Square-free orders for CM elliptic curves modulo $p$ in short intervals
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909874227511296 |
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| author | Wong, Peng-Jie |
| author_facet | Wong, Peng-Jie |
| contents | Let $E$ be a CM elliptic curve over $\Bbb{Q}$. We refine the work of Cojocaru on the asymptotic formulae for the number of primes $p\le x$ for which the reduction modulo $p$ of $E$ is of square-free order. Also, we derive an unconditional short interval variant for the asymptotics. Compared to the estimate derived from the generalised Riemann hypothesis, the presented result is valid for even shorter intervals. Furthermore, we improve the short interval variant of the cyclicity problem for CM elliptic curves previously obtained by the author. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_05904 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Square-free orders for CM elliptic curves modulo $p$ in short intervals Wong, Peng-Jie Number Theory 11N45, 11G05, 11N36 Let $E$ be a CM elliptic curve over $\Bbb{Q}$. We refine the work of Cojocaru on the asymptotic formulae for the number of primes $p\le x$ for which the reduction modulo $p$ of $E$ is of square-free order. Also, we derive an unconditional short interval variant for the asymptotics. Compared to the estimate derived from the generalised Riemann hypothesis, the presented result is valid for even shorter intervals. Furthermore, we improve the short interval variant of the cyclicity problem for CM elliptic curves previously obtained by the author. |
| title | Square-free orders for CM elliptic curves modulo $p$ in short intervals |
| topic | Number Theory 11N45, 11G05, 11N36 |
| url | https://arxiv.org/abs/2408.05904 |