Divisibility Estimates for Gaussian Norms and Almost-Prime Orders of a CM Elliptic Curve

SeungJu Lee

Let $E:y^2=x^3-x$. We prove that $\#E({\mathbf{F}}_p)/8$ is squarefree and has at most two prime factors for $\gg X/(\log X{)}^2$ primes $p\le X$. The proof follows the Gaussian CM sie…