On the number of elliptic curves with CM cover large algebraic fields

Frey, Gerhard; Jarden, Moshe

doi:10.5802/aif.2163

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]

Frey, Gerhard ¹ ; Jarden, Moshe ²

¹ Essen University, Institute for Experimental Mathematics, Ellernstrasse 29, 45326 Essen (Allemagne)
² Tel Aviv University, School of Mathematics, Ramat Aviv, Tel Aviv 69978 (Israël)

Annales de l'Institut Fourier, Tome 55 (2005) no. 7, pp. 2361-2374.

Résumé
Abstract

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 $4$ .

MR Zbl

DOI : 10.5802/aif.2163

Classification : 12E30
Keywords: Elliptic curves with CM, large algebraic fields, absolute Galois group, Haar measure, class number, Elliptic curves with CM, large algebraic fields, absolute Galois group, Haar measure, class number
Mot clés : courbes elliptiques avec CM, grands corps algébriques, groupe de Galois absolu, mesure de Haar, nombre de classes

Affiliations des auteurs :

Frey, Gerhard ¹ ; Jarden, Moshe ²

¹ Essen University, Institute for Experimental Mathematics, Ellernstrasse 29, 45326 Essen (Allemagne)
² Tel Aviv University, School of Mathematics, Ramat Aviv, Tel Aviv 69978 (Israël)

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     title = {On the number of elliptic curves with {CM} cover large algebraic fields},
     journal = {Annales de l'Institut Fourier},
     pages = {2361--2374},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {55},
     number = {7},
     year = {2005},
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     mrnumber = {2207386},
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Frey, Gerhard; Jarden, Moshe. On the number of elliptic curves with CM cover large algebraic fields. Annales de l'Institut Fourier, Tome 55 (2005) no. 7, pp. 2361-2374. doi : 10.5802/aif.2163. http://archive.numdam.org/articles/10.5802/aif.2163/

Bibliographie
Cité par

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

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

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

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

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

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

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

[Jar] Jarden, M. Roots of unity over large algebraic fields, Mathematische Annalen, Volume 213 (1975), pp. 109-127 | DOI | MR | Zbl

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

[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), pp. 409-464 | MR | Zbl

[LeV] LeVeque, W.J. Topic in Number Theory I, Addison-Wesley, Reading, 1958 | Zbl

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

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