J. IGUSA, Some observations on higher degree characters, Amer. J. of Math., 99 (1977), 393-417. Zbl0373.12008MR56 #324
  19. [19] N. IWAHORI et H. MATSUMOTO, Several remarks on projective representations of finite groups, J. Fac. Sci. Univ. Tokyo, 10 (1964), 129-146. Zbl0125.01504MR31 #4841
  20. [20] H. D. KLOOSTERMAN, Asymptotische Formeln für die Fourierkoeffizienten ganzer Modulformen, Hamb. Abh., 5 (1927), 338-352. Zbl53.0346.01JFM53.0346.01
  21. [21] M. I. KNOPP et J. LEHNER, Gaps in the Fourier series of automorphic forms (à paraître). Zbl0473.10022
  22. [22] J. C. LAGARIAS, H. L. MONTGOMERY et A. M. ODLYZKO, A Bound for the Least Prime Ideal in the Chebotarev Density Theorem, Invent. Math., 54 (1979), 271-296. Zbl0401.12014MR81b:12013
  23. [23] J. C. LAGARIAS et A. M. ODLYZKO, Effective Versions of the Chebotarev Density Theorem, Algebraic Number Fields (A. Fröhlich edit.), New York, Academic Press, 1977, 409-464. Zbl0362.12011MR56 #5506
  24. [24] S. LANG et H. TROTTER, Frobenius Distributions in GL2-Extensions, Lect. Notes in Math., 504, Springer-Verlag, 1975. Zbl0329.12015MR58 #27900
  25. [25] R. P. LANGLANDS, Automorphic representations, Shimura varieties and motives. Ein Märchen, Proc. Symp. Pure Math., 33, Amer. Math. Soc., 1979, t. 2, 205-246. Zbl0447.12009MR83f:12010
  26. [26] M. LAZARD, Groupes analytiques p-adiques, Publ. Math. I.H.E.S., 26 (1965), 1-219. Zbl0139.02302MR35 #188
  27. [27] W. LI, Newforms and Functional Equations, Math. Ann., 212 (1975), 285-315. Zbl0278.10026MR51 #5498
  28. [28] B. MAZUR, Modular curves and the Eisenstein ideal, Publ. Math. I.H.E.S., 47 (1978), 33-186. Zbl0394.14008MR80c:14015
  29. [29] B. MAZUR, Rational Isogenies of Prime Degree, Invent. Math., 44 (1978), 129-162. Zbl0386.14009MR80h:14022
  30. [30] J. ŒSTERLÉ, Versions effectives du théorème de Chebotarev sous l'hypothèse de Riemann généralisée, Astérisque, 61 (1979), 165-167. Zbl0418.12005
  31. [31] A. OGG, Abelian curves of 2-power conductor, Proc. Camb. Phil. Soc., 62 (1966), 143-148. Zbl0163.15403MR34 #1320
  32. [32] R. RANKIN, Contribution to the theory of Ramanujan's function τ(n) and similar arithmetical functions, I, II, Proc. Camb. Phil. Soc., 35 (1939), 351-372. Zbl0021.39201MR1,69dJFM65.0353.01
  33. [33] K. RIBET, Galois Representations attached to Eigenforms with Nebentypus, Lect. Notes in Math., 601, Springer-Verlag, 1977, 17-52. Zbl0363.10015MR56 #11907
  34. [34] P. ROBBA, Lemmes de Schwarz et lemmes d'approximations p-adiques en plusieurs variables, Invent. Math., 48 (1978), 245-277. Zbl0392.12020MR80e:32021
  35. [35] I. ŠAFAREVIČ, Corps de nombres algébriques (en russe), Proc. Int. Congr. Math., Stockholm (1962), 163-176 (trad. anglaise : Amer. Math. Transl. (2), vol. 31 (1963), 25-39). 
  36. [36] L. SCHŒNFELD, Sharper Bounds for the Chebyshev Functions θ(x) and ѱ(x), II, Math. of Comp., 30 (1976), 337-360. Zbl0326.10037MR56 #15581b
  37. [37] S. SEN, Ramification in p-adic Lie Extensions, Invent. Math., 17 (1972), 44-50. Zbl0242.12012MR47 #8490
