Some Results in the Theory of Low-Lying Zeros of Families of L-Functions

While Random Matrix Theory has successfully modeled the limiting behavior of many quantities of families of L-functions, especially the distributions of zeros and values, the theory frequently cannot see the arithmetic of the family. In some situations this requires an extended theory that inserts arithmetic factors that depend on the family, while in other cases these arithmetic factors result in contributions which vanish in the limit, and are thus not detected. In this chapter we review the general theory associated with one of the most important statistics, the n-level density of zeros near the central point. According to the Katz–Sarnak density conjecture, to each family of L-functions there is a corresponding symmetry group (which is a subset of a classical compact group) such that the behavior of the zeros near the central point as the conductors tend to infinity agrees with the behavior of the eigenvalues near 1 as the matrix size tends to infinity. We show how these calculations are done, emphasizing the techniques, methods, and obstructions to improving the results, by considering in full detail the family of Dirichlet characters with square-free conductors. We then move on and describe how we may associate a symmetry constant with each family, and how to determine the symmetry group of a compound family in terms of the symmetries of the constituents. These calculations allow us to explain the remarkable universality of behavior, where the main terms are independent of the arithmetic, as we see that only the first two moments of the Satake parameters survive to contribute in the limit. Similar to the Central Limit Theorem, the higher moments are only felt in the rate of convergence to the universal behavior. We end by exploring the effect of lower order terms in families of elliptic curves. We present evidence supporting a conjecture that the average second moment in one-parameter families without complex multiplication has, when appropriately viewed, a negative bias, and end with a discussion of the consequences of this bias on the distribution of low-lying zeros, in particular relations between such a bias and the observed excess rank in families.