Frey, Gerhard; Jarden, Moshe
On the number of elliptic curves with CM cover large algebraic fields  [ Sur le nombre de courbes elliptiques avec CM sur des grands corps algèbriques ]
Annales de l'institut Fourier, Tome 55 (2005) no. 7 , p. 2361-2374
Zbl 1090.11036 | MR 2207386
doi : 10.5802/aif.2163
URL stable : http://www.numdam.org/item?id=AIF_2005__55_7_2361_0

Classification:  12E30
Mots clés: courbes elliptiques avec CM, grands corps algébriques, groupe de Galois absolu, mesure de Haar, nombre de classes
Les courbes elliptiques avec CM révèlent un nouveau phénomène dans la théorie des grands corps algébriques. Plutôt que de tracer une ligne entre 0 et 1 ou entre 1 et 2, elles fournissent un exemple où la ligne va de 2 à 3, ainsi qu’un exemple de ligne allant de 3 à 4
Elliptic curves with CM unveil a new phenomenon in the theory of large algebraic fields. Rather than drawing a line between 0 and 1 or 1 and 2 they give an example where the line goes beween 2 and 3 and another one where the line goes between 3 and 4.

Bibliographie

[BoS] Borevich, Z.I.; Shafarevich, I.R. Number Theory, Academic Press (1966) MR 195803 | Zbl 0145.04902

[FoM] Fouvry, E.; Murty, M.R. On the distribution of supersingular primes, Canadian Journal of Mathematics, 48 (1996), p. 81--104 Article  MR 1382477 | Zbl 0864.11030

[FrJ] Fried, M.D.; Jarden, M. Field Arithmetic, 2nd Edition, revised and enlarged by Moshe Jarden, Springer (2005) MR 2102046 | Zbl 1055.12003

[FyJ] Frey, G.; Jarden, M. Approximation theory and the rank of abelian varieties over large algebraic fields, Proceedings of the London Mathematical Society, 28 (1974), p. 112--128 Article  MR 337997 | Zbl 0275.14021

[Gol] Goldstein, L.J. Analytic Number Theory, Prentice-Hall, Englewood Cliffs (1971) MR 498335 | Zbl 0226.12001

[JaJ] Jacobson, M.; Jarden, M. Finiteness theorems for torsion of abelian varieties over large algebraic fields, Acta Arithmetica, 98 (2001), p. 15--31 Article  MR 1831454 | Zbl 01605208

[Jan] Janusz, G.J. Algebraic Number Fields, Academic Press, New York (1973) MR 366864 | Zbl 0307.12001

[Jar] Jarden, M. Roots of unity over large algebraic fields, Mathematische Annalen, 213 (1975), p. 109--127 Article  MR 364205 | Zbl 0278.12102

[Lan] Lang, S. Elliptic Functions, Addison-Wesley, Reading (1973) MR 409362 | Zbl 0316.14001

[LaO] Lagarias, J.C.; Odlyzko, A.M. Effective versions of the Chebotarev density theorem, Algebraic Number Fields, A. Fröhlich, Academic Press, Proceedings of a symposium organised by the London Mathematical Society and held in Durham University (1997), p. 409--464 MR 447191 | Zbl 0362.12011

[LeV] Leveque, W.J. Topic in Number Theory I, Addison-Wesley, Reading (1958) Zbl 0070.03803

[Shi] Shimura, G. Introduction to the Arithmetic Theory of Automorphic Functions, Iwanami Shoten Publishers and Princeton University Press (1971) MR 1291394 | Zbl 0221.10029