Lacunary Wronskians on genus one curves

Greg W. Anderson

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

Let X be a nonsingular projective curve of genus one defined over an algebraically closed field of characteristic 0. Let D be a divisor of X of degree n > 1 and let O be a (closed) point of X. As is well known, there exists a unique morphism φD,O : X → X such that φD,O (P) = Q if and only if the divisor nP - D - O + Q is principal. Our main result is a simple explicit description of the map φD,O in terms of Wronskians and certain Wronskian-like determinants lacunary in the sense that derivatives of some orders are skipped. Further, for n = 2, 3 we interpret our main result as a syzygy from classical invariant theory, thus reconciling our work with a circle of ideas treated in two papers by Weil and a recent paper by An, Kim, Marshall, Marshall, McCallum and Perlis.

Original languageEnglish (US)
Pages (from-to)197-214
Number of pages18
JournalJournal of Number Theory
Volume115
Issue number2
DOIs
StatePublished - Dec 2005

Keywords

  • Genus one
  • Jacobians
  • Syzygies

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