The rational points on the Jacobians of modular curves
Sbornik. Mathematics, Tome 30 (1976) no. 4, pp. 478-500 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this paper a theorem on the rank of the group of rational points of an Abelian variety with a sufficiently large ring of endomorphisms is proved. It is applied to construct nontrivial factors of the Jacobians of modular curves with finite groups of rational points and to prove finiteness theorems for modular curves. Bibliography: 20 titles.
@article{SM_1976_30_4_a3,
     author = {V. G. Berkovich},
     title = {The rational points on the {Jacobians} of modular curves},
     journal = {Sbornik. Mathematics},
     pages = {478--500},
     year = {1976},
     volume = {30},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1976_30_4_a3/}
}
TY  - JOUR
AU  - V. G. Berkovich
TI  - The rational points on the Jacobians of modular curves
JO  - Sbornik. Mathematics
PY  - 1976
SP  - 478
EP  - 500
VL  - 30
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/SM_1976_30_4_a3/
LA  - en
ID  - SM_1976_30_4_a3
ER  - 
%0 Journal Article
%A V. G. Berkovich
%T The rational points on the Jacobians of modular curves
%J Sbornik. Mathematics
%D 1976
%P 478-500
%V 30
%N 4
%U http://geodesic.mathdoc.fr/item/SM_1976_30_4_a3/
%G en
%F SM_1976_30_4_a3
V. G. Berkovich. The rational points on the Jacobians of modular curves. Sbornik. Mathematics, Tome 30 (1976) no. 4, pp. 478-500. http://geodesic.mathdoc.fr/item/SM_1976_30_4_a3/

[1] A. Atkin, J. Lehner, “Hecke operators on $\Gamma_0(m)$”, Math. Ann., 185 (1970), 134–160 | DOI | MR | Zbl

[2] A. Borel, Lineinye algebraicheskie gruppy, izd-vo «Mir», Moskva, 1972 | MR

[3] K. Doi, M. Jamaychi, “On the Hecke operators for $\Gamma_0(N)$ and class fields over quadratic number fields”, J. Math. Soc. Japan, 25:4 (1973), 629–643 | DOI | MR | Zbl

[4] A. Grothendieck, “Modèles de Nèron et monodromie”, Springer Lecture Notes, SGA 7, Expose IX, 288, 1972, 313–523 | MR | Zbl

[5] Ch. Kertis, I. Rainer, Teoriya predstavlenii konechnykh grupp i assotsiativnykh algebr, izd-vo «Nauka», Moskva, 1969 | MR

[6] B. Mazur, “Ratsionalnye tochki abelevykh mnogoobrazii nad bashnyami chislovykh polei”, Matematika, 17:2 (1973), 3–57 | MR | Zbl

[7] B. Mazur, Modular curves and the Eisenstein ideal, preprint | MR

[8] B. Mazur, Mordel's conjecture for modular curves over, preprint

[9] B. Mazur, J.-P. Serre, “Points rationnels des courbes modulaires $X_0(N)$”, Seminaire Bourbaki, no. 469, 1974/75

[10] Yu. I. Manin, “$p$-kruchenie ellipticheskikh krivykh ravnomerno ogranicheno”, Izv. AN SSSR, seriya matem., 33 (1969), 459–465 | MR | Zbl

[11] Yu. I. Manin, “Parabolicheskie tochki i dzeta-funktsii modulyarnykh krivykh”, Izv. AN SSSR, seriya matem., 36 (1972), 19–66 | MR | Zbl

[12] M. E. Novodvorskii, I. I. Pyatetskii-Shapiro, “Nekotorye zamecheniya o kruchenii ellipticheskikh krivykh”, Matem. sb., 82(124) (1970), 309–316 | MR | Zbl

[13] A. Ogg, “Survey of modular functions of one variable. Modular Functions of One Variable, IV”, Springer Lecture Notes, 320 (1973), 1–35 | MR | Zbl

[14] A. Ogg, “Hyperelliptic modular curves”, Bull. Soc. Math. France, 102 (1974), 449–462 | MR | Zbl

[15] K. Ribet, “Endomorphisms of semi-stable abelian varieties over number fields”, Ann. Math., 101:3 (1975), 555–562 | DOI | MR | Zbl

[16] Zh.-P. Serr, Dzh. Teit, “Khoroshaya reduktsiya abelevykh mnogoobrazii”, Matematika, 15:5 (1971), 140–165 | Zbl

[17] G. Shimura, Vvedenie v arifmeticheskuyu teoriyu avtomorfnykh funktsii, izd-vo «Mir», Moskva, 1973 | MR

[18] G. Shimura, “On the factors of the jacobian variety of a modular function field”, J. Math. Soc. Japan, 25:3 (1973), 523–544 | DOI | MR | Zbl

[19] Dzh. Teit, “$p$-delimye gruppy”, Matematika, 13:2 (1969), 3–25 | MR

[20] Dzh. Teit, F. Oort, “Skhemy grupp prostogo poryadka”, Matematika, 16:1 (1972), 165–183