Cyclicity and exponent of elliptic curves modulo $p$ in arithmetic progressions

TL;DR

This paper investigates the cyclicity and exponent of elliptic curves modulo primes in arithmetic progressions, providing new asymptotic formulas and refined estimates that extend and improve previous results in the field.

Contribution

It extends recent work by proving an unconditional asymptotic for CM elliptic curves and refines conditional estimates, advancing understanding of elliptic curve behavior in arithmetic progressions.

Findings

Unconditional asymptotic formulas for cyclicity of CM elliptic curves.

Log-power savings in conditional estimates for small moduli.

Extended estimates for the average exponent of elliptic curves modulo primes.

Abstract

In this article, we study the cyclicity problem of elliptic curves $E/\Bbb{Q}$ modulo primes in a given arithmetic progression. We extend the recent work of Akbal and G\"ulo\u{g}lu by proving an unconditional asymptotic for such a cyclicity problem over arithmetic progressions for CM elliptic curves $E$, which also presents a generalisation of the previous works of Akbary, Cojocaru, M.R. Murty, V.K. Murty, and Serre. In addition, we refine the conditional estimates of Akbal and G\"ulo\u{g}lu, which gives log-power savings (for small moduli) and consequently improves the work of Cojocaru and M.R. Murty. Moreover, we study the average exponent of $E$ modulo primes in a given arithmetic progression and obtain several conditional and unconditional estimates, extending the previous works of Freiberg, Kim, Kurlberg, and Wu.