Opposing Average Congruence Class Biases in the Cyclicity and Koblitz Conjectures for Elliptic Curves

TL;DR

This paper investigates the distribution of primes related to elliptic curves, revealing an opposing bias in the average behavior of constants in the cyclicity and Koblitz conjectures for primes in arithmetic progressions.

Contribution

It adapts existing methods to formulate the Koblitz conjecture for primes in AP and uncovers the opposing biases of the constants involved in these conjectures.

Findings

Constants in the cyclicity and Koblitz conjectures are oppositely biased on average.

Refined theorems to analyze moments of these constants for primes in arithmetic progressions.

Provided Magma code for computing the discussed constants.

Abstract

The cyclicity and Koblitz conjectures ask about the distribution of primes of cyclic and prime-order reduction, respectively, for elliptic curves over $\mathbb{Q}$. In 1976, Serre gave a conditional proof of the cyclicity conjecture, but the Koblitz conjecture (refined by Zywina in 2011) remains open. The conjectures are now known unconditionally "on average" due to work of Banks--Shparlinski and Balog--Cojocaru--David. Recently, there has been a growing interest in the cyclicity conjecture for primes in arithmetic progressions (AP), with relevant work by Akbal--G\"ulo\u{g}lu and Wong. In this paper, we adapt Zywina's method to formulate the Koblitz conjecture for primes in AP and refine a theorem of Jones to establish results on the moments of the constants in both the cyclicity and Koblitz conjectures for AP. In doing so, we uncover a somewhat counterintuitive phenomenon: On average, these two constants are oppositely biased over congruence classes. Finally, in an accompanying repository, we give Magma code for computing the constants discussed in this paper.