Conditional lower bounds on the distribution of central values in families of $L$-functions

We establish a general principle that any lower bound on the non-vanishing of central $L$-values obtained through studying the one-level density of low-lying zeros can be refined to show that most such $L$-values have the typical size conjectured by Keating and Snaith. We illustrate this technique in the case of quadratic twists of a given elliptic curve, and similar results would hold for the many examples studied by Iwaniec, Luo, and Sarnak in their pioneering work on $1$-level densities.


Introduction
Selberg [11,12] (see [8] for a recent treatment) established that if t is chosen uniformly from [0, T ] then the values log |ζ( 1 2 + it)| are distributed approximately like a Gaussian random variable with mean 0 and variance 1  2 log log T .More recently, Keating and Snaith [6] have conjectured that central values in families of L-functions have an analogous log-normal distribution with a prescribed mean and variance depending on the "symmetry type" of the family.This is a powerful conjecture which gives more precise versions of conjectures on the non-vanishing of L-values; for example, it refines Goldfeld's conjecture (towards which remarkable progress has been made with the work of Smith [13]) that the rank in families of quadratic twists of an elliptic curve is 0 for almost all twists with even sign of the functional equation.In [7] we enunciated a general principle which shows the upper bound (in a sense to be made precise below) part of the Keating-Saith conjecture in any family where somewhat more than the first moment can be computed.In this paper, we consider the complementary problem of obtaining lower bounds in the Keating-Saith conjecture, which is intimately tied up with questions on the non-vanishing of L-values.One analytic approach, conditional on the Generalized Riemann Hypothesis, towards such non-vanishing results is based on computing the 1-level density for low lying zeros in families of L-functions, and our goal in this paper is to show how this approach (in the situations where it succeeds in producing a positive proportion of non-vanishing) may be refined to give corresponding lower bounds towards the Keating-Snaith conjectures.In a later paper, we shall consider similar refinements of the mollifier method, which is another analytic approach that in many Date: August 2, 2023.The first author was partially supported by DMS-1902063.The second author is partially supported by an NSF grant, and a Simons Investigator award from the Simons Foundation.
cases establishes non-vanishing results unconditionally.Algebraic approaches such as Smith's work [13] on Goldfeld's conjecture are capable of establishing definitive non-vanishing results (or, for other examples, see Rohrlich [9,10] and Chinta [2]), but we are unable to refine these methods to show that the non-zero values that are produced in fact have the typical size predicted by the Keating-Snaith conjectures.
To illustrate our method, we treat the family of quadratic twists of an elliptic curve E defined over Q with conductor N, where the 1-level density of low lying zeros has been studied by many authors, notably Heath-Brown [3].Let the associated L-function be where the coefficients a(n) are normalized such that |a(n)| ≤ d(n).Since elliptic curves are known to be modular, L(s, E) has an analytic continuation to the entire complex plane and satisfies the functional equation where ǫ E , the root number, is ±1 and Throughout the paper, let d denote a fundamental discriminant coprime to 2N, and let χ d = ( d • ) denote the associated primitive quadratic character.Let E d denote the quadratic twist of E by d, and let its associated L-function be is entire and satisfies the functional equation Note that, by Waldspurger's theorem, and in this paper, we shall restrict attention to those twists with root number 1. Put therefore The Keating-Snaith conjectures predict that for d ∈ E, the quantity log L( 12 , E d ) has an approximately normal distribution with mean − 1 2 log log |d| and variance log log |d|.To state this precisely, let α < β be real numbers, and for any X ≥ 20, let us define (1) Then the Keating-Snaith conjecture states that, for fixed intervals (α, β) and as X → ∞, (2) Here we interpret log L( 1 2 , E d ) to be negative infinity if L( 1 2 , E d ) = 0, and the conjecture implies in particular that L( 12 , E d ) = 0 for almost all d ∈ E. Towards this conjecture, we established in [7] that N (X; α, ∞) is bounded above by the right hand side of the conjectured relation (2).Complementing this, we now establish a conditional lower bound for N (X; α, β).
Theorem 1. Assume the Generalized Riemann Hypothesis for the family of twisted Lfunctions L(s, E × χ) for all Dirichlet characters χ.Then for fixed intervals (α, β) and as X → ∞ we have Above we have assumed GRH for all character twists of L(s, E); this is largely for convenience, and would allow us to restrict d in progressions.With more effort one could relax the assumption to GRH for the family of quadratic twists L(s, E d ).Note that the factor 1   4   in our theorem matches the proportion of quadratic twists with non-zero L-value obtained in Heath-Brown's work [3].
While we have described results for the family of quadratic twists of an elliptic curve, the method is very general and applies to many situations where 1-level densities of low lying zeros in families have been analyzed and yield a positive proportion of non-vanishing for the central values.The work of Iwaniec, Luo, and Sarnak [5] gives many such examples, and the technique described here refines their non-vanishing corollaries, showing that the non-zero L-values that are produced have the typical size conjectured by Keating and Snaith.For instance, consider the family of symmetric square L-functions L(s, sym 2 f ) where f ranges over Hecke eigenforms of weight k for the full modular group (denote the set of such eigenforms by H k ), with k ≤ K (thus there are about K 2 /48 such L-values).Assuming GRH in this family, Iwaniec, Luo, and Sarnak (see Corollary 1.8 of [5]), showed that at least a proportion 8 9 of these L-values are non-zero.We may refine this to say that for any fixed interval (α, β) and as We end the introduction by mentioning the recent work of Bui, Evans, Lester, and Pratt [1] who establish "weighted" (where the weight is a mollified central value) analogues of the Keating-Snaith conjecture.This amounts to a form of conditioning on non-zero value since central values that are zero are assigned a weight equal to zero.The use of such a weighted measure allows [1] to establish a full asymptotic, however as a side effect they have little control over the nature of the weight.Acknowledgments.We are grateful to Emmanuel Kowalski for a careful reading of the paper, and helpful comments.The first author was partially supported by DMS-1902063.The second author is partially supported by an NSF grant, and a Simons Investigator award from the Simons Foundation.The paper was completed while KS was a Senior Fellow at the Institute for Theoretical Studies, ETH Zürich, whom he thanks for their excellent working conditions, and warm hospitality.

