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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2504.18732 |
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| _version_ | 1866908338115051520 |
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| author | Xie, Likun |
| author_facet | Xie, Likun |
| contents | In 1988, Koblitz conjectured the infinitude of primes p for which |E(F_p)| is prime for elliptic curves E over Q, drawing an analogy with the twin prime conjecture. He also proposed studying the primality of |E(F_{p^l})| / |E(F_p)|, in parallel with the primality of (p^l - 1)/(p - 1).
Motivated by these problems and earlier work on |E(F_p)|, we study the infinitude of primes p such that |E(F_{p^l})| / |E(F_p)| has a bounded number of prime factors for primes l >= 2, considering both CM and non-CM elliptic curves over Q. In the CM case, we focus on the curve y^2 = x^3 - x to address gaps in the literature and present a more concrete argument. The result is unconditional and applies Huxley's large sieve inequality for the associated CM field. In the non-CM case, analogous results follow under GRH via the effective Chebotarev density theorem.
For the CM curve y^2 = x^3 - x, we further apply a vector sieve to combine the almost prime properties of |E(F_p)| and |E(F_{p^2})| / |E(F_p)|, establishing a lower bound for the number of primes p <= x for which |E(F_{p^2})| / 32 is a square-free almost prime. We also study cyclic subgroups of finite index in E(F_p) and E(F_{p^2}) for CM curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_18732 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Almost Prime Orders of Elliptic Curves Over Prime Power Fields Xie, Likun Number Theory In 1988, Koblitz conjectured the infinitude of primes p for which |E(F_p)| is prime for elliptic curves E over Q, drawing an analogy with the twin prime conjecture. He also proposed studying the primality of |E(F_{p^l})| / |E(F_p)|, in parallel with the primality of (p^l - 1)/(p - 1). Motivated by these problems and earlier work on |E(F_p)|, we study the infinitude of primes p such that |E(F_{p^l})| / |E(F_p)| has a bounded number of prime factors for primes l >= 2, considering both CM and non-CM elliptic curves over Q. In the CM case, we focus on the curve y^2 = x^3 - x to address gaps in the literature and present a more concrete argument. The result is unconditional and applies Huxley's large sieve inequality for the associated CM field. In the non-CM case, analogous results follow under GRH via the effective Chebotarev density theorem. For the CM curve y^2 = x^3 - x, we further apply a vector sieve to combine the almost prime properties of |E(F_p)| and |E(F_{p^2})| / |E(F_p)|, establishing a lower bound for the number of primes p <= x for which |E(F_{p^2})| / 32 is a square-free almost prime. We also study cyclic subgroups of finite index in E(F_p) and E(F_{p^2}) for CM curves. |
| title | Almost Prime Orders of Elliptic Curves Over Prime Power Fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2504.18732 |