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While Random Matrix Theory has successfully modeled many quantities offamilies of L-functions, it frequently cannot see the family's arithmetic. Insome situations this requires an extended theory that inserts arithmeticfactors depending on the family, while in other cases these factors result incontributions which vanish in the limit, and are thus not detected. We reviewthe general theory associated to one of the most important statistics, then-level density of zeros near the central point. According to the Katz-Sarnakdensity conjecture, to each family of L-functions there is a correspondingsymmetry group such that the behavior of zeros near the central point as theconductors tend to infinity agrees with the behavior of eigenvalues near 1 asthe 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 a family of Dirichlet characters. We thendescribe how we may associate a symmetry constant to each family, and how todetermine the symmetry group of a compound family in terms of the symmetries ofthe constituents. These calculations explain the remarkable universality ofbehavior, where the main terms are independent of the arithmetic (only thefirst two moments of the Satake parameters contribute in the limit; similar tothe Central Limit Theorem, the higher moments are only felt in the rate ofconvergence). We end by exploring lower order terms in families of ellipticcurves. We present evidence supporting a conjecture that the average secondmoment in one-parameter families without complex multiplication has, whenappropriately viewed, a negative bias, and end with a discussion of theconsequences of this bias on the distribution of low-lying zeros, in particularrelations between such a bias and the observed excess rank in families.

Some Results in the Theory of Low-lying Zeros: Determining the 1-level  density, identifying the group symmetry and the arithmetic of moments of  Satake parameters