Geodesic
Behavior of the curve $x^3+y^3=1$ in a cyclotomic $\Gamma$-extension
Sbornik. Mathematics, Tome 19 (1973) no. 1, pp. 117-130 
M. I. Bashmakov; N. Zh. Al'-Nader. Behavior of the curve $x^3+y^3=1$ in a cyclotomic $\Gamma$-extension. Sbornik. Mathematics, Tome 19 (1973) no. 1, pp. 117-130. http://geodesic.mathdoc.fr/item/SM_1973_19_1_a7/
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     title = {Behavior of the curve $x^3+y^3=1$ in a~cyclotomic $\Gamma$-extension},
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Voir la notice de l'article provenant de la source Math-Net.Ru

This article proves that the group of rational points on the curve in the title remains finite when the $3^n$th roots of unity are adjoined. Here the 3-component of the Tate–Shafarevich group remains finite, and exact formulas are given for its order. Bibliography: 2 titles.

[1] Yu. I. Manin, “Krugovye polya i modulyarnye krivye”, Uspekhi mat. nauk, XXVI:6 (162) (1971), 7–71 | MR

[2] I. R. Shafarevich, “Rasshireniya s zadannymi tochkami vetvleniya”, Inst. Hautes Etudes Sci. Publ. Math., 18