  38. [38] J.-P. SERRE, Corps locaux, Paris, Hermann, 1980, 3e éd. (trad. anglaise, Local Fields, GTM 67, Springer-Verlag, 1979). Zbl0423.12017
  39. [39] J.-P. SERRE, Classification des variétés analytiques p-adiques compactes, Topology, 3 (1965), 409-412. Zbl0141.37403MR31 #3421
  40. [40] J.-P. SERRE, Abelian ℓ-adic representations and elliptic curves, New York, Benjamin Publ., 1968. Zbl0186.25701MR41 #8422
  41. [41] J.-P. SERRE, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math., 15 (1972), 259-331. Zbl0235.14012MR52 #8126
  42. [42] J.-P. SERRE, Divisibilité de certaines fonctions arithmétiques, L'Ens. Math., 22 (1976), 227-260. Zbl0355.10021MR55 #7958
  43. [43] J.-P. SERRE, Modular forms of weight one and Galois representations, Algebraic Number Theory (A. Fröhlich edit.), New York, Academic Press, 1977, 193-268. Zbl0366.10022MR56 #8497
  44. [44] G. SHIMURA, Introduction to the arithmetic theory of automorphic functions, Publ. Math. Soc. Japan, vol. II, Princeton Univ. Press, 1971. Zbl0221.10029
  45. [45] H. STARK, Some effective cases of the Brauer-Siegel theorem, Invent. Math., 23 (1974), 135-152. Zbl0278.12005MR49 #7218
  46. [46] H. P. F. SWINNERTON-DYER, On ℓ-adic representations and congruences for coefficients of modular forms, Lecture Notes in Math., 350, Springer-Verlag, 1973, 1-55. Zbl0267.10032MR53 #10717a
  47. [47] N. TSCHEBOTAREFF, Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehören, Math. Ann., 95 (1926), 191-228. Zbl51.0149.04JFM51.0149.04
  48. [48] A. WEIL, Zeta-functions and Mellin transforms, Proc. Bombay Coll. on Alg. Geometry, Bombay, Tata Institute, 1968, 409-426 (= Œuvres Sci., III, 179-196). Zbl0193.49104MR41 #6857
  49. [49] H. WEYL, On the volume of tubes, Amer. J. of Math., 61 (1939), 461-472 (= Ges. Abh., III, 658-669). Zbl0021.35503JFM65.0796.01
  50. [50] E. WIRSING, Das asymptotische Verhalten von Summen über multiplikative Funktionen, Math. Ann., 143 (1961), 75-102. Zbl0104.04201MR24 #A1241

Citations in EuDML Documents

top
  1. Koopa Tak-Lun Koo, William Stein, Gabor Wiese, On the generation of the coefficient field of a newform by a single Hecke eigenvalue
  2. Raf Cluckers, Adriaan Herremans, The fundamental theorem of prehomogeneous vector spaces modulo p m (With an appendix by F. Sato)
  3. Ami Fischman, On the image of Λ -adic Galois representations
  4. Hans Roskam, Artin's primitive root conjecture for quadratic fields
  5. Rajiv Gupta, M. Ram Murty, Primitive points on elliptic curves
  6. D. W. Masser, Small values of the quadratic part of the Néron-Tate height on an abelian variety
  7. Daniel Bertrand, Groupes algébriques et équations différentielles linéaires
  8. Alexander F. Brown, Eknath P. Ghate, Endomorphism algebras of motives attached to elliptic modular forms
  9. Fedor Bogomolov, Yuri Zarhin, Ordinary reduction of K3 surfaces
  10. Pierre Dèbes, Nour Ghazi, Galois Covers and the Hilbert-Grunwald Property

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.