# Linear independence in linear systems on elliptic curves

### Bradley W. Brock

Center for Communications Research, Princeton, USA### Bruce W. Jordan

Baruch College, CUNY, New York, USA### Bjorn Poonen

Massachusetts Institute of Technology, Cambridge, USA### Anthony J. Scholl

University of Cambridge, UK### Joseph L. Wetherell

Center for Communications Research, San Diego, USA

## Abstract

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 $E$.

## Cite this article

Bradley W. Brock, Bruce W. Jordan, Bjorn Poonen, Anthony J. Scholl, Joseph L. Wetherell, Linear independence in linear systems on elliptic curves. Comment. Math. Helv. 96 (2021), no. 2, pp. 199–213

DOI 10.4171/CMH/511