Low-lying zeros of elliptic curve L-functions: Beyond the ratios conjecture
LLOW-LYING ZEROS OF ELLIPTIC CURVE L -FUNCTIONS:BEYOND THE RATIOS CONJECTURE DANIEL FIORILLI, JAMES PARKS AND ANDERS SÖDERGREN
Abstract.
We study the low-lying zeros of L -functions attached to quadratic twists of a givenelliptic curve E defined over Q . We are primarily interested in the family of all twists coprime tothe conductor of E and compute a very precise expression for the corresponding -level density.In particular, for test functions whose Fourier transforms have sufficiently restricted support, weare able to compute the -level density up to an error term that is significantly sharper than thesquare-root error term predicted by the L -functions Ratios Conjecture. Introduction
The connection between zeros of L -functions and eigenvalues of random matrices first appearedin Montgomery’s seminal paper on the pair correlation of zeros of the Riemann zeta function [Mo],where he proved that for suitably restricted test functions the pair correlation of the zeros of ζ ( s ) equals the pair correlation of the eigenvalues of random matrices from the Gaussian UnitaryEnsemble (GUE). This work was later complemented by extensive numerical calculations of thezeros of ζ ( s ) by Odlyzko [O1, O2] that gave outstanding evidence for the agreement between localstatistics of these zeros and the corresponding GUE statistics. It has also been shown that the n -level correlations of the zeros of ζ ( s ) agree with the corresponding GUE statistics, again for suitablyrestricted test functions [H, RS1].In the work of Rudnick and Sarnak [RS2], it is shown that the n -level correlations of zeros ofprimitive automorphic L -functions all agree with the GUE statistics. However, it was predicted byKatz and Sarnak [KaS2, KaS3] that by looking at low-lying zeros of families of L -functions, oneshould expect different statistics, which correspond to statistics of eigenvalues coming from scalinglimits of certain compact Lie groups, specifically one of U ( N ) , O ( N ) , SO (2 N + 1) , SO (2 N ) and Sp (2 N ) .Our purpose in the present paper is to study the low-lying zeros of the L -functions attached tothe family of quadratic twists of a given elliptic curve E over Q . We assume that E is given inglobal minimal Weierstrass form as E : y = x + ax + b, (1.1)where a, b ∈ Z . The discriminant of E equals ∆ E := − a + 27 b ) and is necessarily non-zero. We denote the conductor of E by N E and recall that for p > we have p | N E if and only if p | ∆ E . In general the conductor of an elliptic curve is a rather subtle object.However, for our purposes it will be enough to note that for all elliptic curves over Q the conductoris at least ; in particular we have ( N E , > (see, e.g., [C]). Date : September 18, 2018.The first author was supported by an NSERC Postdoctoral Fellowship. The second author was supported bya PIMS Postdoctoral Fellowship. The third author was supported by a Postdoctoral Fellowship from the SwedishResearch Council, by the National Science Foundation under agreement No. DMS-1128155, as well as by a grant fromthe Danish Council for Independent Research and FP7 Marie Curie Actions-COFUND (grant id: DFF-1325-00058). a r X i v : . [ m a t h . N T ] M a y DANIEL FIORILLI, JAMES PARKS AND ANDERS SÖDERGREN
We now recall the definition of the L -function of E . The trace of the Frobenius endomorphism isgiven, for p (cid:45) N E , by a p ( E ) = p + 1 − E p ( F p ) , where E p ( F p ) is the number of projective pointson the reduction of E modulo p . Extending the definition of a p ( E ) to the set of primes p | N E bysetting a p ( E ) := if E has split multiplicative reduction at p, − if E has non-split multiplicative reduction at p, if E has additive reduction at p, the L -function of E is defined as the Euler product L ( s, E ) := (cid:89) p | N E (cid:18) − a p ( E ) p s +1 / (cid:19) − (cid:89) p (cid:45) N E (cid:18) − a p ( E ) p s +1 / + 1 p s (cid:19) − = (cid:89) p (cid:18) − α E ( p ) p s (cid:19) − (cid:18) − β E ( p ) p s (cid:19) − , (cid:60) ( s ) > . (1.2)Here, for all p (cid:45) N E , α E ( p ) and β E ( p ) are complex numbers satisfying β E ( p ) = α E ( p ) , | α E ( p ) | = | β E ( p ) | = 1 and α E ( p ) + β E ( p ) = a p ( E ) / √ p . Moreover, in the remaining cases, that is when p | N E , α E ( p ) and β E ( p ) satisfy α E ( p ) = a p ( E ) / √ p and β E ( p ) = 0 . Thus L ( s, E ) satisfies the Ramanujan-Petersson conjecture; in particular we have | α E ( p ) | , | β E ( p ) | ≤ for all primes p . Note that with theabove normalization the critical strip of L ( s, E ) is < (cid:60) ( s ) < . Expanding the product (1.2), wedefine the sequence { λ E ( n ) } ∞ n =1 as the coefficients in the resulting Dirichlet series: L ( s, E ) = ∞ (cid:88) n =1 λ E ( n ) n s , (cid:60) ( s ) > . By the impressive work of Wiles [W], Taylor and Wiles [TW], and Breuil, Conrad, Diamond, andTaylor [BCDT], we know that there exists a cuspidal newform f E of weight and level N E such that L ( s, E ) = L ( s, f E ) , that is, L ( s, E ) is a modular L -function. In particular, it follows that L ( s, E ) has an analytic continuation to the complex plane and that L ( s, E ) satisfies the functional equation Λ( s, E ) := (cid:18) √ N E π (cid:19) s + Γ( s + ) L ( s, E ) = (cid:15) E Λ(1 − s, E ) , (1.3)where (cid:15) E = ± is the root number of E .We are interested in the quadratic twists E d : dy = x + ax + b, (1.4)of the fixed elliptic curve E . It is clear that we can, by a change of variables, assume that d issquare-free. We furthermore restrict our attention to twists by integers d satisfying ( d, N E ) = 1 and note that for such d the conductor of E d equals N E d = N E d . We let L ( s, E d ) denote the L -function of E d defined by an Euler product as in (1.2). As above, L ( s, E d ) is entire and satisfiesthe functional equation Λ( s, E d ) := (cid:18) √ N E | d | π (cid:19) s + Γ( s + ) L ( s, E d ) = (cid:15) E d Λ(1 − s, E d ) , (1.5)where the root number of E d satisfies (cid:15) E d = (cid:15) E d | d | χ d ( N E ) (cf. [IK, Prop. 14.20]; see also [IK, p.538]). We let χ d ( n ) denote the quadratic character defined in terms of the Kronecker symbol by χ d ( n ) := (cid:18) dn (cid:19) . We will use the symbols χ d ( · ) and ( d · ) interchangeably throughout the paper. We stress that, aslong as we keep d square-free, χ d is a real primitive character (see [D, Sect. 5]). OW-LYING ZEROS OF ELLIPTIC CURVE L -FUNCTIONS 3 Remark 1.1.
It is useful to note that L ( s, E d ) equals the Rankin-Selberg L -function L ( s, f E ⊗ χ d ) := ∞ (cid:88) n =1 λ E ( n ) χ d ( n ) n s , (cid:60) ( s ) > . Remark 1.2.
