# One-level density for holomorphic cusp forms of arbitrary level

- Owen Barrett
^{1}, - Paula Burkhardt
^{2}, - Jonathan DeWitt
^{3}, - Robert Dorward
^{4}and - Steven J. Miller
^{5}Email author

**Received: **13 June 2016

**Accepted: **14 July 2017

**Published: **5 December 2017

## Abstract

In 2000 Iwaniec, Luo, and Sarnak proved for certain families of *L*-functions associated to holomorphic newforms of square-free level that, under the Generalized Riemann Hypothesis, as the conductors tend to infinity the one-level density of their zeros matches the one-level density of eigenvalues of large random matrices from certain classical compact groups in the appropriate scaling limit. We remove the square-free restriction by obtaining a trace formula for arbitrary level by using a basis developed by Blomer and Milićević, which is of use for other problems as well.

## Keywords

## Mathematics Subject Classification

## 1 Introduction

Montgomery [1] conjectured that the pair correlation of critical zeros up to height *T* of the Riemann zeta function \(\zeta (s)\) coincides with the pair correlation of eigenvalues of random unitary matrices of dimension *N* in the appropriate limit as \(T,N\rightarrow \infty \). This remarkable connection initiated a new branch of number theory concerned with relating the statistics of zeros of \(\zeta (s)\), and of *L*-functions more generally, to those of eigenvalues of random matrices. While additional support for this agreement was obtained by the work of Hejhal [2] on the triple correlation of \(\zeta (s)\), Rudnick and Sarnak [3] on the *n*-level correlation for cuspidal automorphic forms, and Odlyzko [4, 5] on the spacings between adjacent zeros of \(\zeta (s)\), the story cannot end here as these statistics are insensitive to the behavior of any finite set of zeros. As the zeros at and near the central point play an important role in a variety of problems, this led Katz and Sarnak [6, 7] to develop a new statistic which captures this behavior.

### Definition 1.1

*L*(

*s*,

*f*) be an

*L*-function with zeros in the critical strip \(\rho _f = 1/2 + i \gamma _f\) (note \(\gamma _f \in \mathbb {R}\) if and only if the Generalized Riemann Hypothesis holds for

*f*), and let \(\phi \) be an even Schwartz function whose Fourier transform has compact support. The

**one-level density**is

*L*-functions as the conductors tend to infinity. Specifically, let \(\mathcal {F}_N\) be a sub-family of \(\mathcal {F}\) with suitably restricted conductors; often one takes all forms of conductor

*N*, or conductor at most

*N*, or conductor in the range [

*N*, 2

*N*]. If the symmetry group is \(\mathcal {G}\), then we expect

*L*-functions, elliptic curves, cuspidal newforms, Maass forms, number field

*L*-functions, and symmetric powers of \(\mathrm{GL}_2\) automorphic representations) agree with the scaling limits of a random matrix ensemble; see [6–29] for some examples, and [10, 27, 30] for discussions on how to determine the underlying symmetry. For additional readings on connections between random matrix theory, nuclear physics and number theory see [31–39].

We concentrate on extending the results of Iwaniec, Luo, and Sarnak in [18]. One of their key results is a formula for unweighted sums of Fourier coefficients of holomorphic newforms of a given weight and level. This formula writes the unweighted sums in terms of weighted sums to which one can apply the Petersson trace formula; it is instrumental in performing any averaging over holomorphic newforms, since one can interchange summation and replace the average of Fourier coefficients with Kloosterman sums and Bessel functions, which are amenable to analysis.

A drawback of their formula is that it may only be applied to averages of newforms of square-free level. One reason is that the development of such a formula depends essentially on the construction of an explicit orthonormal basis for the space of cusp forms of a given weight and level, which they only computed in the case of square-free level. In 2011, Rouymi [40] complemented the square-free calculations of Iwaniec, Luo, and Sarnak, finding an orthonormal basis for the space of cusp forms of prime power level, and applying this explicit basis towards the development of a similar sum of Fourier coefficients over all newforms with level equal to a fixed prime power.

In 2015, Blomer and Milićević [41] extended the results of Iwaniec, Luo, and Sarnak and Rouymi by writing down an explicit orthonormal basis for the space of cusp forms (holomorphic or Maass) of a fixed weight and, novelly, arbitrary level.

