Low-lying zeros in families of holomorphic cusp forms: the weight aspect

We study low-lying zeros of $L$-functions attached to holomorphic cusp forms of level $1$ and large weight. In this family, the Katz--Sarnak heuristic with orthogonal symmetry type was established in the work of Iwaniec, Luo and Sarnak for test functions $\phi$ satisfying the condition supp$(\widehat \phi) \subset(-2,2)$. We refine their density result by uncovering lower-order terms that exhibit a sharp transition when the support of $\widehat \phi$ reaches the point $1$. In particular the first of these terms involves the quantity $\widehat \phi(1)$ which appeared in previous work of Fouvry--Iwaniec and Rudnick in symplectic families. Our approach involves a careful analysis of the Petersson formula and circumvents the assumption of GRH for $\text{GL}(2)$ automorphic $L$-functions. Finally, when supp$(\widehat \phi)\subset (-1,1)$ we obtain an unconditional estimate which is significantly more precise than the prediction of the $L$-functions Ratios Conjecture.


Introduction
Katz and Sarnak [KS] conjectured that the distribution of low-lying zeros in a family F of Lfunctions is governed by a certain random matrix model G(F) called the symmetry type of F. This symmetry type has been determined in many families; see for example [FI,HR,ILS,M1,OS,Y], as well as the references in [SST]. Sarnak, Shin and Templier [SST] recently refined the Katz-Sarnak heuristics and introduced invariants which allow for a conjectural determination of the symmetry type.
In the current paper we focus on the family of classical holomorphic cusp forms of level 1 and large even weight k. As in [ILS,Chapter 10], this will ease the exposition and allow for a more transparent analysis. For this family, the predictions of Katz and Sarnak were confirmed in the influential work of Iwaniec, Luo and Sarnak [ILS] for a certain class of test functions, under the assumption of the Riemann Hypothesis for Dirichlet L-functions and holomorphic cusp form Lfunctions. Our first goal is to relax these conditions by only assuming the Riemann Hypothesis for Dirichlet L-functions. Our second and main goal is to refine the Iwaniec-Luo-Sarnak density results by determining lower-order terms up to an arbitrary negative power of log k.
More precisely, we fix a basis B k of Hecke eigenforms in the space H k of holomorphic modular forms of level 1 and even weight k. We normalize so that for every the first coefficient satisfies a f (1) = 1. Hence, the Hecke eigenvalues of f are given by λ f (n) = a f (n) and for ℜ(s) > 1 the L-function of f takes the form This classically extends to an entire function and satisfies a functional equation relating the values at s to those at 1 − s. In the sums over f ∈ B k to be considered in this paper, we will scale each term with the harmonic weight where (f, f ) := SL 2 (Z)\H y k−2 |f (z)| 2 dxdy.
We now state our main theorem which, in the case when the level N = 1, refines the estimate in [ILS,Theorem 1.3] by weakening its assumptions and obtaining lower-order terms which contain a phase transition as the support of φ reaches 1.
Theorem 1.1. Let φ be an even Schwartz test function for which supp( φ) ⊂ (−2, 2). Assuming the Riemann Hypothesis for Dirichlet L-functions, we have the estimate where the constants R j,h and S j,h appearing in the lower-order terms only depend on the weight function h (see (6.8), (6.9), and (6.10)).
We deduce Theorem 1.1 from a power-saving formula for the 1-level density (see (6.1)), which we combine with an asymptotic evaluation of the resulting terms (see Theorem 6.6). We were able to circumvent the use of the Riemann Hypothesis for holomorphic cusp form L-functions by refining the estimate [ILS,Corollary 2.2] on the remainder in the Petersson trace formula (see Proposition 3.3). The specific estimate we obtain is the following: (1.4) In particular, when (m, n) = 1 this estimate is nontrivial in the range mn ≤ k 4−ε , whereas [ILS,Corollary 2.2] is nontrivial up to mn ≤ k 10 3 −ε . The terms involving φ (j) (1) in (1.3) are responsible for a sharp transition at 1 in these orthogonal families and are analogous to those obtained in symplectic families in [FI,FPS2,FPS3,R,Wax]. Indeed, in the family of real Dirichlet characters considered in [FPS2], after applying the explicit formula and treating the resulting sums over primes by repeatedly using the Poisson summation formula, one obtains lower-order terms involving φ (j) (1). This work was inspired by the function field case considered in [R], in which, using Poisson summation, the 1-level density is turned into an average of the trace of the Frobenius class in the hyperelliptic ensemble, from which a transition term is isolated using the explicit formula. Transition terms also surface in predictions coming from the L-function Ratios Conjecture [FPS3,Wax]; in this case one needs to compute averages of ratios of local factors at infinity. In the current situation, these terms come from a significantly different source, namely from a careful analysis of averages of Bessel functions and Kloosterman sums coming from the Petersson trace formula. Independently of the use of different methods, this seems to indicate that a transition in lower-order terms should exist whenever the symmetry type of a family is even or odd orthogonal, or symplectic.
Interestingly, averaging over all even values of the weight k, we find that where W = W (O) := 1 2 + δ 0 and H(K) := H + (K) + H − (K). Hence, as expected we see that there is no transition at 1 in this mixed signs family (see also [M2,Theorem 1.6]). We should point out that for similar reasons, there is no transition in mixed sign families of holomorphic cusp forms of fixed weight and of large level [M2,RR].
Remark 1.2. One can compute explicitly the constants R j,h and S j,h in Theorem 1.1. In particular, the first of these are given by where θ(t) := p≤t log p.
We now state our results for test functions whose Fourier transform is supported in the interval (−1, 1). Under this restriction our estimates are substantially more precise. Indeed, we do not need the GRH assumption, the error term is exponentially small in the weight k and we do not need the average over k (we set X = k 2 in D k (φ; X)). Theorem 1.3. Let φ be an even Schwartz test function for which supp( φ) ⊂ (−1, 1). Then the (unaveraged) 1-level density satisfies the estimate (1) We emphasize that Theorem 1.3 is unconditional. Moreover, the error term in (1.5) is exponentially small, in particular this is significantly more precise than predictions from the Ratios Conjecture [CFZ,CS,M2]. This comes from exponential bounds on the Bessel functions occuring in the Petersson trace formula (see (3.2)).
(3) An estimate for the 1-level density D k (φ; k 2 ) was previously obtained by Miller [M2,Lemmas 4.2 and 4.4] with the error term O ε (k σ 2 − 5 6 +ε ), under the same conditions. Lower-order terms in the level aspect were previously studied in [M2,MM,RR]. In an upcoming paper we will refine the Iwaniec-Luo-Sarnak and Miller-Montague estimates for fixed weight and large level and isolate a transition term of the same type as in Theorem 1.1.
The paper is divided as follows. In Sections 2 and 3 we discuss prerequisites, establish (1.4) and discard higher prime powers in the explicit formula. Section 4 is dedicated to the proof of Theorem 1.3. Finally, in Section 5 we apply estimates on averages of Bessel functions to isolate a transition term, which we carefully evaluate in Section 6.

