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).
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机译：在广义黎曼假设下，我们获得一个条件，该条件是$ p $元素的素有限域$ mathbb {F} _p $上不同椭圆曲线$ E $的数量的下界，使得判别式$ D（E包含$ E $超过$ mathbb {F} _p $的同态环的二次数字段的）$小。对于几乎所有素数，我们还获得了类似的无条件界。这些下限与F.Luca和I.E.Shparlinski（2007）的上限互补。
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