Congruence obstructions in the refined Koblitz conjecture

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve. Zywina's refinement of Koblitz's conjecture predicts that there are infinitely many primes $p$ of good reduction for which $\#E_p(\mathbb{F}_p)$ is prime precisely when $E$ has no congruence obstruction. For non-CM elliptic curves over $\mathbb{Q}$, we classify all primitive congruence obstructions. In particular, we show that a primitive obstruction of composite level can occur only at $6$, $10$, $14$, $15$, or $30$ and determine the corresponding Galois images. As a consequence, for non-CM elliptic curves, the existence of any congruence obstruction is detected by the mod $210$ Galois image, so the positivity of the Koblitz--Zywina constant is determined at level $210$. We also determine which primitive obstruction levels can occur simultaneously. Finally, we prove, assuming GRH, that if $E$ has no congruence obstruction, then for every $0<κ<1/8$ there are infinitely many primes $p$ of good reduction for which the least prime divisor of $\#E_p(\mathbb{F}_p)$ is greater than $κ\log p$.