Acknowledgments
We would like to thank the Institut français de Suède, the Laboratoire de Mathématiques d'Orsay and the Centro Internazionale per la Ricerca Matematica in Trento for supporting this project and providing excellent research conditions. The first author was supported by a Postdoctoral Fellowship at the University of Ottawa. The second author was supported by an NSERC discovery grant at the University of Ottawa. The third author was supported by a grant from the Swedish Research Council (grant 2016-03759).

Explicit formula
We begin by recalling the explicit formula for holomorphic cusp form L-functions in the case where the level equals 1.
Lemma 2.1. Let φ be an even Schwartz test function. We have the formula Here, α f (p), β f (p) are the local coefficients of the L-function Proof. For f ∈ B k , the formula [ILS,(4.11)] reads Summing over f ∈ B k against the weight ω f we obtain the desired formula.
We now estimate the integral involving the logarithmic derivative of the gamma function in (2.1).
Lemma 2.2. Let ε > 0 and let φ be an even Schwartz test function. In the range k ≤ X 5 , we have the estimate The result follows from extending the integral to R.

The Petersson trace formula and related estimates
In order to handle the term involving sums over prime powers in (2.1) we will apply the Petersson trace formula. For m, n ∈ Z and c ∈ N we define the Kloosterman sum We will repeatedly use the classical Weil bound (see for instance [IK,Corollary 11.12]) where J k−1 is the Bessel function.
We recall the following bound on the Bessel function.
In [ILS,Chapter 2], this bound is shown to imply the estimate which is non-trivial in the range mn ≤ k 10 3 −ε . By a more careful decomposition of the the sum over c in (3.1), we establish a more precise estimate which is non-trivial in the wider range mn ≤ k 4−ε .
Proposition 3.3. Let ε > 0, and let m, n, k ∈ N, with 2 | k. We have the estimate In the range mn ≤ k 2 /(4πe) 2 , we have the exponentially precise estimate Proof. We bound the rightmost term in the statement of Lemma 3.1 by combining the Weil bound with Lemma 3.2, as follows: We first bound S 4 . To do so, note that by Stirling's approximation. Note that S 1 , S 2 and S 3 are all empty whenever mn ≤ k 2 /(4πe) 2 and hence (3.2) follows. We now assume that mn > k 2 /(4πe) 2 . A computation similar to the one above shows that (the second term accounts for the possibility that the sum contains only one term).
As for S 1 , we compute that

