The frequency of higher rank in a family of quadratic twists

Random matrix theory can be used to predict the frequency of rank 2 curves in a family of quadratic twists, but it does not seem to be able to tell us what the prediction is for quadratic twists of rank 3 or higher.

The reason that the RMT approach fails is that the height of a generating point is a factor in the formula for the central value of the derivative of the L-function of a rank 1 curve. We don't have a prediction for the distribution of the heights of the generating points, and so can't predict (within the collection of curves of odd rank) the distribution the critical derivatives of the $L$-functions. Since the frequency of vanishing is determined by analyzing the tail of that distribution, we are unable to predict how often the derivative of the L-series is 0.

In [ arXiv:math/0010056] (to appear in Exp. Math.), Rubin and Silverberg show that for several infinite families of elliptic curves $E$, the number of quadratic twists $E_d$, $\vert d\vert < X$, with rank at least $3$ is $\gg X^{1/6}$. (Assuming the Parity Conjecture, the same result holds with $3$ replaced by $4$.) Some of these examples were previously obtained by Stewart and Top [ MR 95m:11055].

Rubin and Silverberg ask whether it is possible to find a hyperelliptic curve over $Q$ whose jacobian contains $r \ge 4$ copies of a fixed elliptic curve $E$. This would give a lower bound for the frequency of quadratic twists of $E$ of rank at least $r$.

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