细化Koblitz猜想中的同余障碍

中文摘要 AI 辅助

设$E/\mathbb{Q}$为一条椭圆曲线。Zywina对Koblitz猜想的细化预测：当且仅当$E$没有同余障碍时，存在无穷多个具有良好约化的素数$p$，使得$\\#E_p(\mathbb{F}_p)$为素数。对于$\mathbb{Q}$上的非CM椭圆曲线，我们分类了所有本原同余障碍。特别地，我们证明了复合级别的本原障碍只能出现在$6$、$10$、$14$、$15$或$30$处，并确定了相应的伽罗瓦像。作为推论，对于非CM椭圆曲线，任何同余障碍的存在性都由模$210$的伽罗瓦像检测，因此Koblitz–Zywina常数的正性在级别$210$处确定。我们还确定了哪些本原障碍级别可以同时出现。最后，在假设GRH的前提下，我们证明：若$E$没有同余障碍，则对于每个$0<\kappa<1/8$，存在无穷多个具有良好约化的素数$p$，使得$\\#E_p(\mathbb{F}_p)$的最小素因子大于$\kappa\log p$。

英文摘要

Let $E/\mathbb{Q}$ be an elliptic curve. Zywina's refinement of Koblitz's conjecture predicts that there are infinitely many primes $p$ of good reduction for which $\#E_p(\mathbb{F}_p)$ is prime precisely when $E$ has no congruence obstruction. For non-CM elliptic curves over $\mathbb{Q}$, 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 $\#E_p(\mathbb{F}_p)$ is greater than $κ\log p$.

发表机构

• Wake Forest University（维克森林大学）
• University of Delaware（特拉华大学）
• The University of Maine（缅因大学）

机构由 AI 辅助整理，请以论文原文为准。