In what follows (cf., e.g., (1.7)) we will consider quadratic twists E d and their L -functions L ( s, E d ) also for non-square-free d . Writing d (cid:48) for the square-free part of d , note that L ( s, E d ) and L ( s, E d (cid:48) ) have the same nontrivial zeros. This observation will always allow us toreduce the study of quadratic twists by general d to quadratic twists by square-free d (cid:48) .The L -functions coming from elliptic curves are in many ways analogous to the Riemann zetafunction. In particular they are expected to satisfy the following Riemann hypothesis. Hypothesis 1.3 (Elliptic Curve Riemann Hypothesis, ECRH) . For any elliptic curve E over Q ,the nontrivial zeros of L ( s, E ) have real part equal to . Throughout this paper, φ will denote an even Schwartz test function satisfying φ ( R ) ⊂ R . TheFourier transform of φ is defined as (cid:98) φ ( ξ ) := (cid:90) R φ ( t ) e − πiξt dt. The quantity we are interested in is the weighted -level density of low-lying zeros of the familyof L -functions attached to the quadratic twists E d of the fixed elliptic curve E . Given a (large)positive number X and a test function φ , we introduce the -level density for the single L -function L ( s, E d ) as the sum D X ( E d ; φ ) := (cid:88) γ d φ (cid:18) γ d L π (cid:19) , with γ d := − i ( ρ d − ) , where ρ d runs over the nontrivial zeros of L ( s, E d ) . Moreover, L is definedby L := log (cid:18) N E X (2 πe ) (cid:19) . (1.6)Note that L is chosen so that for d ≈ X , the sequence γ d L π of normalized low-lying zeros arisingfrom (say) γ d ≤ has essentially constant mean spacing (recall that by a Riemann-von Mangoldtype theorem as in for example [IK, Thm. 5.8] L ( s, E d ) has approximately log( N E d / (2 πe ) ) / π zeros in the region < (cid:61) ( ρ d ) ≤ ). We further remark that ECRH asserts that γ d ∈ R . The quantity D X ( E d ; φ ) is very hard to understand for individual elliptic curves. However, whenwe consider averages over families of quadratic twists the situation becomes much more tractable.To allow a maximally detailed analysis of the the -level density, we first consider the family { E d : ( d, N E ) = 1 } which clearly contains an abundance of repetitions. Hence we introduce thefollowing weighted -level density D ( φ ; X ) := 1 W ( X ) (cid:88) ( d,N E )=1 w (cid:18) dX (cid:19) D X ( E d ; φ ) , (1.7) It is not essential for our results to assume ECRH at this point. However, ECRH is of course crucial for enablinga spectral interpretation of our results. This family contains elliptic curves with both signs in the functional equation. We choose not to separate thefamily into even and odd signs in order to keep the statement of our main results as concise as possible. A similaranalysis can give the corresponding results also for the even and odd families. Allowing repetitions in the family in order to make the analysis manageable is not a new strategy; cf., e.g.,[Y2, FM].
DANIEL FIORILLI, JAMES PARKS AND ANDERS SÖDERGREN where w ( t ) is an even nonnegative Schwartz test function having positive total mass (which morallyrestricts the sum to d (cid:28) X ) and W ( X ) := (cid:88) ( d,N E )=1 w (cid:18) dX (cid:19) . (1.8)Note that any given zero occurring in (1.7) will be repeated infinitely many times. However, since φ and w are rapidly decaying, most zeros will not be given a large total weight in the outer sum.Indeed, a zero that is given a large total weight necessarily originates from a curve with smallconductor, but such zeros are on average quite far from the real line and thus cannot give a largecontribution to the -level density.Quantities analogous to D ( φ ; X ) have been studied by Goldfeld [G], Brumer [B], Heath-Brown[H-B] and Young [Y2], in order to obtain conditional bounds on the average rank of elliptic curvesin certain families. One can reinterpret their results as asymptotic estimates for D ( φ ; X ) when thesupport of (cid:98) φ is appropriately restricted; the larger the allowable support is, the better the resultingupper bound on the average rank becomes.To predict an asymptotic for D ( φ ; X ) , Katz and Sarnak [KaS2, KaS3] associate a given (natural)family F of L -functions defined over a number field with a corresponding family of L -functionsdefined over a suitable function field. By an analysis of the function field family, they predict thatthe low-lying zeros of the L -functions in F behave like the eigenvalues near in a related compactLie group G ( F ) of either unitary, orthogonal or symplectic matrices. In our case the symmetrygroup G ( F ) is O ( N ) , and the Katz-Sarnak prediction takes the form lim X →∞ D ( φ ; X ) = (cid:98) φ (0) + φ (0)2 = (cid:90) R φ ( t ) W O ( t ) dt, (1.9)where W O := 1 + δ . The analogous prediction on a closely related family has been checked byKatz and Sarnak [KaS1], for a restricted class of test functions.The Katz-Sarnak prediction on D ( φ ; X ) is given in terms of statistics of random matrices, andone can ask whether random matrix theory can predict other features of zeros of L -functions, suchas possible lower order terms in (1.9). It turns out that for test functions φ whose Fourier transformhas restricted support, Young [Y1] has shown that, in certain families of elliptic curves, lowerorder terms of order (log X ) − do exist in the -level density. Moreover, these terms cannot beexplained using random matrix theory. Such lower order terms have also been found in familiesof quadratic twists of a fixed elliptic curve [HMM]. The limitations of random matrix theory formaking predictions on statistics of L -functions have also been observed in other contexts, mostnotably in predictions for moments [KeS1, KeS2].An extremely powerful conjecture was put forward by Conrey, Farmer and Zirnbauer [CFZ],which predicts estimates for averages of quotients of (products of) L -functions evaluated at certainvalues. A variant of this conjecture implies a formula for D ( φ ; X ) which contains the Katz-Sarnakprediction, lower order terms and an error term of size at most O ε ( X − / ε ) (see [HKS]). Othervariants of the conjecture imply very precise estimates for many other L -function statistics [CS].The Ratios Conjecture’s prediction in our family contains the following modified weight function,on which we will expand in Section 2. Given the even nonnegative Schwartz weight w ( t ) , we define (cid:101) w ( x ) = (cid:101) w E ( x ) := (cid:88) n ≥ n,N E )=1 w ( n x ) . (1.10)Moreover, throughout this paper the symbol (cid:80) ∗ d indicates that the sum is restricted to square-freevalues of d . We now state a precise consequence of the Ratios Conjecture (Conjecture A.5). Theproof of Theorem 1.4 will be given in Appendix A. OW-LYING ZEROS OF ELLIPTIC CURVE L -FUNCTIONS 5 Theorem 1.4.
Fix ε > . Let E be an elliptic curve defined over Q with conductor N E . Let w bea nonnegative Schwartz function on R which is not identically zero and let φ be an even Schwartzfunction on R whose Fourier transform has compact support. Assuming GRH and Conjecture A.5(the Ratios Conjecture for our family), the -level density for the zeros of the family of L -functionsattached to the quadratic twists of E coprime to N E is given by D ( φ ; X ) = (cid:98) φ (0) LW ( X ) (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) log (cid:18) N E d (2 π ) (cid:19) + 12 π (cid:90) R φ (cid:18) tL π (cid:19) (cid:20) Γ (cid:48) (1 + it )Γ(1 + it )+ Γ (cid:48) (1 − it )Γ(1 − it ) + 2 (cid:18) − ζ (cid:48) (1 + 2 it ) ζ (1 + 2 it ) + L (cid:48) (cid:0) it, Sym E (cid:1) L (cid:0) it, Sym E (cid:1) + A α,E ( it, it ) (cid:19) − it (cid:21) dt (1.11) + φ (0)2 + O ε (cid:0) X − / ε (cid:1) , where ∗ indicates that we are summing over square-free d , the functions L , W and (cid:101) w are defined by (1.6) , (1.8) and (1.10) respectively, L (cid:0) s, Sym E (cid:1) is the symmetric square L -function of E (cf. [S] ),and the function A α,E is defined by (A.14) (see also (A.7) , (A.8) , (A.12) and (A.13) ). The impliedconstant in the error term depends on E , φ and w . Remark 1.5.
