by Ritabrata Munshi (Piscataway, NJ) 1. Introduction. Let E be an elliptic curve over Q of conductor N , root number ω, and defined by the Weierstrass equation y 2 = f (x). For any square-free d relatively prime to N , the root number of the twisted curve E d : dy 2 = f (x) is given by ω d = ωχ d (−N ), where χ d denotes the quadratic character associated with the field Q( √ d). Then depending on N and ω one may appropriately choose a congruence class modulo 4N such that for each d in the

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... ence class the root number ω d is either always 1, or always −1. In the light of the deep results of Kolyvagin, Gross and Zagier [3], it is natural to study the central values {L(1, E d )} for the family of twists with fixed root number 1, as in this case the nonvanishing of L(1, E d ) implies that there are at most finitely many Q-rational points on the elliptic curve E d . On the other hand, when the root number is −1, then the central values vanish and one studies the sequence of derivatives {L (1, E d )} with the hope of proving that some or many of them are nonvanishing. Of course, then it follows that the corresponding elliptic curve has Mordell-Weil rank one, and consequently infinitely many rational points. Such nonvanishing results are well known when the only restriction on d is a congruence condition. This was first established by Murty and Murty [11] , and almost simultaneously, but independently and via a totally different method, by Bump, Friedberg and Hoffstein [1]. Later Iwaniec [7] gave a quantitative version of this result. Goldfeld has proposed a strong conjecture regarding the distribution of r d = ord s=1 L(s, E d ) in the family of all quadratic twists {E d } (with d square-free and (d, N ) = 1). Let n(X) denote the number of curves in the above family with d < X. Then the conjecture predicts that d≤X r d =i 2000 Mathematics Subject Classification: 11G40, 11N36, 11F67.