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

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.

Appendix: Biases in Second Moments in Additional Families

By Megumi Asada, Eva Fourakis, Steven J. Miller and Kevin Yang

This appendix describes work in progress on investigating biases in the second moments of other families. It is thus a companion to Sect. 4.2. Fuller details and proofs will be reported by the authors in [AFMY]; our purpose below is to quickly describe results on analogues of the Bias Conjecture.

1.1 Dirichlet Families

Let q be prime, and let $\mathcal{F}_{q}$ be the family of nontrivial Dirichlet characters of level q. In this family, the second moment is given by

$$\displaystyle{ M_{2}(\mathcal{F}_{q};X) =\sum _{p<X}\ \sum _{\chi \in \mathcal{F}_{q}}\chi ^{2}(p). }$$

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Denote the amalgamation of families by $\mathcal{F}_{Y } = \cup _{Y/2<q<Y }\mathcal{F}_{q}$, with the naturally defined second moment.

Computing $M_{2}(\mathcal{F}_{q},X)$ is straightforward from the orthogonality relations, which as we’ve seen earlier yields a quantity related to the classical problem on the distribution of primes in residue classes. Approximating carefully π(X) and π(X, q, a) via the Prime Number Theorem, one can deduce the following.

Theorem 4.8.

The family $\mathcal{F}_{q}$ has positive bias, independent of q, in the second moments of the Fourier coefficients of the L-functions.

Remark 4.9.

Note that the behavior of Dirichlet L-functions is very different than that from families of elliptic curves.

Now, suppose q ≠ ℓ is a prime such that q ≡ 1(ℓ). Let $\mathcal{F}_{q,\ell}$ be the family of non-trivial ℓ-torsion Dirichlet characters of level q, which is nonempty by the stipulated congruence condition. In this family, the second moment is given by

$$\displaystyle{ M_{2}(\mathcal{F}_{q,\ell};X) =\sum _{p<X}\sum _{\chi \in \mathcal{F}_{q,\ell}}\chi ^{2}(p). }$$

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Define $\mathcal{F}_{Y }:= \cup _{Y/2<q<Y }\mathcal{F}_{q,\ell_{q}}$ for any choice of suitable ℓ _q for each q.

Theorem 4.10.

The family $\mathcal{F}_{q,\ell}$ has zero bias independent of q and ℓ. Thus, $\mathcal{F}_{Y }$ exhibits zero bias in the second moments of the Fourier coefficients of the L-functions.

1.2 Families of Holomorphic Cusp Forms

Let S _k, q (χ ₀) denote the space of cuspidal newforms of level q, weight k and trivial nebentypus, endowed with the structure of a Hilbert space via the Petersson inner product. Let B _k, q (χ ₀) be any orthonormal basis of S _k, q (χ ₀) and let $\mathcal{F}_{X}:= \cup _{k<X:k\equiv 0(2)}\mathcal{B}_{k,q=1}(\chi _{0})$. In this family, the second moment is given by the weighted Fourier coefficients¹⁰:

$$\displaystyle{ M_{2}(\mathcal{F}_{X};\delta ) =\sum _{p<X^{\delta }}\ \sum _{k<X:k\equiv 0(2)}\ \sum _{f\in B_{k,q}(\chi _{0})}\vert \psi _{f}(p)\vert ^{2}, }$$

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where $\psi _{f}(p) = \frac{\left (\varGamma (k-1)\right )^{\frac{1} {2} }} {(4\pi p)^{\frac{k-1} {2} }} \lambda _{f}(p)\sqrt{\log p}$, with λ _f (p) the Hecke eigenvalue of f for the Hecke operator T _p . Let $\mathcal{F}_{X;\varepsilon } = \cup _{q<X^{\varepsilon }}\mathcal{F}_{X}$ be the amalgamation of families with the second moment

$$\displaystyle{ M_{2}(\mathcal{F}_{X;\varepsilon };\delta ) =\sum _{p<X^{\delta }}\ \sum _{q<X^{\varepsilon }}\ \sum _{k<X:k\equiv 0(2)}\ \sum _{f\in B_{k,q}(\chi _{0})}\ \vert \psi _{f}(p)\vert ^{2}. }$$

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The Petersson Formula provides an explicit method of computing $M_{2}(\mathcal{F}_{X};\delta )$ via Kloosterman sums and Bessel functions. Averaging over the level and weight to obtain asymptotic approximations as in [ILS], we prove the following theorem in [AFMY].

Theorem 4.11.

The family $\mathcal{F}_{X}$ has negative bias, independent of the level q of $\frac{1} {2}$ , in the second moments of the Fourier coefficients of the L-functions. Thus, $\mathcal{F}_{X;\varepsilon }$ exhibits negative bias.

Let us now let H _k, q ^∗(χ ₀) denote a basis of newforms of Petersson norm 1 for prime level q and even weight k. We consider another weighted second moment, given by

$$\displaystyle{ M_{2}^{\mathrm{weighted}}(\mathcal{F}_{ X};\delta ) =\sum _{p<X^{\delta }}\ \sum _{k<X:k\equiv 0(2)}\ \sum _{f\in H_{k,q}^{{\ast}}(\chi _{0})} \frac{\varGamma (k)} {(4\pi )^{k}}\vert \lambda _{f}(p)\vert ^{2}. }$$

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Similarly, let $\mathcal{F}_{X;\varepsilon } = \cup _{q<X^{\varepsilon }}\mathcal{F}_{X}$ be the amalgamation of these families with the weighted second moment

$$\displaystyle{ M_{2}^{\mathrm{weighted}}(\mathcal{F}_{ X;\varepsilon };\delta ) =\sum _{p<X^{\delta }}\ \sum _{q<X^{\varepsilon }}\ \sum _{k<X:k\equiv 0(2)}\ \sum _{f\in H_{k,q}^{{\ast}}(\chi _{0})}\ \frac{\varGamma (k)} {(4\pi )^{k}}\vert \lambda _{f}(p)\vert ^{2}. }$$

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We prove the following in [AFMY].

Theorem 4.12.

The family $\mathcal{F}_{X}$ has positive bias dependent on the level q. Moreover, the family $\mathcal{F}_{X,\varepsilon }$ exhibits positive bias as well.

If we now consider the unweighted second moment given by

$$\displaystyle{ M_{2}(\mathcal{F}_{X};\delta ) =\sum _{p<X^{\delta }}\ \sum _{k<X:k\equiv 0(2)}\ \sum _{f\in H_{k,q}^{{\ast}}(\chi _{0})}\lambda _{f}^{2}(p), }$$

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we prove the following in [AFMY] as well.

Theorem 4.13.

Assume δ < 1 and ɛ = 1. The family $\mathcal{F}_{X}$ has positive bias dependent on q. Moreover, the family $\mathcal{F}_{X;\varepsilon }$ exhibits positive unweighted bias as well.