On the proportion of rank 0 twists of elliptic curves

Abstract

Let E be an elliptic curve defined over Q, let E d denote its dth quadratic twist, and r E d : = rank E d ( Q ) . We prove, that, for any positive integer k there are pairwise non-isogenous elliptic curves E 1 , … , E k such that r E 1 p = ⋯ = r E k p = 0 for a positive proportion of primes p. To cite this article: A. Dąbrowski, C. R. Acad. Sci. Paris, Ser. I 346 (2008).