Compositio Mathematica

Rank 0 Quadratic Twists of a Family of Elliptic Curves

Gang Yu a1
a1 Department of Mathematics, The University of Michigan, Ann Arbor, MI 48105 U.S.A e-mail:

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In this paper, we consider a family of elliptic curves over ${\open Q}$ with 2-torsion part ${\open Z}_2$. We prove that, for every such elliptic curve, a positive proportion of quadratic twists have Mordell–Weil rank 0.

Key Words: elliptic curves; 2-descent procedure; character sums.