|
Répartition asymptotique des valeurs propres de l'opérateur de Hecke 
Author:
Jean-Pierre Serre
Journal:
J. Amer. Math. Soc. 10 (1997), 75-102
MSC (1991):
Primary 11F11
MathSciNet review:
1396897
Full-text PDF Free Access
References |
Similar Articles |
Additional Information
- 1.
A.
O. L. Atkin and J.
Lehner, Hecke operators on Γ₀(𝑚), Math.
Ann. 185 (1970), 134–160. MR 0268123
(42 #3022)
- 2.
N.
Bourbaki, Éléments de mathématique. Fasc.
XIII. Livre VI: Intégration. Chapitres 1, 2, 3 et 4:
Inégalités de convexité, Espaces de Riesz, Mesures sur
les espaces localement compacts, Prolongement d’une mesure, Espaces
𝐿^{𝑝}, Deuxième édition revue et
augmentée. Actualités Scientifiques et Industrielles, No.
1175, Hermann, Paris, 1965 (French). MR 0219684
(36 #2763)
- 3.
Armand
Brumer, The rank of 𝐽₀(𝑁),
Astérisque 228 (1995), 3, 41–68. Columbia
University Number Theory Seminar (New York, 1992). MR 1330927
(96f:11083)
- 4.
P.
Cartier, Harmonic analysis on trees, Harmonic analysis on
homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll.,
Williamstown, Mass., 1972), Amer. Math. Soc., Providence, R.I., 1973,
pp. 419–424. MR 0338272
(49 #3038)
- 5.
Henri
Cohen, Trace des opérateurs de Hecke sur
Γ₀(𝑁), Séminaire de Théorie des
Nombres (1976–1977), Lab. Théorie des Nombres, CNRS, Talence,
1977, pp. Exp. No. 4, 9 (French). MR 0562292
(58 #27771)
- 6.
H. Cohen, Sur les
tels que soit -isogène à un produit de courbes ellliptiques, Bordeaux, 1994.
- 7.
David
A. Cox and Walter
R. Parry, Genera of congruence subgroups in 𝑄-quaternion
algebras, J. Reine Angew. Math. 351 (1984),
66–112. MR
749678 (85i:11029)
- 8.
Torsten
Ekedahl and Jean-Pierre
Serre, Exemples de courbes algébriques à jacobienne
complètement décomposable, C. R. Acad. Sci. Paris
Sér. I Math. 317 (1993), no. 5, 509–513
(French, with English and French summaries). MR 1239039
(94j:14029)
- 9.
J.-H.
Evertse and J.
H. Silverman, Uniform bounds for the number of solutions to
𝑌ⁿ=𝑓(𝑋), Math. Proc. Cambridge Philos.
Soc. 100 (1986), no. 2, 237–248. MR 848850
(87k:11034), http://dx.doi.org/10.1017/S0305004100066068
- 10.
G.
H. Hardy and E.
M. Wright, An introduction to the theory of numbers, Oxford,
at the Clarendon Press, 1954. 3rd ed. MR 0067125
(16,673c)
- 11.
Ki-ichiro
Hashimoto, Zeta functions of finite graphs and representations of
𝑝-adic groups, Automorphic forms and geometry of arithmetic
varieties, Adv. Stud. Pure Math., vol. 15, Academic Press, Boston,
MA, 1989, pp. 211–280. MR 1040609
(91i:11057)
- 12.
M.
N. Huxley, A note on polynomial congruences, Recent progress
in analytic number theory, Vol. 1 (Durham, 1979) Academic Press, London,
1981, pp. 193–196. MR 637347
(83e:10005)
- 13.
Jun-ichi
Igusa, Kroneckerian model of fields of elliptic modular
functions, Amer. J. Math. 81 (1959), 561–577.
MR
0108498 (21 #7214)
- 14.
Yasutaka
Ihara, On discrete subgroups of the two by two projective linear
group over 𝔭-adic fields, J. Math. Soc. Japan
18 (1966), 219–235. MR 0223463
(36 #6511)
- 15.
Yasutaka
Ihara, Some remarks on the number of rational points of algebraic
curves over finite fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math.
28 (1981), no. 3, 721–724 (1982). MR 656048
(84c:14016)
- 16.
Alexander
Lubotzky, Discrete groups, expanding graphs and invariant
measures, Progress in Mathematics, vol. 125, Birkhäuser
Verlag, Basel, 1994. With an appendix by Jonathan D. Rogawski. MR 1308046
(96g:22018)
- 17.
A.
Lubotzky, R.
Phillips, and P.
Sarnak, Ramanujan graphs, Combinatorica 8
(1988), no. 3, 261–277. MR 963118
(89m:05099), http://dx.doi.org/10.1007/BF02126799
- 18.
F.
I. Mautner, Spherical functions over 𝔓-adic fields. I,
Amer. J. Math. 80 (1958), 441–457. MR 0093558
(20 #82)
F.