One can, for any K ∈ N , rewrite (1.11) in the form D ( φ ; X ) = (cid:98) φ (0) + φ (0)2 + K − (cid:88) j =1 c j ( φ, w, E )(log X ) j + O φ,w,E,K (cid:18) X ) K (cid:19) , (1.12)for some constants c j ( φ, w, E ) . Indeed, Lemma 2.8 implies that the first term on the right-handside of (1.11) is of the desired form. As for the second, making the change of variables u = tL/ π ,truncating the resulting integral at the points u = ±√ L and expanding the expression in squarebrackets into Taylor series around zero gives the desired expansion.One of our main objectives is to give an estimate for D ( φ ; X ) with an error term of size atmost O ε ( X − / ε ) , for test functions whose Fourier transforms have small support. Our first maintheorem shows that we can obtain such an estimate with an error term that is significantly sharperthan the error term appearing in the Ratios Conjecture’s prediction (cf. Theorem 1.4). Theorem 1.6.
Fix ε > . Let E be an elliptic curve defined over Q with conductor N E . Let w bea nonnegative Schwartz function on R which is not identically zero and let φ be an even Schwartzfunction on R whose Fourier transform satisfies σ = sup ( supp (cid:98) φ ) < . Then the -level density forthe zeros of the family of L -functions attached to the quadratic twists of E coprime to N E is givenby D ( φ ; X ) = (cid:98) φ (0) LW ( X ) (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) log (cid:18) N E d (2 π ) (cid:19) − L (cid:90) ∞ (cid:32) (cid:98) φ ( x/L ) e − x − e − x − (cid:98) φ (0) e − x x (cid:33) dx − L ∞ (cid:88) (cid:96) =1 (cid:88) p (cid:0) α E ( p ) (cid:96) + β E ( p ) (cid:96) (cid:1) log pp (cid:96) (cid:98) φ (cid:18) (cid:96) log pL (cid:19) (cid:18) ψ N E ( p ) p (cid:19) − + O ε (cid:0) X η ( σ )+ ε (cid:1) , (1.13) In this paper, GRH denotes the Riemann Hypothesis for ζ ( s ) , L ( s, E ) and L ( s, Sym E ) for every elliptic curve E over Q . Note that the first two terms on the right-hand side of (1.12) coincide with the Katz-Sarnak prediction.
DANIEL FIORILLI, JAMES PARKS AND ANDERS SÖDERGREN where ∗ indicates that we are summing over square-free d , ψ N E is the principal Dirichlet charactermodulo N E , the functions L , W and (cid:101) w are defined by (1.6) , (1.8) and (1.10) respectively, and η ( σ ) = (cid:40) − σ if m +2 ≤ σ < m +1 , − m − m +1 if m +1 ≤ σ < m − , for each m ≥ . The implied constant in the error term depends on E , φ and w . The techniques used to prove Theorem 1.6 are inspired by the work of Katz and Sarnak [KaS1].The main tools we use, which were pioneered by Iwaniec [I] in this context, are Poisson summationand the Pólya-Vinogradov inequality. The key to obtaining an error term sharper than the RatiosConjecture’s prediction is to allow repetitions in our family, and to use the smooth cutoff function w . Remark 1.7.
Theorem A.2 shows that the sum of the second and third terms appearing on theright-hand side of (1.13) matches the sum of the second and third terms appearing on the right-handside of (1.11), up to an error which is at most O ε ( X − ε ) . Therefore, Theorem 1.6 agrees with theRatios Conjecture’s prediction, but is even more precise. This result should be compared with themain theorem of [FM], in which the authors obtain an estimate for the -level density in the familyof all Dirichlet L -functions, which is more precise than the Ratios Conjecture’s prediction. Remark 1.8.
It is possible to improve the estimate in Theorem 1.6 in certain ranges of σ by usingBurgess’s bound. We have chosen to carry out this improvement in a separate paper [FPS].In the next theorem we show that ECRH implies a formula for D ( φ ; X ) with a sharper errorterm, which in particular doubles the allowable support for (cid:98) φ . Theorem 1.9.
Fix ε > . Let E be an elliptic curve defined over Q with conductor N E . Let w bea nonnegative Schwartz function on R which is not identically zero and let φ be an even Schwartzfunction on R whose Fourier transform satisfies σ = sup ( supp (cid:98) φ ) < . Then, assuming ECRH, the -level density for the zeros of the family of L -functions attached to the quadratic twists of E coprimeto N E is given by D ( φ ; X ) = (cid:98) φ (0) LW ( X ) (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) log (cid:18) N E d (2 π ) (cid:19) − L (cid:90) ∞ (cid:32) (cid:98) φ ( x/L ) e − x − e − x − (cid:98) φ (0) e − x x (cid:33) dx − L ∞ (cid:88) (cid:96) =1 (cid:88) p (cid:0) α E ( p ) (cid:96) + β E ( p ) (cid:96) (cid:1) log pp (cid:96) (cid:98) φ (cid:18) (cid:96) log pL (cid:19) (cid:18) ψ N E ( p ) p (cid:19) − + O ε (cid:0) X θ ( σ )+ ε (cid:1) , (1.14) where ∗ indicates that we are summing over square-free d , ψ N E is the principal Dirichlet charactermodulo N E , the functions L , W and (cid:101) w are defined by (1.6) , (1.8) and (1.10) respectively, and θ ( σ ) = − σ if m +2 ≤ σ < m , − m if m ≤ σ < m − , − σ if ≤ σ < . The implied constant in the error term depends on E , φ and w . In Figure 1, we compare the exponent of X in the error terms of Theorems 1.6 and 1.9 by plotting η ( σ ) and θ ( σ ) as functions of σ = sup ( supp (cid:98) φ ) .Finally, we study the weighted -level density averaged over square-free values of d : D ∗ ( φ ; X ) := 1 W ∗ ( X ) (cid:88) ∗ ( d,N E )=1 w (cid:18) dX (cid:19) D X ( E d ; φ ) , (1.15) OW-LYING ZEROS OF ELLIPTIC CURVE L -FUNCTIONS 7 where W ∗ ( X ) := (cid:88) ∗ ( d,N E )=1 w (cid:18) dX (cid:19) . (1.16)This quantity is more natural to study than D ( φ ; X ) , since there are no repetitions. However,the estimate we obtain for the error term is weaker, and we are not able to surpass the RatiosConjecture’s prediction in this case. Theorem 1.10.
Fix ε > . Let E be an elliptic curve defined over Q with conductor N E . Let w bea nonnegative Schwartz function on R which is not identically zero and let φ be an even Schwartzfunction on R whose Fourier transform satisfies σ = sup ( supp (cid:98) φ ) < . Then, assuming the RiemannHypothesis (RH) and ECRH, the -level density for the zeros of the family of L -functions attachedto the square-free quadratic twists of E coprime to N E is given by D ∗ ( φ ; X ) = (cid:98) φ (0) LW ∗ ( X ) (cid:88) ∗ ( d,N E )=1 w (cid:18) dX (cid:19) log (cid:18) N E d (2 π ) (cid:19) − L (cid:90) ∞ (cid:32) (cid:98) φ ( x/L ) e − x − e − x − (cid:98) φ (0) e − x x (cid:33) dx − L ∞ (cid:88) (cid:96) =1 (cid:88) p (cid:0) α E ( p ) (cid:96) + β E ( p ) (cid:96) (cid:1) log pp (cid:96) (cid:98) φ (cid:18) (cid:96) log pL (cid:19) (cid:18) ψ N E ( p ) p (cid:19) − + O ε (cid:0) X σ − + ε (cid:1) , where ∗ indicates that we are summing over square-free d and the functions L and W ∗ are definedby (1.6) and (1.16) respectively. The implied constant in the error term depends on E , φ and w . Remark 1.11.
