标题： 阿贝尔簇中的曲线次数 显示英文标题

标题： Degrees of Curves in Abelian Varieties

摘要： 曲线$C$在极化阿贝尔簇$(X,\lambda)$中的次数是整数$d=C\cdot\lambda$。 当 $C$ 生成 $X$ 时，我们找到一个关于 $d$ 的下界，该下界依赖于 $n$ 和极化度 $\lambda$ 。 最小可能的度数是 $d=n$ ，并且仅在它的雅可比簇中具有主极化的光滑曲线上得到（Ran，Collino）。 情况$d=n+1$和$d=n+2$被研究了。 此外，当$X$是单的时，利用 Smyth 关于全正代数整数的迹的结果，可以证明如果$d\le 1.7719\, n$，则$C$是光滑的且$X$同构于其雅可比簇。 我们还得到了$C$的几何亏格的一个上界，该上界用其次数来表示。 显示英文摘要

摘要： The degree of a curve $C$ in a polarized abelian variety $(X,\lambda)$ is the integer $d=C\cdot\lambda$. When $C$ generates $X$, we find a lower bound on $d$ which depends on $n$ and the degree of the polarization $\lambda$. The smallest possible degree is $d=n$ and is obtained only for a smooth curve in its Jacobian with its principal polarization (Ran, Collino). The cases $d=n+1$ and $d=n+2$ are studied. Moreover, when $X$ is simple, it is shown, using results of Smyth on the trace of totally positive algebraic integers, that if $d\le 1.7719\, n$, then $C$ is smooth and $X$ is isomorphic to its Jacobian. We also get an upper bound on the geometric genus of $C$ in terms of its degree.