Notation and statements of the key propositions
We begin by introducing some notation, as in our paper [7], and then describing three key propositions which underlie the proof of the main theorem.Let N 0 denote the lcm of 8 and N. Let κ be ±1, and let a mod N 0 denote a residue class with a ≡ 1 or 5 mod 8.We assume that κ and a are such that for any fundamental discriminant d with sign κ and with We write below where we may write a(p) = α p + α p for a complex number α p of magnitude 1 (unique up to complex conjugation), and then For fundamental discriminants d ∈ E with |d| ≤ 3X, and a parameter 3 ≤ x define (3) Let h denote a smooth function with compactly supported Fourier transform and such that |h(x)| ≪ (1 + x 2 ) −1 for all x ∈ R. For concreteness, one could simply consider h to be the Fejer kernel given by ( 4) Lastly, let Φ denote a smooth, non-negative function compactly supported in [ 1 2 , 5  2 ] with Φ(x) = 1 for x ∈ [1, 2], and we put Φ(s) = ∞ 0 Φ(x)x s dx.Below all implied constants will be allowed to depend on N, h, and Φ, which are considered fixed.
Our first proposition connects log L( To analyze sums over the zeros we shall use the following proposition, whose proof is based on the explicit formula.The ideas behind this proposition are also familiar, and in this setting (and in the case ℓ = 1 below) may be traced back to the work of Heath-Brown [3].
Proposition 2. Let h be a smooth function with h(x) ≪ (1 + x 2 ) −1 and whose Fourier transform is compactly supported in [−1, 1].Let L ≥ 1 be a real number, and ℓ be a positive integer coprime to N 0 , and assume that e L ℓ 2 ≤ X 2 .If ℓ is neither a square, nor a prime times a square, then If ℓ is a square then Finally if ℓ is q times a square, for a prime number q, then (7) Finally, to understand the distribution of P(d; x) both when d is chosen uniformly over discriminants d ∈ E, and when d ∈ E is weighted by contributions from low-lying zeros, we shall use the method of moments, drawing upon the following proposition.Proposition 3. Let k be any fixed non-negative integer.Let X be large, and put x = X 1/ log log log X .Then (8) d∈E(κ,a) where M k denotes the k-th Gaussian moment: Further, for any parameter L ≥ 1 with e L ≤ X 2 we have,

