Almost Prime Orders of Elliptic Curves Over Prime Power Fields

Abstract: In 1988, Koblitz conjectured the infinitude of primes p for which |E(F_p)| is prime for elliptic curves E over Q, drawing an analogy with the twin prime conjecture. He also proposed studying the primality of |E(F_{p^l})| / |E(F_p)|, in parallel with the primality of (p^l - 1)/(p - 1). Motivated by these problems and earlier work on |E(F_p)|, we study the infinitude of primes p such that |E(F_{p^l})| / |E(F_p)| has a bounded number of prime factors for primes l >= 2, considering both CM and non-CM elliptic curves over Q. In the CM case, we focus on the curve y^2 = x^3 - x to address gaps in the literature and present a more concrete argument. The result is unconditional and applies Huxley's large sieve inequality for the associated CM field. In the non-CM case, analogous results follow under GRH via the effective Chebotarev density theorem. For the CM curve y^2 = x^3 - x, we further apply a vector sieve to combine the almost prime properties of |E(F_p)| and |E(F_{p^2})| / |E(F_p)|, establishing a lower bound for the number of primes p <= x for which |E(F_{p^2})| / 32 is a square-free almost prime. We also study cyclic subgroups of finite index in E(F_p) and E(F_{p^2}) for CM curves.