One can remove the assumption of RH in Theorem 1.10 by proving an unconditionalversion of Lemma 2.10 with a weaker error term. Note that this modification does not affect theoverall result.
Remark 1.12.
Using similar techniques to those used to obtain Theorem 1.10 but splitting thefamily according to the sign of the functional equation, one can improve both the allowable supportof (cid:98) φ and the quality of the error term in the main theorem of [HMM].1.1. Acknowledgments.
The present work was initiated during the American Mathematical Soci-ety’s MRC program
Arithmetic Statistics in Snowbird, Utah, under the supervision of Nina Snaith.We are grateful to the AMS and the organizers of the program for financial support and encourage-ment. In particular we are grateful to Brian Conrey and Nina Snaith for inspiring discussions andhelpful remarks. We also thank David Zywina for enlightening discussions.2.
Preliminaries
We begin this section by showing that the Mellin Transform M w ( s ) has very nice analyticalproperties, which will be useful later. Recall that w : R −→ R is a fixed nonnegative even Schwartzfunction which is not identically zero. Lemma 2.1.
The Mellin transform M w ( s ) := (cid:90) ∞ + x s − w ( x ) dx Although in [HMM, Theorem 1.1] the authors claim an error term of X − − σ (log X ) , their proof only producesthe weaker error term X − + σ (log X ) , which yields a nontrivial result for σ < . In [HMM, (2.35)], the restrictionon the sum over primes should read p (cid:96) +1 ≤ X σ , since in [HMM, (2.31)], (cid:98) g (log p k +1 / L ) is zero outside this range.Accounting for this, [HMM, (2.35)] results in the error term X − + σ (log X ) . Note also that the main term in [HMM,Theorem 1.1] is not correct as stated. Indeed, the third term on the right-hand side of [HMM, (1.5)] is the integralof a function which has a simple pole on the contour of integration. DANIEL FIORILLI, JAMES PARKS AND ANDERS SÖDERGREN
Figure 1.
A comparison of our unconditional results (light blue), our results underECRH (dark blue), our results for D ∗ ( φ ; X ) (dark green), the results of [HMM] (lightgreen) and the Ratios Conjecture’s prediction (dashed line). is a holomorphic function of s = σ + it except for possible simple poles at non-positive integers, andsatisfies the bound M w ( s ) (cid:28) n,w (1 + | t | ) − n (2.1) for any fixed n ∈ N , uniformly for σ in any compact subset of R , provided s is bounded away fromthe set { , − , − , . . . } .Proof. Note first that since w is Schwartz, the integral (cid:82) ∞ + x s − w ( x ) dx converges absolutely anduniformly on any compact subset of { s ∈ C : (cid:60) ( s ) > } . To give an analytic continuation of M w ( s ) ,we integrate by parts: M w ( s ) = x s w ( x ) s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ + − s (cid:90) ∞ + x s w (cid:48) ( x ) dx = − s (cid:90) ∞ + x s w (cid:48) ( x ) dx, which by absolute and uniform convergence of the integral shows that M w ( s ) is a holomorphicfunction for σ > − except possibly for a simple pole at s = 0 . Iterating this process n times weobtain the formula M w ( s ) = ( − n s ( s + 1) · · · ( s + n − (cid:90) ∞ + x s + n − w ( n ) ( x ) dx. OW-LYING ZEROS OF ELLIPTIC CURVE L -FUNCTIONS 9 This shows that (2.1) holds, and that M w ( s ) extends to a holomorphic function on C except forpossible simple poles at s = 0 , − , − , . . . . Here we used that the integral (cid:82) ∞ + x s + n − w ( n ) ( x ) dx converges absolutely and uniformly on compact subsets of (cid:60) ( s ) > − n , due to the fact that w isSchwartz. (cid:3) Remark 2.2.
The proof of Lemma 2.1 shows that M w ( s ) is holomorphic at s = − n when w ( n ) (0) =0 , and has a simple pole at this point otherwise. Lemma 2.3.
Define the even smooth function (cid:101) w : R \ { } → R by (cid:101) w ( x ) = (cid:101) w E ( x ) := (cid:88) n ≥ n,N E )=1 w ( n x ) . (2.2) Then (cid:101) w ( x ) decays rapidly as x → ∞ , and we have that M (cid:101) w ( s ) = (cid:89) p | N E (cid:18) − p s (cid:19) ζ (2 s ) M w ( s ) . Remark 2.4.
Note that (cid:101) w ( x ) blows up near x = 0 . Indeed, it follows from (2.2) and our assump-tions on w ( x ) that (cid:101) w ( x ) (cid:16) x − / as x → . Proof of Lemma 2.3.
For (cid:60) ( s ) large enough, we have (cid:90) ∞ + x s − (cid:101) w ( x ) dx = (cid:88) n ≥ n,N E )=1 (cid:90) ∞ + x s − w ( n x ) dx = (cid:88) n ≥ n,N E )=1 n − s (cid:90) ∞ + u s − w ( u ) du = (cid:89) p | N E (cid:18) − p s (cid:19) ζ (2 s ) M w ( s ) . The result follows by analytic continuation. (cid:3)
Weighted character sums.
The following estimate is central in our analysis of D ( φ ; X ) . Lemma 2.5.
Fix n ∈ N and ε > . We have the estimate (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) (cid:18) dn (cid:19) = κ ( n ) X (cid:98) w (0) (cid:89) p | n (cid:18) p (cid:19) − (cid:89) p | N E (cid:18) − p (cid:19) (cid:89) p | ( n,N E ) (cid:18) p (cid:19) + O ε,w (cid:0) | n | − κ ( n )2 + ε ( XN E ) ε (cid:1) , where κ ( n ) := (cid:40) if n = (cid:3) , otherwise. Remark 2.6.
In particular, taking n = 1 in Lemma 2.5 gives W ( X ) = (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) = X (cid:98) w (0) (cid:89) p | N E (cid:18) − p (cid:19) + O ε,w (cid:0) ( XN E ) ε (cid:1) , an estimate which will be useful in the proof of Theorem 1.6 and in Appendix A. Remark 2.7.
Lemma 2.5 applies equally well when N E is replaced by any nonzero integer (notnecessarily the conductor of an elliptic curve). Proof of Lemma 2.5.
