Cyclicity and exponent of elliptic curves modulo $p$ in arithmetic progressions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913345121026048 |
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| author | Wong, Peng-Jie |
| author_facet | Wong, Peng-Jie |
| contents | In this article, we study the cyclicity problem of elliptic curves $E/\Bbb{Q}$ modulo primes in a given arithmetic progression. We extend the recent work of Akbal and Güloğlu by proving an unconditional asymptotic for such a cyclicity problem over arithmetic progressions for CM elliptic curves $E$, which also presents a generalisation of the previous works of Akbary, Cojocaru, M.R. Murty, V.K. Murty, and Serre. In addition, we refine the conditional estimates of Akbal and Güloğlu, which gives log-power savings (for small moduli) and consequently improves the work of Cojocaru and M.R. Murty. Moreover, we study the average exponent of $E$ modulo primes in a given arithmetic progression and obtain several conditional and unconditional estimates, extending the previous works of Freiberg, Kim, Kurlberg, and Wu. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_05594 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Cyclicity and exponent of elliptic curves modulo $p$ in arithmetic progressions Wong, Peng-Jie Number Theory 11G05, 11N36, 11G15, 11R45 In this article, we study the cyclicity problem of elliptic curves $E/\Bbb{Q}$ modulo primes in a given arithmetic progression. We extend the recent work of Akbal and Güloğlu by proving an unconditional asymptotic for such a cyclicity problem over arithmetic progressions for CM elliptic curves $E$, which also presents a generalisation of the previous works of Akbary, Cojocaru, M.R. Murty, V.K. Murty, and Serre. In addition, we refine the conditional estimates of Akbal and Güloğlu, which gives log-power savings (for small moduli) and consequently improves the work of Cojocaru and M.R. Murty. Moreover, we study the average exponent of $E$ modulo primes in a given arithmetic progression and obtain several conditional and unconditional estimates, extending the previous works of Freiberg, Kim, Kurlberg, and Wu. |
| title | Cyclicity and exponent of elliptic curves modulo $p$ in arithmetic progressions |
| topic | Number Theory 11G05, 11N36, 11G15, 11R45 |
| url | https://arxiv.org/abs/2307.05594 |