Low-lying Zeros of Families of Elliptic Curves

The random matrix model predicts that many statistics associated to zeros of a family of L-functions can be modeled (or predicted) by the distribution of eigenvalues of large random matrices in one of the classical linear groups. If the statistics of a family of L-functions are modeled by the eigenvalues of the group G, then we say that G is the symmetry group (or symmetry type) associated to the family. The statistic of interest to us in this work is the density of zeros near the central point (also known as the 1-level density). The random matrix model predicts that the distribution of these zeros should be modeled by the eigenvalues nearest 1 for one of the symmetry types G (unitary, symplectic, and orthogonal). All of the different groups G have distinct behavior in this regard. Therefore, computing the 1-level density gives a theoretical way to predict the symmetry type of a family. The 1-level density has been studied for a wide variety of families of L-functions; see [R], [KS1], [ILS], [Mil] for example. It is standard to assume the Generalized Riemann Hypothesis (GRH) to study the 1-level density and we do so throughout this work. In particular, it is necessary to use GRH for the application of obtaining a bound on the average analytic rank from a density theorem. In some cases the use of GRH improves the range of the density theorem, which translates to an improved bound on the average rank. Besides these crucial applications of GRH, we have freely assumed GRH even when it could be removed with extra work since it simplifies arguments in some nonessential places. It is especially interesting to investigate the 1-level density for families of Lfunctions attached to elliptic curves over the rationals since zeros at the central point have important arithmetic information (by the conjecture of Birch and SwinnertonDyer). These investigations have been the main focus of this work.