I. Mautner, Spherical functions over 𝔟-adic fields.
II, Amer. J. Math. 86 (1964), 171–200. MR 0166305
(29 #3582)
- 19.
Brendan
D. McKay, The expected eigenvalue distribution of a large regular
graph, Linear Algebra Appl. 40 (1981), 203–216.
MR 629617
(84h:05089), http://dx.doi.org/10.1016/0024-3795(81)90150-6
- 20.
J.-F.
Mestre, La méthode des graphes. Exemples et
applications, fundamental units of algebraic number fields (Katata,
1986) Nagoya Univ., Nagoya, 1986, pp. 217–242 (French). MR 891898
(88e:11025)
- 21.
Kenneth
A. Ribet, Twists of modular forms and endomorphisms of abelian
varieties, Math. Ann. 253 (1980), no. 1,
43–62. MR
594532 (82e:10043), http://dx.doi.org/10.1007/BF01457819
- 22.
Jürgen
Rohlfs and Birgit
Speh, On limit multiplicities of representations with cohomology in
the cuspidal spectrum, Duke Math. J. 55 (1987),
no. 1, 199–211. MR 883670
(88k:22010), http://dx.doi.org/10.1215/S0012-7094-87-05511-6
- 23.
Gordan
Savin, Limit multiplicities of cusp forms, Invent. Math.
95 (1989), no. 1, 149–159. MR 969416
(90c:22035), http://dx.doi.org/10.1007/BF01394147
- 24.
René
Schoof and Marcel
van der Vlugt, Hecke operators and the weight distributions of
certain codes, J. Combin. Theory Ser. A 57 (1991),
no. 2, 163–186. MR 1111555
(92g:94017), http://dx.doi.org/10.1016/0097-3165(91)90016-A
- 25.
Jean-Pierre
Serre, Arbres, amalgames, 𝑆𝐿₂,
Société Mathématique de France, Paris, 1977 (French).
Avec un sommaire anglais; Rédigé avec la collaboration de
Hyman Bass; Astérisque, No. 46. MR 0476875
(57 #16426)
- 26.
Jean-Pierre
Serre, Sur le nombre des points rationnels d’une courbe
algébrique sur un corps fini, C. R. Acad. Sci. Paris
Sér. I Math. 296 (1983), no. 9, 397–402
(French, with English summary). MR 703906
(85b:14027)
- 27.
Freydoon
Shahidi, Symmetric power 𝐿-functions for
𝐺𝐿(2), Elliptic curves and related topics, CRM Proc.
Lecture Notes, vol. 4, Amer. Math. Soc., Providence, RI, 1994,
pp. 159–182. MR 1260961
(95c:11066)
- 28.
Goro
Shimura, Introduction to the arithmetic theory of automorphic
functions, Publications of the Mathematical Society of Japan, No. 11.
Iwanami Shoten, Publishers, Tokyo, 1971. Kan\cflex o Memorial Lectures, No.
1. MR
0314766 (47 #3318)
- 29.
Goro
Shimura, On the factors of the jacobian variety of a modular
function field, J. Math. Soc. Japan 25 (1973),
523–544. MR 0318162
(47 #6709)
- 30.
Allan
J. Silberger, 𝑃𝐺𝐿₂ over the
𝑝-adics: its representations, spherical functions, and Fourier
analysis, Lecture Notes in Mathematics, Vol. 166, Springer-Verlag,
Berlin, 1970. MR
0285673 (44 #2891)
- 31.
Michael
A. Tsfasman, Some remarks on the asymptotic number of points,
Coding theory and algebraic geometry (Luminy, 1991) Lecture Notes in
Math., vol. 1518, Springer, Berlin, 1992, pp. 178–192. MR 1186424
(93h:11064), http://dx.doi.org/10.1007/BFb0088001
- 32.
M. A. Tsfasman et S. G. Vl\u{a}du\c{t}, Asymptotic properties of zeta functions, Prépubl. de l'I.M.L.,
96-12, C.N.R.S., Marseille (1996).
- 1.
- A. O. L. Atkin et J. Lehner, Hecke operators on
, Math. Ann. 185 (1970), 134-160. MR 42:3022
- 2.
- N. Bourbaki, Intégration, chap. I-IV,
édition, Hermann, Paris, 1965. MR 36:2763
- 3.
- A. Brumer, The rank of
, Astérisque 228 (1995), 41-68. MR 96f:11083
- 4.
- P. Cartier, Harmonic analysis on trees, A.M.S. Proc. Sympos. Pure Math. 26 (1973), 419-424. MR 49:3038
- 5.
- H. Cohen, Trace des opérateurs de Hecke sur
, Séminaire de théorie des nombres de Bordeaux 1976-1977, exposé 4. MR 58:27771
- 6.
- H. Cohen, Sur les
tels que soit -isogène à un produit de courbes ellliptiques, Bordeaux, 1994.
- 7.
- D. A. Cox et W. R. Parry, Genera of congruence subgroups in
-quaternion algebras, J. Crelle 351 (1984), 66-112. MR 85i:11029
- 8.
