# Article

Full entry | Fulltext not available (moving wall 24 months)
Keywords:
elliptic curve; complex multiplication field; Frobenius discriminant
Summary:
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).
References:
[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 2076566 | Zbl 1085.11030
[2] Cojocaru, A. C., Duke, W.: Reductions of an elliptic curve and their Tate-{S}hafarevich groups. Math. Ann. 329 (2004), 513-534. DOI 10.1007/s00208-004-0517-2 | MR 2127988 | Zbl 1062.11039
[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 10.4153/CJM-2005-045-7 | MR 2178556 | Zbl 1094.11021
[4] Iwaniec, H., Kowalski, E.: Analytic Number Theory. Amer. Math. Soc. Colloquium Publications 53 American Mathematical Society, Providence (2004). MR 2061214 | Zbl 1059.11001
[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 0916721 | Zbl 0629.10006
[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 10.4153/CMB-2007-039-2 | MR 2344175 | Zbl 1146.11034
[8] Montgomery, H. L.: Topics in Multiplicative Number Theory. Lecture Notes in Mathematics 227 Springer, Berlin (1971). MR 0337847 | Zbl 0216.03501
[9] Shparlinski, I. E.: Tate-{S}hafarevich groups and Frobenius fields of reductions of elliptic curves. Q. J. Math. 61 (2010), 255-263. DOI 10.1093/qmath/hap001 | MR 2646088 | Zbl 1196.11080
[10] Silverman, J. H.: The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics 106 Springer, New York (2009). MR 2514094 | Zbl 1194.11005

Partner of