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On hyperelliptic curves of odd degree and genus $g$ with $6$ torsion points of order $2g+1$статья
- Автор: Fedorov Gleb Vladimirovich
- Журнал: Doklady Mathematics
- Год издания: 2024
- Издательство: Maik Nauka/Interperiodica Publishing
- Местоположение издательства: Russian Federation
- Аннотация: Let a hyperelliptic curve $\mathcal{C}$ of genus $g$ defined over an algebraically closed field $K$ of characteristic $0$, given by the equation $y^2 = f(x)$, where the polynomial $f(x) \in K[x]$ is square-free and has odd degree $2g+1$. The curve $\mathcal{C}$ contains a single ``infinite'' point $\mathcal{O}$, which is the Weierstrass point. There is a classical embedding of $\mathcal{C}(K)$ into the group of $K$-points $J(K)$ of the Jacobian variety $J$ of the curve $\mathcal{C}$, identifying the point $\mathcal{O}$ with the unit element of the group $J(K)$. For $2 \le g \le 5$, the article explicitly found representatives of birational equivalence classes such hyperelliptic curves $\mathcal{C}$ with a marked unique point at infinity $\mathcal{O}$ that the set $\mathcal{C}(K) \cap J(K)$ contains at least 6 torsion points of order $2g+1$. It was previously known that for $g = 2$ there are exactly $5$ such equivalence classes, and for $g \ge 3$ an upper bound was known that depended only on the genus of $g$.
- Добавил в систему: Федоров Глеб Владимирович