Zero statistics near the central point

In order to make predictions about central values of $L$-functions at the central point using random matrix theory, it is necessary to have a model for the statistical behaviour of the zeros of the $L$-function near that point. In various families of elliptic curve $L$-functions [S. J. Miller] has observed for low conductor repulsion of the zeros from the central point. In the limit of large conductor this repulsion disappears and the zeros should follow the distribution of eigenvalues from random matrices in $SO(2N)$. It is a topic of further investigation to understand the provenance of the finite-conductor behaviour and to model it effectively.

This behaviour is seen in families of predominantly rank zero curves, as well as in families of higher rank and a sound model may lead to insight into the frequency of curves of various rank occuring within families of elliptic curves using methods similar to those described in Frequency of Vanishing.

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