- T. Ekedahl et J.-P. Serre, Exemples de courbes algébriques à jacobienne complètement décomposable, C. R. Acad. Sci. Paris 317 (1993), 509-513. MR 94j:14029
- 9.
- J.-H. Evertse et J. H. Silverman, Uniform bounds for the number of solutions to
, Math. Proc. Cambridge Philos. Soc. 100 (1986), 237-248. MR 87k:11034
- 10.
- G. Hardy et E. Wright, An Introduction to the Theory of Numbers,
édition, Oxford, 1954. MR 16:673c
- 11.
- K. Hashimoto, Zeta functions of finite graphs and representations of
-adic groups, Adv. Studies in Pure Math., vol. 15, Kinokuniya Company Ltd. et Acad. Press, Tokyo, 1989, pp. 211-280. MR 91i:11057
- 12.
- M. N. Huxley, A note on polynomial congruences, Recent Progress in Analytic Number Theory (Durham, 1979), Academic Press, 1981, pp. 193-196. MR 83e:10005
- 13.
- J. Igusa, Kroneckerian model of fields of elliptic modular functions, Amer. J. Math. 81 (1959), 561-577. MR 21:7214
- 14.
- Y. Ihara, On discrete subgroups of the two by two projective linear group over
-adic fields, J. Math. Soc. Japan 18 (1966), 219-235. MR 36:6511
- 15.
- Y. Ihara, Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Tokyo 28 (1982), 721-724. MR 84c:14016
- 16.
- A. Lubotzky, Discrete Groups, Expanding Graphs and Invariant Measures, Progress in Math., vol. 125, Birkhäuser Verlag, 1994. MR 96g:22018
- 17.
- A. Lubotzky, R. Phillips et P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988), 261-277. MR 89m:05099
- 18.
- F. I. Mautner, Spherical functions over
-adic fields I, Amer. J. Math. 80 (1958), 441-457; II, ibid., 86 (1964), 171-200. MR 20:82; MR 29:3582
- 19.
- B. D. McKay, The expected eigenvalue distribution of a large regular graph, Linear Algebra and its Applications 40 (1981), 203-216. MR 84h:05089
- 20.
- J-F. Mestre, La méthode des graphes. Exemples et applications, Taniguchi Symp., Kyoto (1986), pp. 217-242. MR 88e:11025
- 21.
- K. Ribet, Twists of modular forms and endomorphisms of abelian varieties, Math. Ann. 253 (1980), 43-62. MR 82e:10043
- 22.
- J. Rohlfs et B. Speh, On limit multiplicities of representations with cohomology in the cuspidal spectrum, Duke Math. J. 55 (1987), 199-212. MR 88k:22010
- 23.
- G. Savin, Limit multiplicities of cusp forms, Invent. Math. 95 (1989), 149-159. MR 90c:22035
- 24.
- R. Schoof et M. van der Vlugt, Hecke operators and weight distribution of certain codes, J. Combinatorial Theory, série A, 57 (1991), 163-186. MR 92g:94017
- 25.
- J-P. Serre, Arbres, Amalgames,
, Astérisque 46, S.M.F., 1977 (trad. anglaise: Trees, Springer-Verlag, 1980). MR 57:16426
- 26.
- J-P. Serre, Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini, C. R. Acad. Sci. Paris 296 (1983), 397-402 (= Oe. 128). MR 85b:14027
- 27.
- F. Shahidi, Symmetric power
-functions for , C.R.M. Proc., vol. 4, A.M.S., 1994, pp. 159-182. MR 95c:11066
- 28.
- G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Iwanomi Shoten et Princeton University Press, Princeton, 1971. MR 47:3318
- 29.
- G. Shimura, On the factors of the jacobian variety of a modular function field, J. Math. Soc. Japan 25 (1973), 523-544. MR 47:6709
- 30.
- A. J. Silberger,
over the -adics: its representations, spherical functions and Fourier analysis, Lect. Notes in Math., vol. 166, Springer-Verlag, 1970. MR 44:2891
- 31.
- M. A. Tsfasman, Some remarks on the asymptotic number of points, Lect. Notes in Math., vol. 1518, Springer-Verlag, 1992, pp. 178-192. MR 93h:11064
- 32.
- M. A. Tsfasman et S. G. Vl\u{a}du\c{t}, Asymptotic properties of zeta functions, Prépubl. de l'I.M.L.,
96-12, C.N.R.S., Marseille (1996).
Similar Articles
Retrieve articles in Journal of the American Mathematical Society
with MSC (1991):
11F11
Retrieve articles in all journals
with MSC (1991):
11F11
Additional Information
Jean-Pierre Serre
Affiliation:
Collège de France, 3 rue d’Ulm, F-75231 Paris Cedex 05, France
Email:
serre@dmi.ens.fr
DOI:
http://dx.doi.org/10.1090/S0894-0347-97-00220-8
PII:
S 0894-0347(97)00220-8
Received by editor(s):
March 1, 1996
Article copyright:
© Copyright 1997 American Mathematical Society
|