First note that (cid:101) w ( t ) is even, and hence (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) (cid:18) dn (cid:19) = (cid:18) (cid:18) − n (cid:19)(cid:19) (cid:88) ∗ d ≥ d,N E )=1 (cid:101) w (cid:18) dX (cid:19) (cid:18) dn (cid:19) . (2.3)Using Mellin inversion, we write I n ( X ) : = (cid:88) d ≥ d,N E )=1 µ ( d ) (cid:101) w (cid:18) dX (cid:19) (cid:18) dn (cid:19) = 12 πi (cid:90) (cid:60) ( s )=2 (cid:88) d ≥ d,N E )=1 µ ( d ) (cid:0) dn (cid:1) d s X s M (cid:101) w ( s ) ds = 12 πi (cid:90) (cid:60) ( s )=2 (cid:89) p (cid:45) N E (cid:18) (cid:0) pn (cid:1) p s (cid:19) (cid:89) p | N E (cid:18) − p s (cid:19) ζ (2 s ) M w ( s ) X s ds (by Lemma 2.3) = 12 πi (cid:90) (cid:60) ( s )=2 (cid:89) p | N E (cid:18) (cid:0) pn (cid:1) p s (cid:19) − L ( s, (cid:0) · n (cid:1) ) L (2 s, (cid:0) · n (cid:1) ) (cid:89) p | N E (cid:18) − p s (cid:19) ζ (2 s ) M w ( s ) X s ds = 12 πi (cid:90) (cid:60) ( s )=2 L (cid:0) s, (cid:16) · n (cid:17) (cid:1) (cid:89) p | N E (cid:18) (cid:0) pn (cid:1) p s (cid:19) − (cid:18) − p s (cid:19) (cid:89) p | n (cid:18) − p s (cid:19) − M w ( s ) X s ds. We first consider the case when n is not a square. In this case L ( s, (cid:0) · n (cid:1) ) is holomorphic at s = 1 ,and thus we can shift the contour of integration to the left: I n ( X ) = 12 πi (cid:90) (cid:60) ( s )= ε L (cid:0) s, (cid:16) · n (cid:17) (cid:1) (cid:89) p | N E (cid:18) (cid:0) pn (cid:1) p s (cid:19) − (cid:18) − p s (cid:19) (cid:89) p | n (cid:18) − p s (cid:19) − M w ( s ) X s ds. Here < ε < / is fixed (note that the integrand might have poles on the line (cid:60) ( s ) = 0 ). We thenapply the convexity bound [IK, (5.20)], which for non-principal χ reads L ( s, χ ) (cid:28) ε ( q ( | s | + 1)) −(cid:60) ( s )2 + ε (0 ≤ (cid:60) ( s ) ≤ , where q is the conductor of χ . Combining this with Lemma 2.1 yields the bound I n ( X ) (cid:28) ε,w (cid:90) R ( | n | ( | t | + 1)) − ε + ε (cid:89) p | N E (cid:18) − p ε (cid:19) − (cid:18) p ε (cid:19) (cid:89) p | n (cid:18) − p ε (cid:19) − ( | t | + 1) − X ε dt (cid:28) ε | n | + ε N εE X ε . Indeed, we have the bound (cid:89) p | N E (cid:18) − p ε (cid:19) − (cid:28) ε (cid:89) p | N E p> /ε (cid:18) − p ε (cid:19) − ≤ (cid:89) p | N E p> /ε (cid:28) ε N ε/ E , and the two other products over primes are bounded in a similar fashion. The proof of this case iscompleted by combining these estimates with (2.3).As for the case where n is a square, we again shift the contour of integration to the left, pickingup a residue at s = 1 . Note that in this case L (cid:0) s, (cid:16) · n (cid:17) (cid:1) = (cid:89) p | n (cid:18) − p s (cid:19) ζ ( s ) , OW-LYING ZEROS OF ELLIPTIC CURVE L -FUNCTIONS 11 and hence the contribution of the pole at s = 1 is given by (cid:89) p | n (cid:18) p (cid:19) − (cid:89) p | N E (cid:32) (cid:0) pn (cid:1) p (cid:33) − (cid:18) − p (cid:19) M w (1) X. By Lemma 2.1, the shifted integral is (cid:28) ε,w (cid:90) R ( | t | + 1) − ε + ε (cid:89) p | n (cid:18) p ε (cid:19) × (cid:89) p | N E (cid:18) − p ε (cid:19) − (cid:18) p ε (cid:19) (cid:89) p | n (cid:18) − p ε (cid:19) − ( | t | + 1) − X ε dt (cid:28) ε ( | n | XN E ) ε . The proof is finished by combining this estimate with (2.3) and by noting that M w (1) = (cid:98) w (0) / . (cid:3) The next lemma is used to understand the first main term in the -level density (see for example(1.11) or (1.13)). Lemma 2.8.
Fix ε > , and assume the Riemann Hypothesis (RH). We have the estimate W ( X ) (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) log | d | = log X + 2 (cid:98) w (0) (cid:90) ∞ w ( x ) log xdx + (cid:88) p | N E pp − ζ (cid:48) (2) ζ (2) − (cid:89) p | N E (cid:18) − p / (cid:19) (cid:18) − p (cid:19) − M w ( ) M w (1) ζ ( ) X − / + O ε,w (cid:0) N εE X − / ε (cid:1) . Proof.
The proof follows closely that of Lemma 2.5. One writes (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) log | d | = − πi (cid:90) (cid:60) ( s )=2 (cid:18) (cid:88) d ≥ d,N E )=1 µ ( d ) d s (cid:19) (cid:48) (cid:89) p | N E (cid:18) − p s (cid:19) ζ (2 s ) M w ( s ) X s ds, (2.4)and pulls the contour of integration to the left until the line (cid:60) ( s ) = 1 / ε . Note that under RH,the only poles of the function Z ( s ) : = (cid:18) (cid:88) d ≥ d,N E )=1 µ ( d ) d s (cid:19) (cid:48) (cid:89) p | N E (cid:18) − p s (cid:19) ζ (2 s )= (cid:89) p | N E (cid:18) − p s (cid:19) (cid:20) ζ (cid:48) ( s ) − ζ ( s ) ζ (cid:48) (2 s ) ζ (2 s ) + ζ ( s ) (cid:88) p | N E log pp s + 1 (cid:21) in the region / < (cid:60) ( s ) ≤ are at s = 1 and at s = 1 / . The value of the residues of the integrandin (2.4) are obtained from a straightforward computation. (cid:3) Remark 2.9.
One can pull the contour of integration further to the left in (2.4), and obtain anunconditional estimate with an error term of size O ε,E ( X ε ) . This estimate will contain terms of theform ζ ( ρ/ M w ( ρ/ X ρ/ (cid:89) p | N E (cid:18) − p ρ/ (cid:19) , with ρ running over the nontrivial zeros of ζ ( s ) . Remark 2.7 also applies here.
We also prove a version of Lemma 2.5 which will be important in the analysis of D ∗ ( φ ; X ) . Lemma 2.10.
Fix n ∈ N and ε > . Under the Riemann Hypothesis (RH), we have the estimate (cid:88) ∗ ( d,N E )=1 w (cid:18) dX (cid:19) (cid:18) dn (cid:19) = κ ( n ) Xζ (2) (cid:98) w (0) (cid:89) p | n (cid:18) p (cid:19) − (cid:89) p | N E (cid:18) p (cid:19) − (cid:89) p | ( n,N E ) (cid:18) p (cid:19) + O ε,w (cid:0) ( N E ) ε | n | (1 − κ ( n ))+ ε X + ε (cid:1) , where κ ( n ) := (cid:40) if n = (cid:3) , otherwise.Proof. Since w ( t ) is even, we have (cid:88) ∗ ( d,N E )=1 w (cid:18) dX (cid:19) (cid:18) dn (cid:19) = (cid:18) (cid:18) − n (cid:19)(cid:19) (cid:88) ∗ d ≥ d,N E )=1 w (cid:18) dX (cid:19) (cid:18) dn (cid:19) . (2.5)An application of Mellin inversion and a straightforward calculation shows that J n ( X ) := (cid:88) d ≥ d,N E )=1 µ ( d ) w (cid:18) dX (cid:19) (cid:18) dn (cid:19) = 12 πi (cid:90) (cid:60) ( s )=2 Z n,E ( s ) M w ( s ) X s ds, where Z n,E ( s ) := L (cid:0) s, (cid:0) · n (cid:1) (cid:1) ζ (2 s ) (cid:89) p | N E (cid:18) (cid:0) pn (cid:1) p s (cid:19) − (cid:89) p | n (cid:18) − p s (cid:19) − . In the case when n is not a square, L ( s, (cid:0) · n (cid:1) ) is holomorphic at s = 1 . We can thus shift thecontour of integration to the left until the line (cid:60) ( s ) = + ε , since by the Riemann Hypothesis, thezeros of ζ (2 s ) all have real part at most . We apply the estimates [IK, (5.20)], [MV, Thm. 13.23]and Lemma 2.1 together with the rapid decay of M w ( s ) on vertical lines (rather than following theproof of [MV, Thm. 13.24] directly), to obtain, for some C > , J n ( X ) = 12 πi (cid:90) (cid:60) ( s )= + ε |(cid:61) ( s ) |≤ + (cid:90) (cid:60) ( s )= + ε |(cid:61) ( s ) | > Z n,E ( s ) M w ( s ) X s ds (cid:28) ε,C,w ( N E ) ε | n | + ε X + ε + ( | n | N E ) ε (cid:90) | t | > ( | t | + 4) C log((2 ε ) − | t | +4) ( | n | ( | t | + 1)) + ε ( | t | + 1) C log((2 ε ) − )+2 X + ε dt (cid:28) ε,C ( N E ) ε | n | + ε X + ε . In the case where n is a square, the proof is similar, except that the integral we are interested inis given by K n ( X ) := 12 πi (cid:90) (cid:60) ( s )=2 ζ ( s ) ζ (2 s ) (cid:89) p | N E (cid:18) (cid:0) pn (cid:1) p s (cid:19) − (cid:89) p | n (cid:18) p s (cid:19) − M w ( s ) X s ds. Shifting the contour of integration to the left until (cid:60) ( s ) = + ε , we pick up the residue from thesimple pole at s = 1 and arrive at the formula K n ( X ) = M w (1) Xζ (2) (cid:89) p | N E (cid:18) (cid:0) pn (cid:1) p (cid:19) − (cid:89) p | n (cid:18) p (cid:19) − + O ε,w (cid:0) ( | n | N E ) ε X + ε (cid:1) . Remark 2.7 also applies here.
