Product formulas for the 5-division points on the Tate normal form and the Rogers-Ramanujan continued fraction

JOURNAL OF NUMBER THEORY

卷:200

Article

Morton, Patrick¹ 

[1] Indiana Univ Purdue Univ, Dept Math Sci, LD 270, Indianapolis, IN 46202 USA

关键词: Tate normal form;    5-division points;    Rogers-Ramanujan continued fraction;    Watson's method;    Quintic equations;   

DOI  :  10.1016/j.jnt.2018.12.013

来源: Elsevier

【 摘 要 】

Explicit formulas are proved for the 5-torsion points on the Tate normal form E-5 of an elliptic curve having (X, Y) = (0,0) as a point of order 5. These formulas express the coordinates of points in E-5[5] - <(0,0)> as products of linear fractional quantities in terms of fifth roots of unity and a parameter u, where the parameter b which defines the curve E-5 is given as b = (epsilon(5)u(5) - epsilon(-5))/(u(5) + 1) and epsilon = (-1 + root 5)/2. If r(tau) is the Rogers-Ramanujan continued fraction and b = r(5)(tau), then the coordinates of points of order 5 in E-5[5] <(0,0)> are shown to be products of linear fractional expressions in r(5 tau) with coefficients in Q(zeta(5)). (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

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