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2016 | OriginalPaper | Chapter

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

Authors : Blake Mackall, Steven J. Miller, Christina Rapti, Caroline Turnage-Butterbaugh, Karl Winsor

Published in: Families of Automorphic Forms and the Trace Formula

Publisher: Springer International Publishing

Abstract

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.

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The derivation is by doing a contour integral of the logarithmic derivative of the completed L-function times the test function, using the Euler product and shifting contours; see [RudSa] for details.

It is worth noting that these formulas hold without assuming GRH. In that case, however, the zeros no longer lie on a common line and we lose the correspondence with eigenvalues of Hermitian matrices.

A similar absorbtion holds in other families, so long as the Satake parameters satisfy | α _i(p) | ≤ Cp ^δ for some δ < 1∕6.

We comment on this in greater length when we consider the family of all characters with square-free modulus. Briefly, a constancy in the conductors allows us to pass certain sums through the test functions to the coefficients. This greatly simplifies the analysis of the 1-level density; unfortunately cross terms arise in the 2-level and higher cases, and the savings vanish (see [Mil1, Mil2]).

The definition of the 1-level density as a sum of a test function at scaled zeros is well defined even if GRH fails; however, in that case the zeros are no longer on a line and we thus lose the ability to talk about spacings between zeros. Thus in many of the arguments in the subject GRH is only used to interpret the quantities studied, though there are exceptions (in [ILS] the authors use GRH for Dirichlet L-functions to expand Kloosterman sums).

It is easy to handle the case where the conductors are monotone by rescaling the zeros by the average log-conductor; as remarked many times above the general case is more involved.

The Satake parameters | α _π, i | are bounded by p ^δ for some δ, and it is conjectured that we may take δ = 0. While this conjecture is open in general, for many forms there is significant progress towards these bounds with some δ < 1∕2. See, for example, recent work of Kim and Sarnak [Kim, KimSa]. For our purposes, we only need to be able to take δ < 1∕6, as such an estimate and trivial bounding suffices to show that the sum over all primes and all ν ≥ 3 is O(1∕logR).

There are some situations where arithmetic enters. The standard example is that in estimating moments of L-functions one has a product a _k g _k, where a _k is an arithmetic factor coming from the arithmetic of the form and g _k arises from random matrix theory. See, for example, [CFKRS, KeSn1, KeSn2].

These bounds cannot be improved, as Miller [Mil3] found a family where there is a term of size p ^3∕2.

Following [ILS] we can remove the weights, but their presence facilitates the application of the Petersson formula.

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Title: Some Results in the Theory of Low-Lying Zeros of Families of L-Functions
Authors: Blake Mackall
Steven J. Miller
Christina Rapti
Caroline Turnage-Butterbaugh
Karl Winsor
Publisher: Springer International Publishing
Book: Families of Automorphic Forms and the Trace Formula
Print ISBN: 978-3-319-41422-5

Electronic ISBN: 978-3-319-41424-9

Copyright Year: 2016
DOI: https://doi.org/10.1007/978-3-319-41424-9_11

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