Glasnik Matematicki, Vol. 46, No. 2 (2011), 333-338.


Ivica Gusić

Faculty of Chemical Engineering and Technology, University of Zagreb, Marulićev trg 19, 10000 Zagreb, Croatia

Abstract.   Let K be an algebraic number field, and let h(x)=x3+ax be a polynomial over K. We prove that there exists infinitely many b in K such that the equation dy2=x3+ax+b has no solutions over K for infinitely many d in K*/K* 2. The proof is based on recent results of B. Mazur and K. Rubin on the 2-Selmer rank in families of quadratic twists of elliptic curves over number fields. On the other side, it is known that if the parity conjecture is valid, then there exist a number field K and a cubic polynomial f irreducible over K, such that the equation dy2=f(x) has infinitely many solutions for each d in K*.

2000 Mathematics Subject Classification.   11G05, 14G05.

Key words and phrases.   Elliptic curve, quadratic twist, 2-Selmer rank, number field.

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DOI: 10.3336/gm.46.2.05


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