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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.00822 |
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| _version_ | 1866908930886598656 |
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| author | Ando, Chihiro Harashita, Shushi |
| author_facet | Ando, Chihiro Harashita, Shushi |
| contents | For an elliptic curve $E$ over $\mathbb{Q}$ without complex multiplication, Lang and Trotter conjectured that the number of primes $p <X$ at which $E$ has a supersingular reduction is asymptotically equal to $c\sqrt{X}/\log X$, where $c>0$ is a constant depending only on $E$. While it remains an open question, an average estimation related to the Lang-Trotter conjecture was established by Fouvry and Murty. This result is called the Lang-Trotter conjecture on average. We extend the Lang-Trotter conjecture to curves of genus $2$ and obtain a similar result to the Lang-Trotter conjecture on average for the family of curves $C_λ:y^2=x(x-1)(x-λ)(x-(λ-1)/λ)(x-1/ (1-λ))$. These curves are characterized as curves of genus $2$ with reduced automorphism group containing symmetric group $S_3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_00822 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Lang-Trotter conjecture on average for genus-$2$ curves with $S_3$ reduced automorphism group Ando, Chihiro Harashita, Shushi Number Theory For an elliptic curve $E$ over $\mathbb{Q}$ without complex multiplication, Lang and Trotter conjectured that the number of primes $p <X$ at which $E$ has a supersingular reduction is asymptotically equal to $c\sqrt{X}/\log X$, where $c>0$ is a constant depending only on $E$. While it remains an open question, an average estimation related to the Lang-Trotter conjecture was established by Fouvry and Murty. This result is called the Lang-Trotter conjecture on average. We extend the Lang-Trotter conjecture to curves of genus $2$ and obtain a similar result to the Lang-Trotter conjecture on average for the family of curves $C_λ:y^2=x(x-1)(x-λ)(x-(λ-1)/λ)(x-1/ (1-λ))$. These curves are characterized as curves of genus $2$ with reduced automorphism group containing symmetric group $S_3$. |
| title | The Lang-Trotter conjecture on average for genus-$2$ curves with $S_3$ reduced automorphism group |
| topic | Number Theory |
| url | https://arxiv.org/abs/2604.00822 |