The purpose of this article is, first, to leverage the basis of Blomer and Milićević to prove an exact formula for sums of Fourier coefficients of holomorphic newforms over all newforms of a given weight and level, where now the level is permitted to be arbitrary (see below, as well as Proposition 5.2 for a detailed expansion). The basis of Blomer and Milićević requires one to split over the square-free and square-full parts of the level; this splitting combined with the loss of several simplifying assumptions for Hecke eigenvalues and arithmetic functions makes the case where the level is not square-free much more complex. As an application, we use this formula to show the 1-level density agrees only with orthogonal symmetry.

### 1.1 Harmonic averaging

Throughout we assume that \(k,N \ge 1\) with *k* even. By \(H_k^\star (N)\) we always mean a basis of arithmetically normalized Hecke eigenforms in the space orthogonal to oldforms. Explicitly, it is a basis of holomorphic cusp forms of weight *k* and level *N* which are new of level *N* in the sense of Atkin and Lehner [42] and whose elements are eigenvalues of the Hecke operators \(T_n\) with \((n,N) = 1\) and normalized so that the first Fourier coefficient is 1. We let \(\lambda _f(n)\) denote the *n*th Fourier coefficient of an \(f \in H_k^\star (N)\) (see the next section for more details).

*f*, we introduce the renormalized Fourier coefficients

*k*and level

*N*. The importance of \(\Delta _{k,N}(m,n)\) is clarified by the introduction of the Petersson formula in the next section.

Using the orthonormal basis \(\mathscr {B}_k(N)\) of Milićević and Blomer, we then prove the following (unconditional) formula.

### Theorem 1.2

A key part of the proof is a result on weighted sums of products of the Fourier coefficients, which we extract in Lemma 3.1. Note that in many cases, the right-hand side of (1.6) is preferable to the left-hand side, as it is amenable to application of spectral summation formulas such as the Petersson formula (Proposition 2.1) and can be studied via Kloosterman sums, see Proposition 5.2. More generally, this sort of formula has a variety of applications involving the Fourier coefficients of holomorphic cusp forms and *L*-functions. Rouymi uses his basis and formula to study the non-vanishing at the central point of *L*-functions attached to primitive cusp forms; we elect to apply our formula to generalize [18, Theorem 1.1] on the one-level density of families of holomorphic newform *L*-functions by removing the condition that *N* must pass to infinity through the square-free integers.

### 1.2 The Density Conjecture

*L*-function

*L*(

*s*,

*f*) associated to a \(f\in H_k^\star (N)\) as the Dirichlet series

*L*(

*s*,

*f*);

*L*(

*s*,

*f*) may be analytically continued to an entire function on \(\mathbf {C}\) with a functional equation relating

*s*to \(1-s\). We also need similar results for its symmetric square (see [43, 44]):

*f*(we will only consider the case of trivial nebentypus below).

*L*(

*s*,

*f*), and, for technical reasons, \(L(s,{\mathrm{sym}}^2\,f)\) as well as for all Dirichlet

*L*-functions (see Remark 1.5). Then we may write all nontrivial zeros of

*L*(

*s*,

*f*) as

*f*; the ordinates \(\gamma _f\) are counted with their corresponding multiplicities, and \(\phi (x)\) is an even function of Schwartz class such that its Fourier transform

*N*is our asymptotic parameter (and \(\mathscr {F}=\cup _{N\ge 1}\mathscr {F}(N)\)). It is worth mentioning that \(\lim _{N\rightarrow \infty }|H_k^\star (N)| = \infty \). The one-level density is the expectation of \(D_1(f;\phi )\) averaged over our family:

*N*runs over square-free numbers. We prove the following theorem with no conditions on how

*N*tends to infinity; new features emerge from the presence of square factors dividing the level.

### Theorem 1.3

*L*(

*s*,

*f*) and \(L(s,{\mathrm{sym}}^2\,f)\) for \(f \in H_k^\star (N)\) and for all Dirichlet

*L*-functions,

More generally, under the same assumptions the Density Conjecture holds for the family \(H_k^\star (N)\) for any test function \(\phi (x)\) whose Fourier transform is supported inside \((-u,u)\) with \(u < 2\log (kN) / \log (k^2 N)\).

