Twists of elliptic curves

Abstract

If $E$ is an elliptic curve over ${\bb Q}$, then let $E(D)$ denote the $D$-quadratic twist of $E$. It is conjectured that there are infinitely many primes $p$ for which $E(p)$ has rank 0, and that there are infinitely many primes $l$ for which $E(l)$ has positive rank. For some special curves $E$ we show that there is a set $S$ of primes $p$ with density $\frac{1}{3}$; for which if $D = \Pi_{pj}$ is a squarefree integer where $pj \in S$, then $E(D)$ has rank 0. In particular $E(p)$ has rank 0 for every $p \in S$. As an example let $E_1$ denote the curve \[ E_1: y^2 = x^3 + 44x^2 -19360x + 1682384. \]Then its associated set of primes $S_1$ consists of the prime 11 and the primes $p$ for which the order of the reduction of $X0(11)$ modulo $p$ is odd. To obtain the general result we show for primes $p \in S$ that the rational factor of $L(E(p),1)$ is nonzero which implies that $E(p)$ has rank 0. These special values are related to surjective ${\bb Z}/2{\bb Z}$ Galois representations that are attached to modular forms. Another example of this result is given, and we conclude with some remarks regarding the existence of positive rank prime twists via polynomial identities.