1 6 Ju n 20 06 QUADRATIC TWISTS OF PAIRS OF ELLIPTIC CURVES
نویسنده
چکیده
Given two elliptic curves defined over a number field K, not both with j-invariant zero, we show that there are infinitely many D ∈ K × with pairwise distinct image in K × /K × 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.
منابع مشابه
Selmer Companion Curves
We say that two elliptic curves E1, E2 over a number field K are n-Selmer companions for a positive integer n if for every quadratic character χ of K, there is an isomorphism Seln(E χ 1 /K) ∼= Seln(E 2 /K) between the nSelmer groups of the quadratic twists E 1 , E χ 2 . We give sufficient conditions for two elliptic curves to be n-Selmer companions, and give a number of examples of non-isogenou...
متن کاملA Note on Twists of (y^2=x^3+1)
‎‎In the category of Mordell curves (E_D:y^2=x^3+D) with nontrivial torsion groups we find curves of the generic rank two as quadratic twists of (E_1), ‎and of the generic rank at least two and at least three as cubic twists of (E_1). ‎Previous work‎, ‎in the category of Mordell curves with trivial torsion groups‎, ‎has found infinitely many elliptic curves with ...
متن کاملThe rank of hyperelliptic Jacobians in families of quadratic twists
The variation of the rank of elliptic curves over Q in families of quadratic twists has been extensively studied by Gouvêa, Mazur, Stewart, Top, Rubin and Silverberg. It is known, for example, that any elliptic curve over Q admits infinitely many quadratic twists of rank ≥ 1. Most elliptic curves have even infinitely many twists of rank ≥ 2 and examples of elliptic curves with infinitely many t...
متن کاملRanks of Quadratic Twists of Elliptic Curves over Fq(t)
Some notes on the analogy between number theory over Z and Fq[t] and an attempt to translate a paper of Gouvêa and Mazur on ranks of quadratic twists of elliptic curves over Q to elliptic curves over Fq(t).
متن کاملTwists of elliptic curves of rank at least four
We give infinite families of elliptic curves over Q such that each curve has infinitely many non-isomorphic quadratic twists of rank at least 4. Assuming the Parity Conjecture, we also give elliptic curves over Q with infinitely many non-isomorphic quadratic twists of odd rank at least 5.
متن کامل