Let $E$ be an elliptic curve, with identity $O$, and let $C$ be a cyclic subgroup of odd order $N$, over an algebraically closed field $k$ with char $k \nmid N$. For $P \in C$, let $s_P$ be a rational function with divisor $N \cdot P - N \cdot O$. We ask whether the $N$ functions $s_P$ are linearly independent. For generic $(E,C)$, we prove that the answer is yes. We bound the number of exceptional $(E,C)$ when $N$ is a prime by using the geometry of the universal generalized elliptic curve over $X_1(N)$. The problem can be recast in terms of sections of an arbitrary degree $N$ line bundle on $E$.

中文翻译：

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椭圆曲线上线性系统的线性独立性

令$E$为椭圆曲线，单位为$O$，令$C$为奇阶$N$的循环子群，在代数闭域$k$上，字符$k\nmidN$。对于 $P \in C$，令 $s_P$ 是一个有理函数，其除数为 $N \cdot P - N \cdot O$。我们询问 $N$ 函数 $s_P$ 是否线性无关。对于泛型 $(E,C)$，我们证明答案是肯定的。当$N$为素数时，我们通过使用$X_1(N)$上的通用广义椭圆曲线的几何来限制异常$(E,C)$的数量。该问题可以根据 $E$ 上的任意阶 $N$ 线束的部分来重铸。