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
Linear independence in linear systems on elliptic curves cover
Download PDF

A subscription is required to access this article.

Abstract

Let EE be an elliptic curve, with identity OO, and let CC be a cyclic subgroup of odd order NN, over an algebraically closed field kk with char kNk \nmid N. For PCP \in C, let sPs_P be a rational function with divisor NPNON \cdot P - N \cdot O. We ask whether the NN functions sPs_P are linearly independent. For generic (E,C)(E,C), we prove that the answer is yes. We bound the number of exceptional (E,C)(E,C) when NN is a prime by using the geometry of the universal generalized elliptic curve over X1(N)X_1(N). The problem can be recast in terms of sections of an arbitrary degree NN line bundle on EE.

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