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On Abel relations. (Russian) Zbl 0629.14025
Let E be the elliptic curve \(y^ 2=x^ 3+rx+s\), \(\{0_{m,1},0_{m,2}\}\) be a basis of torsion points of order m on E, \((x_{a,b},y_{a,b}):=a0_{m,1}+b0_{m,2}\), \(\epsilon =e^{2\pi i/m}\). Set \(q=(a,b,m)\), \(a=qa_ 1\), \(b=qb_ 1\), \[ H_{a,b}=y^ 2_{a,b}\prod^{m-1}_{t=1}(x_{a,b}-x_{ta_ 1,tb_ 1})\quad (t\neq q,m-q), \] \[ F_{a,b;c,d}=\prod^{m-1}_{t=1}(x_{a,b}- x_{t(a-c)/r,t(b-d)/r)},r=(a-c,b-d,m), \] \[ L_{a,b;c,d}=H_{a,b}/F_{a,b;c,d}. \] The author proves the following nice formula: \(L_{a,b;c,d}=\epsilon^{ad-bc}L_{c,d;a,b}\). The particular cases \(\{a,b;c,d\}=\{a,b;a,0\},\{a,b;0,b\}\) were already proved by the author in J. Sov. Math. 29, 1272-1275 (1985); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 121, 58-61 (1983; Zbl 0539.14021).
Reviewer: Ş.A.Basarab

MSC:
14H45 Special algebraic curves and curves of low genus
14H52 Elliptic curves
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