Geodesic
Small discriminants of complex multiplication fields of elliptic curves over finite fields
Czechoslovak Mathematical Journal, Tome 65 (2015) no. 2, pp. 381-388

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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).
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

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