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arXiv 2005.05473math.NTmath.AG
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椭圆曲线上线性系组的线性无关性

Linear independence in linear systems on elliptic curves

Bradley W. Brock, Bruce W. Jordan, Bjorn Poonen, Anthony J. Scholl, Joseph L. Wetherell

备注：
10 pages
发表信息：
Commentarii Mathematici Helvetici, Volume 96, Issue 2, 2021, pp. 199-213
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摘要：

设$E$是椭圆曲线，单位元为$O$，$C$是代数闭域$k$上的奇数阶循环子群，且$\operatorname{char} k \nmid N$为$k$.对于$P \in C$，令$s_P$为具有除数$N \cdot P - N \cdot O$的有理函数。我们问$N$函数$s_P$是否线性独立。对于通用的$(E,C)$，我们证明了答案是肯定的。利用$X_1(N)$上泛广义椭圆曲线的几何性质，给出了当$N$为素数时例外$(E,C)$的个数的界.这个问题可以用$E$上任意阶$N$线丛的截面来表示。

英文摘要：

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 $\operatorname{char} k \nmid N$. 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 $E$.

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