### Remark 1.4

*N*the sign of the functional equation, \(\epsilon _f\), is given by

### Remark 1.5

We briefly comment on the use of the various Generalized Riemann Hypotheses. First, assuming GRH for *L*(*s*, *f*) yields a nice spectral interpretation of the 1-level density, as the zeros now lie on a line and it makes sense to order them; note, however, that this statistic is well-defined even if GRH fails. Second, GRH for \(L(s,{\mathrm{sym}}^2 f)\) is used to bound certain sums which arise as lower order terms; in [18] (page 80 and especially page 88) the authors remark how this may be replaced by additional applications of the Petersson formula (assuming GRH allows us to trivially estimate contributions from each form, but a bound on average suffices). Finally, GRH for Dirichlet *L*-functions is needed when we follow [18] and expand the Kloosterman sums in the Petersson formula with Dirichlet characters; if we do not assume GRH here we are still able to prove the 1-level density agrees with orthogonal symmetry, but in a more restricted range.

The structure of the paper is as follows. Our main goal is to prove the formula for sums of Hecke eigenvalues and then use this to compute the one-level density. We begin in §2 with a short introduction of the theory of primitive holomorphic cusp forms, as well as the Petersson trace formula and the basis of Blomer and Milićević. In §3 we find a formula for \(\Delta _{k,N}(m,n)\), which we leverage in §4 to find a formula for the arithmetically weighted sums, \(\Delta ^\star _{k,N}(n)\) (see [18, (2.53)]); this is Theorem 1.2. Using our formula, we find bounds for \(\Delta ^\star _{k,N}(n)\) in §5, culminating in the computation of the one-level density in §6 (Theorem 1.3).

## 2 Preliminaries

In this section we introduce some notation and results to be used throughout, much of which can be found in [43].

### 2.1 Hecke eigenvalues and the Petersson inner product

*k*,

*N*are positive integers, with

*k*even. Let \(S_k(N)\) be the linear space spanned by cusp forms of weight

*k*and trivial nebentypus which are Hecke eigenforms for the congruence group \(\Gamma _0(N)\). Each \(f \in S_k(N)\) admits a Fourier development

*N*to indicate we are considering

*f*and

*g*as forms on \(\Gamma _0(N)\), when perhaps \(\langle f,g\rangle _M\) might make sense as well.

*k*and level

*M*(typically we choose

*M*to be a divisor of

*N*). Then

*f*is an eigenfunction of all Hecke operators \(T_M(n)\), where

*f*:

### Proposition 2.1

Though the quantity \(\Delta _{k,N}(m,n)\) is independent of the choice of an orthonormal basis, we would like to compute with the Petersson trace formula using an explicit basis \(\mathscr {B}_k(N)\) to average over newforms. However, as remarked, the spaces \(S_k(L;f)\) do not have a distinguished orthogonal basis. Therefore, to produce a basis \(\mathscr {B}_k(N)\), we need a basis for the spaces \(S_k(L;f)\). Iwaniec, Luo, and Sarnak [18] write down an explicit basis when *N* is square-free. As we will see in the next section, Blomer and Milićević [41] have recently obtained a basis for arbitrary level *N*. Our first key idea, a kind of trace formula for sums of Hecke eigenvalues over newforms in the case *N* is arbitrary, is an explicit computation with this new basis. Our second key idea on the one-level density of the *L*-functions *L*(*s*, *f*) for \(f\in H_k^\star (N)\) uses our first key idea in an essential way to reduce the problem to the one already treated by Iwaniec, Luo, and Sarnak.

*M*:

### 2.2 An orthonormal basis for \(S_k(N)\)

### Proposition 2.2

Note that in our application we are not using the Petersson normalization but instead have normalized our forms to have first coefficient 1; thus for us below we have an orthogonal basis which becomes orthonormal upon dividing the forms by their norm.

In addition, we will also use of the following lemma. Originally stated in the context of square-free level, the same proof holds in general.

### Lemma 2.3

*M*|

*N*and \(f\in H_k^\star (M)\). Then

*L*and

*M*. Specializing to \(s=1\), we find

## 3 A formula for \(\Delta _{k,N}(m,n)\)

In this section we provide the following explicit formula for \(\Delta _{k,N}(m,n)\) in terms of Hecke eigenvalues. We start with a generalization of Lemma 2.7 of [18] to general *N*.

