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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
##### Keywords:
torsion points; elliptic curve