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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2408.13695 |
| Etiquetas: |
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- 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.