How many primes decompose completely in an infinite unramified Galois extension of a global field?

Yasutaka IHARA

doi:10.2969/jmsj/03540693

Yasutaka IHARA

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1983 Volume 35 Issue 4 Pages 693-709

DOI https://doi.org/10.2969/jmsj/03540693

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Published: 1983 Received: February 28, 1983 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: SUBTITLE Details: Wrong : Dedicated to Professor I. R. Šafarevic on his 60-th birthday

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) V. G. Drinfeld and S. G. Vladut, On the number of points of algebraic curves, (Russian), Functional Analysis, 17(1983), 68-69.
2) E. S. Golod and I. R. Šafarevic, On class field towers, (Russian), Izv. Akad Nauk SSSR, 28(1964), 261-272; (English transl.) Amer. Math. Soc. Transl. Ser 2, 48 (1965), 91-102.
3) Y. Ihara, The congruence monodromy problems, J. Math. Soc. Japan, 20 (1968), 107-121.
4) Y. Ihara, Non-abelian classfields over function fields in special cases, Actes du Congrès Internat. Math., Nice 1970, Tome 1, Gauthier-Villars, Paris, 381-389.
5) Y. Ihara, Congruence relations and Shimura curves I, Amer. Math. Soc. Proc. Symp. Pure Math., 33 (1979) , Vol. 2, 291-311; II, J. Fac. Sci. Univ. Tokyo Sect. IA 25 (1979), 301-361.
6) Y. Ihara, Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Univ. Tokyo Sect. IA, 28 (1981), 721-724.
7) Y. Ihara, On unramified extensions of function fields over finite fields, Advanced Studies in Pure Math., 2(1983) (the Proc. of Nagoya Symposium on Galois groups and their representations 1981), Kinokuniya, Tokyo, 89-97.
8) J. C. Lagarias and A. M. Odlyzko, Effective versions of the Chebotarev density theorem (Algebraic Number Fields, A. Fröhlich, Ed., Proc. of the 1975 Durham Symposium), Academic Press, 1977, 409-464.
9) A. M. Odlyzko, Lower bounds for discriminants of number fields, Acta Arith., 29 (1976), 275-297.
10) G. Poitou, (a) Minorations de discriminants (d'après A.M. Odlyzko), Sém. Bourbaki, 1975/76, exp. 479, Lecture Notes in Math., 567, 1977, 136-153.(b) Sur les petits discriminants, Sem. Delange-Pisot-Poitou (théorie des nombres, 18^e année, 1976/77, no. 6).
11) P. Roquette, On class field towers, in Alg. Number Theory, (Cassels and Fröhlich, Ed.), Proc. of an Instr. Conf. 1967, Academic Press, 231-249.
12) I. R. Šafarevic, Extensions with prescribed branch points, (Russian with French Summary), Publ. IHES, 18(1963), 71-95.
13) J-P. Serre, A course in arithmetic, Graduate Texts in Math., 7, Springer, 1973.
14) J-P. Serre, Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini, C.R. Acad. Sci. Paris, 296(1983), Série I, 397-402.
15) H. M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. Math., 23 (1974), 135-152.
16) A. Weil, Courbes algébriques et variétés abéliennes, Hermann, Paris.
17) A. Weil, Sur les “formules explicites” de la theorie des nombres premiers, Comm. Sém. Math. Univ. Lund 1952, Tomes supplémentaire, 252-265.

Right : [1] V. G. Drinfeld and S. G. Vladut, On the number of points of algebraic curves, (Russian), Functional Analysis, 17 (1983), 68-69.
[2] E. S. Golod and I. R. Šafarevic, On class field towers, (Russian), Izv. Akad Nauk SSSR, 28 (1964), 261-272; (English transl.) Amer. Math. Soc. Transl. Ser 2, 48 (1965), 91-102.
[3] Y. Ihara, The congruence monodromy problems, J. Math. Soc. Japan, 20 (1968), 107-121.
[4] Y. Ihara, Non-abelian classfields over function fields in special cases, Actes du Congrès Internat. Math., Nice 1970, Tome 1, Gauthier-Villars, Paris, 381-389.
[5] Y. Ihara, Congruence relations and Shimura curves I, Amer. Math. Soc. Proc. Symp. Pure Math., 33 (1979), Vol. 2, 291-311; II, J. Fac. Sci. Univ. Tokyo Sect. IA 25 (1979), 301-361.
[6] Y. Ihara, Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Univ. Tokyo Sect. IA, 28 (1981), 721-724.
[7] Y. Ihara, On unramified extensions of function fields over finite fields, Advanced Studies in Pure Math., 2 (1983) (the Proc. of Nagoya Symposium on Galois groups and their representations 1981), Kinokuniya, Tokyo, 89-97.
[8] J. C. Lagarias and A. M. Odlyzko, Effective versions of the Chebotarev density theorem (Algebraic Number Fields, A. Fröhlich, Ed., Proc. of the 1975 Durham Symposium), Academic Press, 1977, 409-464.
[9] A. M. Odlyzko, Lower bounds for discriminants of number fields, Acta Arith., 29 (1976), 275-297.
[10] G. Poitou, (a) Minorations de discriminants (d'après A. M. Odlyzko), Sém. Bourbaki, 1975/76, exp. 479, Lecture Notes in Math., 567, 1977, 136-153. (b) Sur les petits discriminants, Sem. Delange-Pisot-Poitou (théorie des nombres, 18^e année, 1976/77, no. 6).
[11] P. Roquette, On class field towers, in Alg. Number Theory, (Cassels and Fröhlich, Ed.), Proc. of an Instr. Conf. 1967, Academic Press, 231-249.
[12] I. R. Šafarevic, Extensions with prescribed branch points, (Russian with French Summary), Publ. IHES, 18 (1963), 71-95.
[13] J-P. Serre, A course in arithmetic, Graduate Texts in Math., 7, Springer, 1973.
[14] J-P. Serre, Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini, C. R. Acad. Sci. Paris, 296 (1983), Série I, 397-402.
[15] H. M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. Math., 23 (1974), 135-152.
[16] A. Weil, Courbes algébriques et variétés abéliennes, Hermann, Paris.
[17] A. Weil, Sur les “formules explicites” de la theorie des nombres premiers, Comm. Sém. Math. Univ. Lund 1952, Tomes supplémentaire, 252-265.

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