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On rational torsion points of central $ℚ$ -curves

Fumio Sairaiji; Takuya Yamauchi — 2008

Journal de Théorie des Nombres de Bordeaux

Let $E$ be a central $ℚ$ -curve over a polyquadratic field $k$ . In this article we give an upper bound for prime divisors of the order of the $k$ -rational torsion subgroup $E_{t o r s} (k)$ (see Theorems 1.1 and 1.2). The notion of central $ℚ$ -curves is a generalization of that of elliptic curves over $ℚ$ . Our result is a generalization of Theorem 2 of Mazur [], and it is a precision of the upper bounds of Merel [] and Oesterlé [].

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