Congruence obstructions in the refined Koblitz conjecture

Abstract

Let E/ℚ be an elliptic curve. Zywina's refinement of Koblitz's conjecture predicts that there are infinitely many primes p of good reduction for which #E_p(𝔽_p) is prime precisely when E has no congruence obstruction. For non-CM elliptic curves over ℚ, we classify all primitive congruence obstructions. In particular, we show that a primitive obstruction of composite level can occur only at $6$, $10$, $14$, $15$, or $30$ and determine the corresponding Galois images. As a consequence, for non-CM elliptic curves, the existence of any congruence obstruction is detected by the mod $210$ Galois image, so the positivity of the Koblitz--Zywina constant is determined at level $210$. We also determine which primitive obstruction levels can occur simultaneously. Finally, we prove, assuming GRH, that if E has no congruence obstruction, then for every 0<κ<1/8 there are infinitely many primes p of good reduction for which the least prime divisor of #E_p(𝔽_p) is greater than κlog p.

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BibTeX RIS

BibTeX

@misc{lee2026congruence,
  title = {{Congruence obstructions in the refined Koblitz conjecture}},
  author = {Sung Min Lee and Jacob Mayle and Rakvi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30573},
  eprint = {2609.30573},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30573},
}

RIS

TY  - UNPB
TI  - Congruence obstructions in the refined Koblitz conjecture
AU  - Sung Min Lee
AU  - Jacob Mayle
AU  - Rakvi
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.30573
UR  - https://arxiv.org/abs/2609.30573
AB  - Let E/ℚ be an elliptic curve. Zywina's refinement of Koblitz's conjecture predicts that there are infinitely many primes p of good reduction for which #E_p(𝔽_p) is prime precisely when E has no congruence obstruction. For non-CM elliptic curves over ℚ, we classify all primitive congruence obstructions. In particular, we show that a primitive obstruction of composite level can occur only at $6$, $10$, $14$, $15$, or $30$ and determine the corresponding Galois images. As a consequence, for non-CM elliptic curves, the existence of any congruence obstruction is detected by the mod $210$ Galois image, so the positivity of the Koblitz--Zywina constant is determined at level $210$. We also determine which primitive obstruction levels can occur simultaneously. Finally, we prove, assuming GRH, that if E has no congruence obstruction, then for every 0<κ<1/8 there are infinitely many primes p of good reduction for which the least prime divisor of #E_p(𝔽_p) is greater than κlog p.
ER  -

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