On Abel’s relations. (Russian) Zbl 0539.14021

Dem’yanenko, V. A.

Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 121, 58-61 (1983).

For an elliptic curve E given in Weierstrass form, the author derives a relationship among the coordinates of the n-torsion points of E similar to Abel’s relations (cf. H. Weber, Lehrbuch der Algebra, Vol. 3, §62). The proof uses identities from former work of the author [cf. Mat. Zametki 14, 827-832 (1973; Zbl 0284.14013) and 33, 111-116 (1982; Zbl 0532.14015)]. As a consequence, the author obtains the following: if E and its n-torsion points are defined over an algebraic number field K such that \(12\equiv 0(mod n),\) then \(D^{1/n}\in K,\) where D is the discriminant of E.

Reviewer: G.Angermüller

MSC:

14H45	Special algebraic curves and curves of low genus
14H52	Elliptic curves
14K15	Arithmetic ground fields for abelian varieties
14G25	Global ground fields in algebraic geometry

Keywords:

Abel relations; n-torsion points