OW-LYING ZEROS OF ELLIPTIC CURVE L -FUNCTIONS 13 The error term comes from the same reasoning as before, except that we used the convexity boundfor ζ ( s ) instead of that of L (cid:0) s, (cid:0) · n (cid:1) (cid:1) . (cid:3) The Explicit Formula.
The fundamental tool to study the -level density is the explicitformula; we will use Mestre’s version [Me]. Recall that L = log( N E X / (2 πe ) ) . Lemma 2.11 (Explicit Formula) . Let φ denote a Schwartz function whose Fourier transform hascompact support. For d square-free with ( d, N E ) = 1 , we have the formula D X ( E d ; φ ) = (cid:98) φ (0) L log (cid:18) N E d (2 π ) (cid:19) − L (cid:90) ∞ (cid:32) (cid:98) φ ( x/L ) e − x − e − x − (cid:98) φ (0) e − x x (cid:33) dx − L (cid:88) p,m ( α E ( p ) m + β E ( p ) m ) χ d ( p m ) log pp m/ (cid:98) φ (cid:18) m log pL (cid:19) . (2.6) Proof.
Recall that D X ( E d ; φ ) = (cid:88) γ d φ (cid:18) γ d L π (cid:19) , where ρ d = + iγ d runs through the non-trivial zeros of L ( s, f E ⊗ χ d ) . Note that since ( d, N E ) = 1 and d is square-free, Proposition 14.20 of [IK] implies that f E ⊗ χ d is a weight newform oflevel N E d .We take Φ( s ) := φ ( − iL π ( s − / in the explicit formula on page 215 of [Me], which applies toany weight newform on Γ ( N E d ) . This yields the formula D X ( E d ; φ ) = F (0) (cid:0) log( N E d ) − π ) (cid:1) − (cid:90) ∞ (cid:18) F ( x ) e − x − e − x − F (0) e − x x (cid:19) dx − (cid:88) p,m ( α E ( p ) m + β E ( p ) m ) χ d ( p m ) log pp m/ F ( m log p ) , (2.7)where the function F ( x ) is such that Φ( s ) = (cid:90) R F ( x ) e x ( s − / dx. In (2.7) we have used the identity α f E ⊗ χ d ( p ) m + β f E ⊗ χ d ( p ) m = ( α E ( p ) m + β E ( p ) m ) χ d ( p m ) . To show this identity, note that since f E ⊗ χ d is a newform, its L -function is given by (cid:88) n ≥ λ E ( n ) χ d ( n ) n s = (cid:89) p (cid:18) − α f E ⊗ χ d ( p ) p s (cid:19) − (cid:18) − β f E ⊗ χ d ( p ) p s (cid:19) − = (cid:89) p (cid:18) − λ E ( p ) χ d ( p ) p s + ψ d N E ( p ) p s (cid:19) − , where ψ k ( n ) is the principal character modulo k . Hence the choice α f E ⊗ χ d ( p ) = α E ( p ) χ d ( p ) and β f E ⊗ χ d ( p ) = β E ( p ) χ d ( p ) is consistent (note that χ d ( p ) is real and the pair ( α f E ⊗ χ d ( p ) , β f E ⊗ χ d ( p )) is defined up to permutation).We have that φ ( tL ) = Φ (cid:18)
12 + 2 πit (cid:19) = (cid:90) R F ( x ) e πitx dx = (cid:98) F ( − t ); hence F ( x ) = (cid:98) φ ( x/L ) /L gives us the desired choice of Φ( s ) , and (2.6) follows. (cid:3) Corollary 2.12.
We have the following formulas for the -level densities we are interested in (see (1.7) and (1.15) ): D ∗ ( φ ; X ) = (cid:98) φ (0) LW ∗ ( X ) (cid:88) ∗ ( d,N E )=1 w (cid:18) dX (cid:19) log (cid:18) N E d (2 π ) (cid:19) − L (cid:90) ∞ (cid:32) (cid:98) φ ( x/L ) e − x − e − x − (cid:98) φ (0) e − x x (cid:33) dx − LW ∗ ( X ) (cid:88) p,m ( α E ( p ) m + β E ( p ) m ) log pp m/ (cid:98) φ (cid:18) m log pL (cid:19) (cid:88) ∗ ( d,N E )=1 w (cid:18) dX (cid:19) (cid:18) dp m (cid:19) , (2.8) D ( φ ; X ) = (cid:98) φ (0) LW ( X ) (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) log (cid:18) N E d (2 π ) (cid:19) − L (cid:90) ∞ (cid:32) (cid:98) φ ( x/L ) e − x − e − x − (cid:98) φ (0) e − x x (cid:33) dx − LW ( X ) (cid:88) p,m ( α E ( p ) m + β E ( p ) m ) log pp m/ (cid:98) φ (cid:18) m log pL (cid:19) (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) (cid:18) dp m (cid:19) . (2.9) (Recall the definition of (cid:101) w given in (1.10) .)Proof. The idea is to sum (2.6) over the desired values of d required to obtain D ∗ ( φ ; X ) and D ( φ ; X ) ,against the smooth weight w ( dX ) . The first identity follows immediately from (2.6). For the secondidentity, note that for any integer (cid:96) ≥ , D X ( E d ; φ ) = D X ( E (cid:96) d ; φ ) . This follows from the fact that L ( s, E d ) and L ( s, E (cid:96) d ) have the same nontrivial zeros, since ( x, y ) (cid:55)→ ( x, (cid:96)y ) induces a bijection between the groups E (cid:96) d ( F p ) and E d ( F p ) for any p (cid:45) (cid:96)dN E . Hence, W ( X ) D ( φ ; X ) = (cid:88) ( d,N E )=1 w (cid:18) dX (cid:19) D X ( E d ; φ ) = (cid:88) ∗ ( d,N E )=1 (cid:88) n ≥ n,N E )=1 w (cid:18) n dX (cid:19) D X ( E d ; φ )= (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) D X ( E d ; φ ) . (2.10)Finally, by applying (2.6) and noting that W ( X ) = (cid:88) ( d,N E )=1 w (cid:18) dX (cid:19) = (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) , (2.11)we deduce (2.9). (cid:3) The prime sum in D ( φ ; X ) The goal of this section is to study the second prime sum appearing in Corollary 2.12, that is theterm − LW ( X ) (cid:88) p,m ( α E ( p ) m + β E ( p ) m ) log pp m/ (cid:98) φ (cid:18) m log pL (cid:19) (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) (cid:18) dp m (cid:19) = S odd + S even , (3.1)where S odd and S even contain respectively the terms with odd and even m . In Appendix A, we willsee that S even appears as is in the Ratios Conjecture’s prediction (see Theorem A.2). Bounding S odd constitutes the heart of the paper, and sets the limit for both the allowable support for thetest function φ as well as the size of the error term in Theorems 1.6 and 1.9. Our analysis is inspiredby that of Katz and Sarnak [KaS1], who used Poisson summation to analyze such a quantity. This OW-LYING ZEROS OF ELLIPTIC CURVE L -FUNCTIONS 15 will be done in Lemma 3.2, but we first show that the terms with odd m ≥ are negligible. In thissection, we do not indicate the dependence on φ and w of the implied constants in the error terms. Lemma 3.1.
