Opposing Average Congruence Class Biases in the Cyclicity and Koblitz Conjectures for Elliptic Curves
Abstract
The cyclicity and Koblitz conjectures ask about the distribution of primes of cyclic and prime-order reduction, respectively, for elliptic curves over . 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.
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Cite
@article{arxiv.2408.16641,
title = {Opposing Average Congruence Class Biases in the Cyclicity and Koblitz Conjectures for Elliptic Curves},
author = {Sung Min Lee and Jacob Mayle and Tian Wang},
journal= {arXiv preprint arXiv:2408.16641},
year = {2025}
}