Making the change of variables
In a similar way we see that S 3 ≪ ε k −1 (mn) In the next lemma we apply Proposition 3.3 in order to discard higher prime powers in the explicit formula (2.1).
Lemma 3.4. Assume that k ∈ 2N, X ∈ R ≥2 and the even Schwartz test function φ are such that X σ < k 4 , where σ :=sup(supp( φ)). Then we have the following estimate on the 1-level density: Assuming the stronger condition X σ < (k/4πe) 2 , we have the more precise estimate (3.4) Proof. The goal of this proof is to estimate the terms p, ν ≥ 2 in (2.1). By the Hecke relations, the sum of those terms is equal to From Proposition 3.3 and (1.1), we see that Similarly, 2

The only terms left are
2 We conclude the proof by applying Lemmas 2.1 and 2.2, and (1.1).

1-level density: Unconditional results
In this section we evaluate the 1-level density D k (φ; X) for test functions satisfying sup(supp(φ)) < 1, unconditionally. We begin by asymptotically evaluating the second term on the right-hand side of (3.3).
Lemma 4.1. Let φ be an even Schwartz test function. For any fixed J ≥ 1, we have the estimate where c 1 := 2 ∞ 1 θ(t) − t t 2 dt + 2 and for j ≥ 2, Proof. Performing summation by parts, we reach the following identity: By the prime number theorem in the form θ(t)−t ≪ t exp(−2c √ log t), we see that for any 0 < ξ < 1, Moreover, taking Taylor series and applying the prime number theorem, we see that The result follows from selecting ξ = (log X) −1+δ for some δ > 0.
We now set X = k 2 and prove Theorem 1.3.
Proof of Theorem 1.3. We apply Proposition 3.3 and obtain that the second prime sum in (3.4) satisfies the bound The proof follows.

1-level density averaged over the weight: Extended support
In this section we study the quantities D + K,h (φ) and D − K,h (φ), that is we average the 1-level density D k (φ; K 2 ) over k ≍ K against the weight h( k−1 K ). Lemma 5.1 ([Iw2, Lemma 5.8], [ILS,Corollary 8.2]). For h a non-negative, smooth function with compact support in R >0 and for any K ≥ 2, we have the estimates In the next lemma we estimate the total weight H ± (K) and a related sum.
Lemma 5.2. For h a non-negative, smooth function with compact support in R >0 and for any K, N ≥ 2, we have the estimates Proof. More generally, we will show that for any a mod 4, (5.2) Now, for any b mod 4, Poisson summation gives The estimate (5.1) follows by orthogonality of additive characters. Similarly, we see that Indeed, integration by parts shows that Finally, the integral on the right-hand side of (5.3) equals and (5.2) follows.
In the next lemma we estimate the average of (3.3) over k. In order to do so, we will apply Lemmas 5.1 and 5.2.
Lemma 5.3. Let φ be an even Schwartz test function and let h be a non-negative, smooth function with compact support in R >0 . Under the condition σ =sup(supp( φ)) < 2 and for K ≥ 2, we have the estimate Proof. From combining Lemmas 3.4 and 5.2, we have that By the Petersson trace formula (Lemma 3.1), the third term is equal to (5.4) Applying Lemma 5.1, we see that Since p ≤ K 4−ε , we see by the rapid decay of that for any A ≥ 1, the first term in this expression is and hence the contribution of this term to (5.4) is As for the sum of the error terms in (5.5), the contribution is ≪ K 2σ−5 , which is an admissible error term. Moreover, applying Lemma 5.1 once more, resulting in a main term as well as the admissible error term O(K 2σ−4 ).
We now end this section by evaluating the second sum over primes in Lemma 5.3, under GRH for Dirichlet L-functions. This term will be responsible for the phase transition at 1, and will be investigated more closely in Section 6.
Lemma 5.4. Let φ be an even Schwartz test function, let h be a non-negative, smooth function with compact support in R >0 , and assume the Riemann Hypothesis for Dirichlet L-functions. Then for any K ≥ 2 we have the estimate where ϕ is Euler's totient function.
Proof. If σ < 1, then for large enough K the left-hand side of (5.6) is identically zero. We may thus assume that σ ≥ 1. The sum over p equals where, by [ILS,Lemma 6.1], Note that our restriction on the support of h implies that c ≍ √ t/K, and hence the restriction on the support of φ implies that for squarefree values of c and for t ≤ K 4−ǫ , the main term in this estimate is always larger than the error term. After a straightforward calculation, we obtain that the left-hand side of (5.6) equals The proof follows.

