Opposing Average Congruence Class Biases in the Cyclicity and Koblitz Conjectures for Elliptic Curves
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arXiv
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| Format: | Preprint |
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2024
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| author | Lee, Sung Min Mayle, Jacob Wang, Tian |
| author_facet | Lee, Sung Min Mayle, Jacob Wang, Tian |
| contents | 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--Güloğlu 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_16641 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Opposing Average Congruence Class Biases in the Cyclicity and Koblitz Conjectures for Elliptic Curves Lee, Sung Min Mayle, Jacob Wang, Tian Number Theory Primary 11G05, Secondary 11F80 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--Güloğlu 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. |
| title | Opposing Average Congruence Class Biases in the Cyclicity and Koblitz Conjectures for Elliptic Curves |
| topic | Number Theory Primary 11G05, Secondary 11F80 |
| url | https://arxiv.org/abs/2408.16641 |