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四次の楕円曲線の媒介変数表示とLanden変換について Note on parametrization of fourth-order elliptic curves and Landen transformations 高橋, 大介 Takahashi, Daisuke A. elliptic curves uniformization Weierstrass sigma functions Jacobi elliptic functions modular group Landen transformation application/pdf Parametrization (Uniformization) of elliptic curves with fourth-order polynomials in the right-hand side, Y2 = X4 − h2X2 − h3X − h4, by elliptic functions is revisited. Various seemingly different but equivalent expressions for the parametrizing function are presented, and if possible, their derivations are provided in several ways. In particular, the proofs based on Weierstrass’s and Jacobi’s functions are constructed independently and hence closed in itself. The reduction to the birationally equivalent elliptic curves with the resolvent cubic in the right-hand side, the integration formulae, the reduction to the Legendre elliptic curves whose fourth-order polynomial is biquadratic, are also discussed. In derivation based on the Weierstrass theory, the starting point is the expression given by Akhiezer. To discuss the transformations specific to the Legendre case, we provide the first and second order transformation formulae for σ1, σ2, and σ3 functions, which reflect their triality. In derivation based on the Jacobi theory, we discuss S4 symmetry and related formulae arising from permutation of the four roots and provide a simplified proof for the Landen transformation based on the coefficient matching of the differential equation. 神奈川工科大学 2022-03-01 jpn departmental bulletin paper VoR https://doi.org/10.34411/00032061 http://hdl.handle.net/10368/00032061 https://kait.repo.nii.ac.jp/records/12133 10.34411/00032061 AA12669200 21882878 神奈川工科大学研究報告.B,理工学編 46 21 30 https://kait.repo.nii.ac.jp/record/12133/files/kkb-046-004.pdf application/pdf 1.1 MB 2022-04-28