Quadratic twists of pairs of elliptic curves

Details Quadratic twists of pairs of elliptic curves Quadratic twists of pairs of elliptic curves

2006-06-16

arxiv.org/abs/math/0606405v1

Mathematics

Number Theory

Scientific paper

Given two elliptic curves defined over a number field K, not both with j-invariant zero, we show that there are infinitely many $D\in K^\times$ with pairwise distinct image in $ K^\times/{K^\times}^2 $, such that the quadratic twist of both curves by D have positive Mordell-Weil rank. The proof depends on relating the values of pairs of cubic polynomials to rational points on another elliptic curve, and on a fiber product construction.

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