Abstract
Given an elliptic curve over \(\mathbb Q\) with complex multiplication by \({\mathcal O}_K\), the ring of integers of the quadratic imaginary field K, we analyze the integer d E =gcd\(\{|E(\mathbb F_p)|:p\) splits in \({\mathcal O}_K\}\), where \(|E(\mathbb F_p)|\) is the size of the group of rational \(\mathbb F_p\) points, and prove that it can be bigger than the common factor that comes from the torsion of the curve. Then, we prove that #{p ≤ x, p splits in \({{\mathcal O}_K}:{\frac {1}{d_E}}|{\it E}({\mathbb F}_{p})|=P_2\}\gg {\it x}/(\log x)^2\) hence extending the results in [16]. This is the best known result in the direction of the Koblitz conjecture about the primality of \(|E(\mathbb F_p)|\).
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Jiménez Urroz, J. (2008). Almost Prime Orders of CM Elliptic Curves Modulo p . In: van der Poorten, A.J., Stein, A. (eds) Algorithmic Number Theory. ANTS 2008. Lecture Notes in Computer Science, vol 5011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79456-1_4
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