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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.13695 |
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| _version_ | 1866909758715330560 |
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| author | Ando, Chihiro |
| author_facet | Ando, Chihiro |
| contents | For an elliptic curve $E$ over $\mathbb{Q}$ without complex multiplication, Lang and Trotter conjecture
\[
\# \{ p<X \mid E \text{ has a supersingular reduction at } p \} \sim \frac{c\sqrt{X}}{\log X}
\]
as $X \rightarrow \infty$, where $c>0$ is a constant depending only on $E$. Fourvy and Murty obtained an average estimation related to the Lang-Trotter conjecture, called the Lang-Trotter conjecture on average. We considered the Lang-Trotter conjecture for curves of genus 2, and obtained a similar result to the Lang-Trotter conjecture on average for the family of curves $C_λ:y^2=x(x-1)(x+1)(x-λ)(x-1/ λ)$. Such curves are characterized as curves of genus two with reduced automorphism group containing the Klein $4$-group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_13695 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Lang-Trotter conjecture on average for genus-$2$ curves with Klein-$4$ reduced automorphism group Ando, Chihiro Number Theory For an elliptic curve $E$ over $\mathbb{Q}$ without complex multiplication, Lang and Trotter conjecture \[ \# \{ p<X \mid E \text{ has a supersingular reduction at } p \} \sim \frac{c\sqrt{X}}{\log X} \] as $X \rightarrow \infty$, where $c>0$ is a constant depending only on $E$. Fourvy and Murty obtained an average estimation related to the Lang-Trotter conjecture, called the Lang-Trotter conjecture on average. We considered the Lang-Trotter conjecture for curves of genus 2, and obtained a similar result to the Lang-Trotter conjecture on average for the family of curves $C_λ:y^2=x(x-1)(x+1)(x-λ)(x-1/ λ)$. Such curves are characterized as curves of genus two with reduced automorphism group containing the Klein $4$-group. |
| title | The Lang-Trotter conjecture on average for genus-$2$ curves with Klein-$4$ reduced automorphism group |
| topic | Number Theory |
| url | https://arxiv.org/abs/2408.13695 |