Geodesic
Small discriminants of complex multiplication fields of elliptic curves over finite fields
Czechoslovak Mathematical Journal, Tome 65 (2015) no. 2, pp. 381-388 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We obtain a conditional, under the Generalized Riemann Hypothesis, lower bound on the number of distinct elliptic curves $E$ over a prime finite field $\mathbb {F}_p$ of $p$ elements, such that the discriminant $D(E)$ of the quadratic number field containing the endomorphism ring of $E$ over $\mathbb {F}_p$ is small. For almost all primes we also obtain a similar unconditional bound. These lower bounds complement an upper bound of F. Luca and I. E. Shparlinski (2007).
We obtain a conditional, under the Generalized Riemann Hypothesis, lower bound on the number of distinct elliptic curves $E$ over a prime finite field $\mathbb {F}_p$ of $p$ elements, such that the discriminant $D(E)$ of the quadratic number field containing the endomorphism ring of $E$ over $\mathbb {F}_p$ is small. For almost all primes we also obtain a similar unconditional bound. These lower bounds complement an upper bound of F. Luca and I. E. Shparlinski (2007).
DOI : 10.1007/s10587-015-0183-4
Classification : 11G20, 11N32, 11R11
Keywords: elliptic curve; complex multiplication field; Frobenius discriminant 
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Shparlinski, Igor E. Small discriminants of complex multiplication fields of elliptic curves over finite fields. Czechoslovak Mathematical Journal, Tome 65 (2015) no. 2, pp. 381-388. doi: 10.1007/s10587-015-0183-4

[1] Cojocaru, A. C.: Questions about the reductions modulo primes of an elliptic curve. Number Theory; Papers from the 7th Conference of the Canadian Number Theory Association, University of Montreal, Canada, 2002. CRM Proc. Lecture Notes 36 American Mathematical Society, Providence (2004), 61-79 H. Kisilevsky et al. | MR | Zbl

[2] Cojocaru, A. C., Duke, W.: Reductions of an elliptic curve and their Tate-{S}hafarevich groups. Math. Ann. 329 (2004), 513-534. | DOI | MR | Zbl

[3] Cojocaru, A. C., Fouvry, E., Murty, M. R.: The square sieve and the Lang-{T}rotter conjecture. Can. J. Math. 57 (2005), 1155-1177. | DOI | MR | Zbl

[4] Iwaniec, H., Kowalski, E.: Analytic Number Theory. Amer. Math. Soc. Colloquium Publications 53 American Mathematical Society, Providence (2004). | MR | Zbl

[5] Konyagin, S. V., Shparlinski, I. E.: Quadratic non-residues in short intervals. (to appear) in Proc. Amer. Math. Soc.

[6] H. W. Lenstra, Jr.: Factoring integers with elliptic curves. Ann. Math. (2) 126 (1987), 649-673. | MR | Zbl

[7] Luca, F., Shparlinski, I. E.: Discriminants of complex multiplication fields of elliptic curves over finite fields. Can. Math. Bull. 50 (2007), 409-417. | DOI | MR | Zbl

[8] Montgomery, H. L.: Topics in Multiplicative Number Theory. Lecture Notes in Mathematics 227 Springer, Berlin (1971). | MR | Zbl

[9] Shparlinski, I. E.: Tate-{S}hafarevich groups and Frobenius fields of reductions of elliptic curves. Q. J. Math. 61 (2010), 255-263. | DOI | MR | Zbl

[10] Silverman, J. H.: The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics 106 Springer, New York (2009). | MR | Zbl

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