Fix ε > . Assuming that σ := sup ( supp (cid:98) φ ) < ∞ , we have the bound − LW ( X ) (cid:88) pm ≥ odd ( α E ( p ) m + β E ( p ) m ) log pp m/ (cid:98) φ (cid:18) m log pL (cid:19) (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) (cid:18) dp m (cid:19) (cid:28) ε N εE X − ε . Proof.
After noting that (cid:16) dp m (cid:17) = (cid:16) dp (cid:17) , this is a direct application of Lemma 2.5, combined with thebounds | α E ( p ) | , | β E ( p ) | ≤ and Remark 2.6. (cid:3) We now adapt the arguments of [KaS1].
Lemma 3.2.
Fix ε > . Assuming that σ := sup ( supp (cid:98) φ ) < ∞ , we have the following: S odd = − XLW ( X ) (cid:88) ≤ k ≤
10 log X (cid:88) (cid:96) | N E µ ( (cid:96) ) (cid:96) (cid:88) p (cid:45) N E (cid:16) − (cid:96)p (cid:17) (cid:15) p λ E ( p ) log pp k (cid:98) φ (cid:18) log pL (cid:19) (cid:88) t ∈ Z (cid:18) tp (cid:19) (cid:98) w (cid:18) Xtp k (cid:96) (cid:19) + O ε (cid:0) N εE X − ε (cid:1) , where (cid:15) p := (cid:40) if p ≡ ,i if p ≡ . Proof.
First note that the terms with p | N E in S odd are negligible, since by Lemma 2.5 we havethe bound LW ( X ) (cid:88) p | N E ( α E ( p ) + β E ( p )) log pp / (cid:98) φ (cid:18) log pL (cid:19) (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) (cid:18) dp (cid:19) (cid:28) ε X (cid:88) p | N E log p ( N E Xp ) ε/ (cid:28) ε N (cid:15)E X − ε . Note also that by the definition of (cid:101) w ( x ) , we have for p (cid:45) N E the identity (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) (cid:18) dp (cid:19) = (cid:88) k ≥ (cid:88) ( d,N E )=1 w (cid:18) dX/p k (cid:19) (cid:18) dp (cid:19) . Recall that α E ( p ) + β E ( p ) = λ E ( p ) . Therefore, by Lemma 3.1 we have that S odd = − LW ( X ) (cid:88) ≤ k ≤
10 log X (cid:88) p (cid:45) N E λ E ( p ) log pp / (cid:98) φ (cid:18) log pL (cid:19) (cid:88) ( d,N E )=1 w (cid:18) dX/p k (cid:19) (cid:18) dp (cid:19) + O ε ( N εE X − ε ) . In the last expression we have removed the terms with k >
10 log X , since by the rapid decay of w ( t ) , their sum is (we write c E := N E / (2 πe ) ) (cid:28) N εE X (cid:88) k>
10 log X (cid:88) p ≤ ( c E X ) σ log pp / (cid:88) ( d,N E )=1 (cid:18) Xdp k (cid:19) (cid:28) N εE X
30 log X (cid:88) p log pp / (cid:28) N εE X − . We now introduce additive characters using Gauss sums. These characters have the advantageof being smooth functions of their argument and will thus allow us to use Poisson summation. For p an odd prime and Y > we write (see [D, Sections 2 and 9] for the definition and properties ofthe Gauss sum τ ( χ ) ) (cid:88) ( d,N E )=1 w (cid:18) dY (cid:19) (cid:18) dp (cid:19) = (cid:88) (cid:96) | N E µ ( (cid:96) ) (cid:88) r ∈ Z w (cid:18) r(cid:96)Y (cid:19) (cid:18) r(cid:96)p (cid:19) (3.2) = (cid:88) (cid:96) | N E µ ( (cid:96) ) (cid:88) r ∈ Z w (cid:18) r(cid:96)Y (cid:19) τ (cid:16)(cid:16) · p (cid:17)(cid:17) (cid:88) b mod p (cid:18) bp (cid:19) e (cid:18) r(cid:96)bp (cid:19) = (cid:88) (cid:96) | N E µ ( (cid:96) ) (cid:15) p p (cid:88) b mod p (cid:18) bp (cid:19) (cid:88) r ∈ Z w (cid:18) r(cid:96)Y (cid:19) e (cid:18) r(cid:96)bp (cid:19) . Our expression for S odd is now S odd = − LW ( X ) (cid:88) ≤ k ≤
10 log X (cid:88) (cid:96) | N E µ ( (cid:96) ) (cid:34) (cid:88) p (cid:45) (cid:96)N E (cid:15) p λ E ( p ) log pp (cid:98) φ (cid:18) log pL (cid:19) × (cid:88) b mod p (cid:18) bp (cid:19) (cid:88) r ∈ Z w (cid:18) r(cid:96)X/p k (cid:19) e (cid:18) r(cid:96)bp (cid:19) (cid:35) + O ε ( N εE X − ε ) . (3.3)Notice that we removed the terms with p | (cid:96) since they are all zero. This can be seen from the lastexpression using the orthogonality of (cid:16) · p (cid:17) , and is even more apparent in (3.2). We are ready toapply Poisson summation in (3.3): (cid:88) r ∈ Z w (cid:18) r(cid:96)Y (cid:19) e (cid:18) r(cid:96)bp (cid:19) = Y(cid:96) (cid:88) s ∈ Z (cid:98) w (cid:18) Y (cid:18) s(cid:96) − bp (cid:19)(cid:19) , which yields the expression S odd = − XLW ( X ) (cid:88) ≤ k ≤
10 log X (cid:88) (cid:96) | N E µ ( (cid:96) ) (cid:96) (cid:34) (cid:88) p (cid:45) (cid:96)N E (cid:15) p λ E ( p ) log pp k (cid:98) φ (cid:18) log pL (cid:19) × (cid:88) b mod p (cid:18) bp (cid:19) (cid:88) s ∈ Z (cid:98) w (cid:18) Xp k (cid:18) s(cid:96) − bp (cid:19)(cid:19) (cid:35) + O ε ( N εE X − ε ) . Note that as s runs through the integers and b runs through a complete residue system modulo p ,the variable t := sp − b(cid:96) runs through all integers (the fact that ( (cid:96), p ) = 1 is crucial here). In otherwords, the following map is a group isomorphism: f (cid:96),p : Z /p Z × Z −→ Z ( b, s ) (cid:55)−→ sp − b(cid:96). Combining this with the fact that (cid:16) bp (cid:17) = (cid:16) − (cid:96) − ( t − sp ) p (cid:17) = (cid:16) − (cid:96)p (cid:17) (cid:16) tp (cid:17) , we obtain S odd = − XLW ( X ) (cid:88) ≤ k ≤
10 log X (cid:88) (cid:96) | N E µ ( (cid:96) ) (cid:96) (cid:34) (cid:88) p (cid:45) (cid:96)N E (cid:16) − (cid:96)p (cid:17) (cid:15) p λ E ( p ) log pp k (cid:98) φ (cid:18) log pL (cid:19) × (cid:88) t ∈ Z (cid:18) tp (cid:19) (cid:98) w (cid:18) Xtp k (cid:96) (cid:19) (cid:35) + O ε ( N εE X − ε ) . (cid:3) OW-LYING ZEROS OF ELLIPTIC CURVE L -FUNCTIONS 17 Lemma 3.3.
