Finite conductor models for zeros near the central point of elliptic curve L-functions

Random Matrix Theory has successfully modeled the behavior of
zeros of elliptic curve L-functions in the limit of large conductors. In
this talk we explore the behavior of zeros near the central point for
one-parameter families of elliptic curves with rank over Q(T) and small
conductors. Zeros of L-functions are conjectured to be simple except
possibly at the central point for deep arithmetic reasons; these families
provide a fascinating laboratory to explore the effect of multiple zeros
on nearby zeros. Though theory suggests the family zeros (those we believe
exist due to the Birch and Swinnerton-Dyer Conjecture) should not interact
with the remaining zeros, numerical calculations show this is not the
case; however, the discrepency is likely due to small conductors, and
unlike excess rank is observed to noticeably decrease as we increase the
conductors. We shall mix theory and experiment and see some surprisingly
results, which leads us to conjecture that a discretized Jacobi ensemble
correctly models the small conductor behavior.