The Square-free Sieve and the Rank of Elliptic Curves

Let E be an elliptic curve over ij. A celebrated theorem of Mordell asserts that E(ij) , the (abelian) group of rational points of E, is finitely generated. By the rank of E we mean the rank of E(ij). Thus the rank of E is positive if and only if E possesses an infinity of rational points. Relatively few general qualitative assertions can be made about the rank as E varies. How large can the rank get? Although we expect that there are elliptic curves over ij with arbitrarily high ranks, this is presently unknown. What is the average size of the rank? Recently Brumer and McGuinness have reported on their study of 310,716 elliptic curves of prime conductor less than 108 where they have found that 20.06% of those curves have even rank;::: 2. (Cf. [BM]; also see forthcoming work of Brumer where, subject to a number of standard conjectures, he shows that 2.3 is an upper bound for the average rank for all elliptic curves over ij ordered in terms of their Faltings height.) What is the behavior of the rank over the family of twists of a given elliptic curve? Here, one has three natural kinds of families of twists: (1) Quadratic twists. One can take any elliptic curve and systematically twist it by all quadratic characters (this is the type of family of elliptic curves we are concerned with in this paper). Specifically, if E is an elliptic curve over ij given by the Weierstrass equation y2 = X3 + A . X + B , and D is any squarefree integer, the (quadratic) twist of E by D, ED' is given by the equation D· y2 = X3 + A. X + B. Put 9/£(D) := rank(ED ). Goldfeld has conjectured [Go] that the average value (suitably defined) of 9/£ is !. Conditional upon some standard conjectures (made explicit below; see Theorem 2), we find lower bounds for the number of times 9/£ is ;::: 2. Specifically for a fixed elliptic curve E/Q and any e > 0, we prove the (conditional) result that for sufficiently large real numbers x, there are at least X 1/ 2e square-free integers D with IDI :::; x such that 9/£(D) is ;::: 2 (and is even). (2) Cubic and quartic twists. One can also take an elliptic curve over ij with j-invariant 0, and twist it by all cubic characters; or one can take an elliptic curve with j-invariant 1728 and twist by quartic characters. In [ZK] data is