منابع مشابه
Elliptic Curves with No Rational Points
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متن کاملConstructing Rational Points on Elliptic Curves Using Heegner Points
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In this paper, we study the Mordell-Weil group of an elliptic curve as a Galois module. We consider an elliptic curve E defined over a number field K whose Mordell-Weil rank over a Galois extension F is 1, 2 or 3. We show that E acquires a point (points) of infinite order over a field whose Galois group is one of Cn×Cm (n = 1, 2, 3, 4, 6, m = 1, 2), Dn×Cm (n = 2, 3, 4, 6, m = 1, 2), A4×Cm (m = ...
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Let x(P ) = AP /B2 P denote the x-coordinate of the rational point P on an elliptic curve in Weierstrass form. We consider when BP can be a perfect power or a prime. Using Faltings’ theorem, we show that for a fixed f > 1, there are only finitely many rational points P with BP equal to an fth power. Where descent via an isogeny is possible, we show that there are only finitely many rational poi...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 1988
ISSN: 0002-9939
DOI: 10.1090/s0002-9939-1988-0958035-0