### Lemma 3.1

*f*be a newform of weight

*k*and level

*M*. Let \(f' = f/||f||_N\) so that \(f'\) is Petersson-normalized with respect to level

*N*(i.e., \(||f'||_N = 1\)) and note that

*d*|

*L*|

*N*and \((m,N) = (n,N) = 1\), \(\ell |(d,m)\) implies \(\ell = 1\) (and similarly for \(\ell |(d,n)\)). Thus the previous equation simplifies to

### Lemma 3.2

*f*be as before, with \(LM = N\). We have

### Proof

Combining Lemma 3.2 with equations (2.35) and (3.8) yields Lemma 3.1.

## 4 An inversion and a change from weighted to pure sums

### Proposition 4.1

### Proof of Proposition 4.1

We first prove (4.2). Note the following: \((m,N)=1\), \((n,N) = 1\) and \(\ell \mid L^\infty \) imply \((m,M) = 1\), \((n,M)=1\), and \((\ell ,m)=1\).

### Theorem 1.2

### Proof

We remove the weights in (4.1) by summing \(m^{-1}\Delta ^\star _{k,N}(m^2,n)\) over all \((m,N)=1\). We will need to replace \(\sum _{\ell \mid N^\infty } \sum _{(m,N)=1}(\ell m)^{-1}\lambda _f(\ell ^2)\lambda _f(m^2)\) with \(\sum _{r\ge 1}r^{-1}\lambda _f(r^2)\); some care is required as we do not have absolute convergence. This can be handled replacing \((\ell m)^{-1}\) and \(r^{-1}\) by \((\ell m)^{-s}\) and \(r^{-s}\), and then taking the limit as *s* tends to 1 from above. We do not need absolute convergence of the series to justify the limit; it is permissible as the Dirichlet series are continuous in the region of convergence and all sums at \(s=1\) exist as the sum of the coefficients grow sub-linearly.

## 5 Estimating tails of pure sums

*L*(

*s*,

*f*); see [18, p. 80] for more details. The lemma below is a straightforward modification of Lemma 2.12 of [18] to the case of general

*N*.

### Lemma 5.1

### Proof

*q*relatively prime to

*nN*. We simplify some of the resulting sums by grouping them with Lemma 3.1. Thus

We now substitute the Petersson formula (Proposition 2.1) for each instance of \(\Delta _{k,M}(m^2,n)\) to obtain an exact formula for \(\Delta _{k,N}'(n)\) in terms of Kloosterman sums; this is a generalization of Proposition 2.12 of [18].

### Proposition 5.2

We recover the bounds for \(|H_k^{\star }(N)|\) given by Martin in [47, Theorem 6(c)], and immediately obtain the following result (for completeness the calculation, which is standard, is given in the appendix of [48]).

### Proposition 5.3

## 6 The Density Conjecture for \(H_k^\star (N)\)

Fix some \(\phi \in \mathscr {S}(\mathbf {R})\) with \(\widehat{\phi }\) supported in \((-u,u)\). We reprise some basic definitions from the introduction.

*f*, we associate the

*L*-function

*L*(

*s*,

*f*), we can write its non-trivial zeros as

*f*which in our case is \(k^2N\). We also introduce a scaling parameter

*R*which we take to satisfy \(1 < R \asymp k^2N\).

*N*being square-free. The Density Conjecture concerns the average over \( H_k^\star (N)\), so we consider the sum

*Q*in the definition (5.4), and since \(\phi \) is of Schwartz class, we may apply Lemma 5.1 with \(X=Y=(kN)^\delta \) for small positive \(\delta \) to find

*p*is never a square. Then, moving the initial summation over \(p\not \mid N\) into the expression, we can rewrite in terms of \(Q_{k;N}^\star (m;c)\):

*N*not square-free)

*L*-functions (they expand the Kloosterman sums with Dirichlet characters). In order to apply this bound we need to secure \(12\pi m P^{1/2}\le kc\), (\(3z\le k\)) so as to satisfy a condition on an estimate for the Bessel function, given in their equation (2.11\('''\)). Noting that \(m\le Y\) and \(c\ge M\ge N/X\), it suffices to have \(12\pi XYP^{1/2}\le kN\). Taking logarithms, this becomes a condition on

*u*, namely

*u*in this range we can apply the estimate (6.16) to find

## Declarations

### Acknowledgements

The first four named authors were supported by NSF Grant DMS1347804 and Williams College; the fifth-named author was partially supported by NSF Grants DMS0850577 and DMS1561945. We thank Jim Cogdell for discussions on the signs of functional equation, Djorde Milićević and Valentin Blomer for conversations on their work, and the referees and editor for many helpful comments and observations.

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## Authors’ Affiliations

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