Deducing the Theorem from the main propositions
We keep the notations introduced in Section 2. Let X be large, and put x = X 1/ log log log X .
and such that there are no zeros Proof.Take Φ to be a smooth approximation to the indicator function of the interval [1,2], and let κ and a mod N 0 be as in Section 2. The first part of Proposition 3 (namely ( 8)) together with the method of moments shows that (10) d∈E(κ,a) Next, take h to be the Fejer kernel given in (4), and L = (2 − δ/2) log X.Then the second part of Proposition 3 together with the method of moments shows that d∈E(κ,a) P(d;x)/ √ log log X∈(α,β) Note that the weights γ d h(γ d L/(2π)) are always non-negative, and if L(s, E d ) has a zero with |γ d | ≤ (log X log log X) −1 then the weight is ≥ 2 + o(1) (since there would be a complex conjugate pair of such zeros, or a double zero at 1 2 ).Combining this with (10), and summing over all the possibilities for κ and a, we obtain the lemma.

Lemma 2. The number of discriminants
Proof.Applying Proposition 2 with ℓ = 1, h given as in (4), and 1 ≤ L ≤ (2 − δ) log X, we obtain (after summing over the possibilities for κ and a) Integrate both sides of this estimate over L in the range log x ≤ L ≤ 2 log x.Since, for any y > 0 and t = 0, 1 y 2y y sin(πtu) πtu 2 , and therefore we may conclude that The lemma follows at once.With these results in place, it is now a simple matter to deduce the main theorem.By Proposition 1 1 we know that for Lemma 1 tells us that for d ∈ G X (α, β) we may arrange for P(d; x)/ √ log log X to lie in the interval (α, β) and for there to be no zeros with |γ d | ≤ (log X log log X) −1 .Lemma 2 now allows us to discard ≪ X/ log log log X elements of G X (α, β) so as to ensure that the contribution of zeros with |γ d | ≥ (log X log log X) −1 is O((log log log X) 3 ).Thus there are which completes the proof.

Proof of Proposition 1
A straight-forward adaptation of Lemma 1 from [14] (itself based on an identity of Selberg) shows that for any σ ≥ 1  2 with L(σ, E d ) = 0, and any x ≥ 3 one has (11) Here ρ d runs over the non-trivial zeros of L(s, E d ), and this identity in fact holds unconditionally.Now assume GRH for L(s, E d ) and write We may restrict attention to the real part of the integral above since all the other terms involved are real, or noting that the zeros ρ d appear in conjugate pairs.Consider first the sum over n in (12).The contribution from prime powers n = p k with k ≥ 3 is plainly O(1).The contribution of the terms n = p is P(d; x) + O(1), where the error term O(1) arises from the primes dividing N 0 .Finally, by Rankin-Selberg theory (see for instance [4]) it follows that (13) Thus the contribution of the sum over n in ( 12) is ( 14) 1).Next we turn to the sum over zeros in (12).If x and larger values of σ.The first range contributes Thus in all cases the sum over zeros in ( 12) is Finally, taking logarithmic derivatives in the functional equation we find that The proposition follows upon combining this with ( 12), (14), and (15).

