On the Constants of the Lang-Trotter Conjecture for CM Elliptic Curves
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914833218142208 |
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| author | Ray, Anish |
| author_facet | Ray, Anish |
| contents | In 2021, Daqing Wan and Ping Xi studied the equivalence of the Lang-Trotter conjecture for CM elliptic curves and the Hardy-Littlewood conjecture for primes represented by a quadratic polynomial. Wan and Xi provided an alternative description of the Lang-Trotter conjecture under the Hardy-Littlewood conjecture. They obtained an explicit constant $\overlineω_{E,r}$ in the asymptotics of the Lang-Trotter conjecture. They further conjectured that this particular constant would be equal to the constant $C_{E,r}$ in the asymptotics of the original Lang-Trotter conjecture. In this paper, we verify the same for $20$ CM elliptic curves, which also establishes the equivalence of the Lang-Trotter Conjecture and the Hardy-Littlewood Conjecture with respect to $r$, for these CM elliptic curves. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_09938 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Constants of the Lang-Trotter Conjecture for CM Elliptic Curves Ray, Anish Number Theory 11G05 (Primary) 11G15, 11F80 (Secondary) In 2021, Daqing Wan and Ping Xi studied the equivalence of the Lang-Trotter conjecture for CM elliptic curves and the Hardy-Littlewood conjecture for primes represented by a quadratic polynomial. Wan and Xi provided an alternative description of the Lang-Trotter conjecture under the Hardy-Littlewood conjecture. They obtained an explicit constant $\overlineω_{E,r}$ in the asymptotics of the Lang-Trotter conjecture. They further conjectured that this particular constant would be equal to the constant $C_{E,r}$ in the asymptotics of the original Lang-Trotter conjecture. In this paper, we verify the same for $20$ CM elliptic curves, which also establishes the equivalence of the Lang-Trotter Conjecture and the Hardy-Littlewood Conjecture with respect to $r$, for these CM elliptic curves. |
| title | On the Constants of the Lang-Trotter Conjecture for CM Elliptic Curves |
| topic | Number Theory 11G05 (Primary) 11G15, 11F80 (Secondary) |
| url | https://arxiv.org/abs/2309.09938 |