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On the number of elliptic curves with CM over large algebraic fields. (English) Zbl 1090.11036
Let \(\mathbb Q\) be the rational number field and let Gal\(({\mathbb Q})\) be its absolute Galois group. For any integer \(e\geq{1}\), let Gal\(({\mathbb Q})^{e}\) be the Cartesian power. Let \(\widetilde{\mathbb Q}\) be the algebraic closure of \(\mathbb Q\). For each \({\sigma}=(\sigma_1, \sigma_2, ..., \sigma_{n})\), let \(\widetilde{Q({\sigma})}\) be the fixed field in \(\widetilde{\mathbb Q}\) of \(\sigma_1, \sigma_2, ..., \sigma_{n}\). For each \({\sigma}\in \)Gal\(({\mathbb Q})^{e}\), we want to know the number of the elliptic curves \(E\) (up to \(C\)-isomorphism) with CM that are defined over \(\widetilde{\mathbb Q({\sigma})}\). We are also interested in knowing the number of elliptic curves \(E\) (up to \(C\)-isomorphism) with CM that are defined over \(\widetilde{\mathbb Q({\sigma})}\), and such that all \(C\)-endomorphisms of \(E\) are defined over \(\widetilde{\mathbb Q({\sigma})}\). The authors cleverly use several fundamental results from their previous work and other authors, to answer these questions. In fact, they show there is a positive integer \(e_0\) such that there are infinitely many elliptic curves \(E\) (up to \(C\)-isomorphism) with CM over \(\widetilde{\mathbb Q({\sigma})}\) if and only if \(e\leq{e_0}\). The authors specifically prove \(e_0=3\) for the first task and \(e_0=2\) for the second one.
11G05 Elliptic curves over global fields
12E30 Field arithmetic
14H52 Elliptic curves
14K22 Complex multiplication and abelian varieties
Full Text: DOI Numdam EuDML
[1] Borevich, Z. I.; Shafarevich, I. R., Number Theory, (1966), Academic Press · Zbl 0145.04902
[2] Fouvry, E.; Murty, M. R., On the distribution of supersingular primes, Canadian Journal of Mathematics, 48, 81-104, (1996) · Zbl 0864.11030
[3] Fried, M. D.; Jarden, M., 2nd Edition, revised and enlarged by Moshe Jarden, Field arithmetic, (2005), Springer · Zbl 1055.12003
[4] Frey, G.; Jarden, M., Approximation theory and the rank of abelian varieties over large algebraic fields, Proceedings of the London Mathematical Society, 28, 112-128, (1974) · Zbl 0275.14021
[5] Goldstein, L. J., Analytic Number Theory, (1971), Prentice-Hall, Englewood Cliffs · Zbl 0226.12001
[6] Jacobson, M.; Jarden, M., Finiteness theorems for torsion of abelian varieties over large algebraic fields, Acta Arithmetica, 98, 15-31, (2001) · Zbl 1124.14304
[7] Janusz, G. J., Algebraic Number Fields, (1973), Academic Press, New York · Zbl 0307.12001
[8] Jarden, M., Roots of unity over large algebraic fields, Mathematische Annalen, 213, 109-127, (1975) · Zbl 0278.12102
[9] Lang, S., Elliptic Functions, (1973), Addison-Wesley, Reading · Zbl 0316.14001
[10] Lagarias, J. C.; Odlyzko, A. M., Effective versions of the chebotarev density theorem, Algebraic Number Fields, 409-464, (1997), A. Fröhlich, Academic Press · Zbl 0362.12011
[11] LeVeque, W. J., Topic in Number Theory I, (1958), Addison-Wesley, Reading · Zbl 0070.03803
[12] Shimura, G., Introduction to the Arithmetic Theory of Automorphic Functions, (1971), Iwanami Shoten Publishers and Princeton University Press · Zbl 0221.10029
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