TY - JOUR

T1 - Lacunary Wronskians on genus one curves

AU - Anderson, Greg W.

N1 - Copyright:
Copyright 2005 Elsevier B.V., All rights reserved.

PY - 2005/12

Y1 - 2005/12

N2 - 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.

AB - 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.

KW - Genus one

KW - Jacobians

KW - Syzygies

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U2 - 10.1016/j.jnt.2004.11.001

DO - 10.1016/j.jnt.2004.11.001

M3 - Article

AN - SCOPUS:28444480678

VL - 115

SP - 197

EP - 214

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 2

ER -