Square-free orders for CM elliptic curves modulo $p$ in short intervals

TL;DR

This paper refines asymptotic formulas for counting primes where CM elliptic curves have square-free order reductions, providing shorter interval results and improvements over previous cyclicity estimates.

Contribution

It advances the understanding of prime distributions related to CM elliptic curves by refining asymptotic formulas and establishing shorter interval results unconditionally.

Findings

Asymptotic formulas for primes with square-free order reductions

Unconditional short interval asymptotics

Improved short interval cyclicity results

Abstract

Let $E$ be a CM elliptic curve over $\Bbb{Q}$. We refine the work of Cojocaru on the asymptotic formulae for the number of primes $p\le x$ for which the reduction modulo $p$ of $E$ is of square-free order. Also, we derive an unconditional short interval variant for the asymptotics. Compared to the estimate derived from the generalised Riemann hypothesis, the presented result is valid for even shorter intervals. Furthermore, we improve the short interval variant of the cyclicity problem for CM elliptic curves previously obtained by the author.