Proof of Proposition 2
The proof of Proposition 2 is based on the explicit formula, which we first recall in our context.Lemma 3. Let h be a function with h(x) ≪ (1 + x 2 ) −1 and with compactly supported Fourier transform h where the sum is over all ordinates of non-trivial zeros 1/2 + iγ d of L(s, E d ).
Applying the explicit formula to the dilated function h L (x) = h(xL) whose Fourier transform is 1  L h(x/L), we obtain We multiply this expression by χ d (ℓ) and sum over d with suitable weights.Thus we find (17) where The term S 1 is relatively easy to handle.If ℓ is a square, it amounts to counting square-free integers d lying in a suitable progression mod N 0 and coprime to ℓ.While if ℓ is not a square, the resulting sum is a non-trivial character sum, which exhibits substantial cancellation.A more general term of this type is handled in Proposition 1 of [7], which we refer to for a detailed proof.Thus when ℓ is not a square we find while if ℓ is a square We now turn to the more difficult term S 2 .First we dispose of terms n (which we may suppose is a prime power) that have a common factor with N 0 .Note that since d is fixed in a residue class mod N 0 , if n is the power of a prime dividing N 0 then χ d (n) is determined by the congruence condition on d.Thus the contribution of these terms is where δ(ℓ = ) denotes 1 when ℓ is a square, and 0 otherwise.Henceforth we restrict attention to the terms in S 2 where (n, N 0 ) = 1.Note that if d ≡ a mod N 0 then d is automatically 1 mod 4, and the condition that d is a fundamental discriminant amounts to d being square-free.We express the square-free condition by Möbius inversion α 2 |d µ(α), and then split the sum into the cases where α > A is large, and when α ≤ A is small, for a suitable parameter A ≤ X.We first handle the case when α > A is large.These terms give α>A µ(α) upon using GRH to estimate the sum over n and then estimating the sum over d trivially.
We are left with the terms with α ≤ A, and writing d = kα 2 we may express these terms as We now apply the Poisson summation formula to the sum over k above, as in Lemma 7 of [7].This transforms the sum over k above to where τ v (nℓ) is a Gauss sum given by The Gauss sum τ v (nℓ) can be described explicitly, see Lemma 6 of [7] which gives an evaluation of from which τ v (nℓ) may be obtained via The term v = 0 in (25) leads to a main term; we postpone its treatment, and first consider the contribution of terms v = 0. Since h is supported in [−1, 1], we may suppose that n ≤ e L .The rapid decay of the Fourier transform Φ(ξ) allows us to restrict attention to the range |v| ≤ ℓe L A 2 X −1+ǫ , with the total contribution to S 2 of terms with larger |v| being estimated by O(1).For the smaller values of v, we interchange the sums over v, performing first the sum over n using GRH.Thus these terms contribute We now claim that (on GRH) the sum over n above is X|v| X ǫ , so that the contribution of the terms with v = 0 is To minimize the combined contributions of the error terms in (28) and (23), we shall choose A = (X/ℓ) 4 , so that the effect of both these error terms is To justify the claim (27) we first use (26) to replace τ v (nℓ) by G v (nℓ) so that we must bound (for both choices of ±) First consider the generic case when n is a prime power with (n, v) = 1.Here (using Lemma 6 of [7]) The rapid decay of Φ(ξ) implies that we may restrict attention above to the range p > X 1−ǫ |v|/(ℓα 2 N 0 ).Then splitting p into progressions modN 0 and using GRH (it is here that we need GRH for twists of L(s, E) by quadratic characters, as well as all Dirichlet characters modulo N 0 ) we obtain the bound which is in keeping with (27).Now consider the non-generic case when n is the power of some prime dividing v.We may assume that n|v 2 (else G v (nℓ) = 0 by Lemma 6 of [7]) and also that n ≥ X 1−ǫ |v|/(ℓα 2 N 0 ) else the Fourier transform Φ is negligible.Using that |G v (nℓ)| ≤ (v, nℓ) 2 (which again follows from Lemma 6 of [7]) we may bound the contribution of these terms by since log v ≪ log X ≪ X ǫ and α ≤ A ≤ √ X.Thus these terms also satisfy the claimed bound (27).Now we handle the main term contribution from v = 0, noting that τ 0 (nℓ) = 0 unless nℓ is a square, in which case it equals φ(nℓ).Thus the main term contribution from v = 0 is Thus this main term only exists if ℓ is a square (so that n is a square), or if ℓ is q times a square for a unique prime q (so that n is an odd power of q).In the case ℓ is a square, writing n = m 2 and performing the sum over α, we obtain that the main term is Using (13) and partial summation we conclude that the main term when ℓ is a square is (30) p) 2 − 2) log p p = − log y + O(1), so that, by partial summation, the contribution of the terms n = p 2 equals p≤ √ x p∤N 0 [14] E d ) with the sum over primes P(d; x) (for suitable x) with an error term given in terms of the zeros of L(s, E d ).Such formulae have a long history, going back to Selberg, and the work here complements an upper bound version that played a key role in[14].Proposition1.Let d be a fundamental discriminant in E, and let 3 ≤ x ≤ |d|.Assume GRH for L(s, E d ), and suppose that L( 1 2 , E d ) is not zero.Let γ d run over the ordinates of the non-trivial zeros of L(s, E d