Evaluation of the transition term
The goal of this section is to evaluate the integral in Lemma 5.4. This will be done using different techniques depending on the range of the variable u. To this end, for a, b ∈ R ≥0 we define Notice that the inner sum is long only when u is larger and far away from 1. By Lemmas 5.3 and 5.4, we see that when σ =sup(supp( φ)) < 2 and under the assumption of GRH for Dirichlet L-functions, (6.1) We now move on to evaluating the integral I 0,∞ . We let δ K be a positive parameter which satisfies δ K ≫ h 1/ log K. Recall that h is supported in R + , and hence for K large enough the integrand in I 0,∞ is zero in the interval [0, 1 − δ K ). Hence, where, as before, σ =sup(supp( φ)). Proof. We first establish the following estimate, for squarefree values of d: To do so, note that The error term here can be improved to O(x 1 2 exp(−c(log x) 3 5 (log log x) − 1 3 )) by replacing (6.4) with the stronger estimate obtained from combining [MV,Exercise 19,§6.2.1] with the Korobov-Vinogradov zero-free region.
Applying this equality iteratively, we reach the identity A summation by parts combined with [NZM,Theorem 8.25] yields that 6 Inserting this estimate into (6.3), we are left with an error term which is and (6.2) follows. The claimed estimate then follows from the convolution identity µ 2 (m) m and a straightforward summation by parts.
We now evaluate the part of the integral I 0,∞ for which u is slightly larger than 1. In this range, the sum over c is fairly long and we can effectively apply Lemma 6.1.
Lemma 6.2 (Range u > 1 + δ K ). Let φ be an even Schwartz test function and let K ≥ 2. We have that Proof. By Lemma 6.1, we have that The desired estimate follows by integrating over u against K u φ(u) and applying Lemma 5.2. 6 The precise value of the constant is deduced from writing S1(x) = 1 2πi (1) ζ(s+1) ζ(2s+2) x s s ds and shifting the contour of integration to the left.
We now evaluate the part of the integral I 0,∞ in which u is close to 1. In this range we can expand φ(u) into Taylor series around u = 1 and recover the transition terms φ (j) (1) (see Lemma 6.5). The resulting integrals are evaluated in Lemma 6.4 by applying the inverse Mellin transform, truncating the resulting integrals and shifting the contours of integration.
where the implied constant is absolute.
Proof. Applying integration by parts, we reach the exact formula The proof follows.
Lemma 6.4. For j ≥ 0, K ≥ 2 and 20(log K) −1 ≤ δ K ≤ 1 2 we have the estimate c≥1 µ 2 (c) ϕ(c) Proof. Define By the restriction on the support of h, the function f K,j also has compact support on R >0 . We conclude that its Mellin tranform ϕ K,j (s) is entire. Moreover, are also entire. For any N ≥ 1, applying [FPS1, Lemma 2.1] yields the crude bound Next, Mellin Inversion gives the formula Hence, by absolute convergence, By applying Lemma 6.3, we obtain the estimate From the rapid decay of Mh(1 − s) on vertical lines (see [FPS1,Lemma 2.1]), we see that As for the other part of the integral, by applying (6.6) we can shift the countour to the left until the line ℜ(s) = − 1 2 + ε 2 , and reach the identity In a similar fashion as before, we see that Putting these estimates together, we conclude that The result follows.
Lemma 6.5 (Range 1 − δ K < u < 1 + δ K ). Let φ be an even Schwartz test function. We have for K ≥ 2 and odd J ≥ 1 that where the constants C j,h are defined in (6.5).
Proof. By definition of I 1−δ K ,1+δ K , we need to evaluate the sum c≥1 µ 2 (c) ϕ(c) Taking Taylor series and applying 7 Lemma 6.4, this is Applying Lemma 6.4 once more, we reach the expression Applying Lemma 5.2, the proof follows.
Collecting the estimates in this section, we reach the following theorem.
We can clearly assume without loss of generality that J is odd. The sum over primes is estimated in Lemma 4.1. Moreover, we recall that for K large enough I 0,1−δ K = 0 and therefore we have that which together with Lemmas 6.2 and 6.5 and the choice δ K = 3(J + 3) log log K/ log K implies the desired result.