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Explicit 4-descents on an elliptic curve

100%

Merriman J., Siksek S., Smart N.

Acta Arithmetica

1996

tom 77

nr 4

385-404

Solutions of cubic equations in quadratic fields

100%

Chakraborty K., Kulkarni M.

Acta Arithmetica

1999

tom 89

nr 1

37-43

Let K be any quadratic field with $𝓞_K$ its ring of integers. We study the solutions of cubic equations, which represent elliptic curves defined over ℚ, in quadratic fields and prove some interesting results regarding the solutions by using elementary tools. As an application we consider the Diophantine equation r+s+t = rst = 1 in $𝓞_K$. This Diophantine equation gives an elliptic curve defined over ℚ with finite Mordell-Weil group. Using our study of the solutions of cubic equations in quadratic fields we present a simple proof of the fact that except for the ring of integers of ℚ(i) and ℚ(√2), this Diophantine equation is not solvable in the ring of integers of any other quadratic fields, which is already proved in [4].

The orders of the reductions of a point in the Mordell-Weil group of an elliptic curve

80%

Cheon J., Hahn S.

Acta Arithmetica

1999

tom 88

nr 3

219-111

Ograniczanie wyników

3 Acta Arithmetica

1 Chakraborty K.

1 Cheon J.

1 Hahn S.

1 Kulkarni M.

1 Merriman J.

1 Siksek S.

1 Smart N.

2 1999

1 1996

Wyniki wyszukiwania

Explicit 4-descents on an elliptic curve

Solutions of cubic equations in quadratic fields

The orders of the reductions of a point in the Mordell-Weil group of an elliptic curve