Fix ε > . We have the bound S := (cid:88) ≤ k ≤
10 log X (cid:88) (cid:96) | N E µ ( (cid:96) ) (cid:96) (cid:88) p (cid:45) N E p ≤ X − ε k +1 (cid:16) − (cid:96)p (cid:17) (cid:15) p λ E ( p ) log pp k (cid:98) φ (cid:18) log pL (cid:19) (cid:88) t ∈ Z (cid:18) tp (cid:19) (cid:98) w (cid:18) Xtp k (cid:96) (cid:19) (cid:28) ε,E X − . Proof.
Letting M = 1 + max(10 , ε − ) , we have by the rapid decay of (cid:98) w that S (cid:28) ε (cid:88) ≤ k ≤
10 log X (cid:88) (cid:96) | N E (cid:96) (cid:88) p (cid:45) N E p ≤ X − ε k +1 log pp k (cid:88) (cid:54) = t ∈ Z (cid:18) (cid:96)tX ε (cid:19) M (cid:28) M,E X − Mε log X. (cid:3) Lemma 3.4.
Fix K ∈ N and ε > . If σ := sup ( supp (cid:98) φ ) < ∞ , then we have the bound S ,K := (cid:88) K ≤ k ≤
10 log X (cid:88) (cid:96) | N E µ ( (cid:96) ) (cid:96) (cid:88) p (cid:45) N E p>X − ε k +1 (cid:16) − (cid:96)p (cid:17) (cid:15) p λ E ( p ) log pp k (cid:98) φ (cid:18) log pL (cid:19) (cid:88) t ∈ Z (cid:18) tp (cid:19) (cid:98) w (cid:18) Xtp k (cid:96) (cid:19) (cid:28) ε,E X − max( K − K +1 , − σ )+ ε . Proof.
We split the sum over p into two parts, cutting at the point X k +1 / . To bound the first ofthese sums, we first note that (cid:88) t ∈ Z (cid:18) tp (cid:19) (cid:98) w (cid:18) Xtp k (cid:96) (cid:19) (cid:28) (cid:88) < | t |≤ p k(cid:96)X (cid:88) | t | > p k(cid:96)X (cid:18) p k (cid:96)Xt (cid:19) (cid:28) E p k X , (3.4)and hence, writing c E := N E / (2 πe ) , (cid:88) K ≤ k ≤
10 log X (cid:88) (cid:96) | N E µ ( (cid:96) ) (cid:96) (cid:88) p (cid:45) N E X − ε k +1
10 log X (cid:88) X − ε k +1
X k +1 / (cid:1) usingthe Pólya-Vinogradov inequality, which reads S ( T ) := (cid:88) ≤ u ≤ T (cid:18) up (cid:19) (cid:28) p log p. We then have (cid:88) t ≥ (cid:18) tp (cid:19) (cid:98) w (cid:18) Xtp k (cid:96) (cid:19) = (cid:90) ∞ (cid:98) w (cid:18) Xtp k (cid:96) (cid:19) dS ( t ) = − (cid:90) ∞ Xp k (cid:96) (cid:98) w (cid:48) (cid:18) Xtp k (cid:96) (cid:19) S ( t ) dt (cid:28) p log p. Treating the terms with t < in a similar way, we conclude that the second part of S ,K , that isthe sum over K (cid:48) ≤ k ≤
10 log X with K (cid:48) := max( K, ( σ − , satisfies (cid:88) K (cid:48) ≤ k ≤
10 log X (cid:88) (cid:96) | N E µ ( (cid:96) ) (cid:96) (cid:88) p (cid:45) N E p>X k +1 / (cid:16) − (cid:96)p (cid:17) (cid:15) p λ E ( p ) log pp k (cid:98) φ (cid:18) log pL (cid:19) (cid:88) t ∈ Z (cid:18) tp (cid:19) (cid:98) w (cid:18) Xtp k (cid:96) (cid:19) (cid:28) E (cid:88) K (cid:48) ≤ k ≤
10 log X (cid:88) p>X k +1 / (log p ) p k +1 / (cid:28) log XX K (cid:48)− / K (cid:48) +1 / = log XX max (cid:0) K − / K +1 / , − σ (cid:1) . This concludes the proof. (cid:3)
Remark 3.5.
In the proof of Lemma 3.4, we could have used Burgess’s bound to improve ourestimate on the sum with p ∈ (cid:2) X k +3 / , X k +1 / (cid:3) . This would have yielded a better overall boundwhen σ ∈ ( k +3 / , k +1 / ) for some k ≥ . However, we have chosen to carry out this improvementin a separate paper [FPS]. Proposition 3.6.
Fix m ∈ N and assume that m +1) ≤ σ = sup ( supp (cid:98) φ ) < m − . Then, for anyfixed ε > , we have the bound S odd (cid:28) ε,E X − max( m − m +1 , − σ )+ ε . Proof.
By Lemma 3.2 and 3.3, we have that S odd = − XLW ( X ) (cid:88) ≤ k ≤
10 log X (cid:88) (cid:96) | N E µ ( (cid:96) ) (cid:96) (cid:88) p (cid:45) N E p>X − ε k +1 (cid:16) − (cid:96)p (cid:17) (cid:15) p λ E ( p ) log pp k (cid:98) φ (cid:18) log pL (cid:19) × (cid:88) t ∈ Z (cid:18) tp (cid:19) (cid:98) w (cid:18) Xtp k (cid:96) (cid:19) + O ε,E (cid:0) X − ε (cid:1) . Note that for X large enough in terms of E , the support of (cid:98) φ imposes the condition p ≤ X m − − η for some fixed η > . Hence, all terms with k ≤ m − in the above sum are identically zero. Wethen apply Lemma 3.4 and obtain the bound S odd (cid:28) ε,E X − max( m − m +1 , − σ )+ ε . (cid:3) Remark 3.7.
Notice that for σ < , the error term O ε,E (cid:0) X − max( m − m +1 , − σ )+ ε (cid:1) is always at most O ε,E ( X − / ε ) , which is sharper than the Ratios Conjecture’s prediction. Moreover, if the supportof (cid:98) φ is very small, then this error term is O E ( X − δ ) with a very small δ .In Proposition 3.9, we give a sharper bound on S odd , which is conditional on ECRH. We firstgive a standard application of ECRH. Lemma 3.8.
Assume ECRH. We have, for m ∈ Z (cid:54) =0 and y ≥ , the estimate S m ( y ) := (cid:88) p ≤ y (cid:18) mp (cid:19) λ E ( p ) log p (cid:28) y (log y ) log(2 | m | yN E ) . OW-LYING ZEROS OF ELLIPTIC CURVE L -FUNCTIONS 19 Proof.
The L -function L ( s, E m ) = L ( s, f E ⊗ χ m ) = ∞ (cid:88) n =1 λ E ( n ) (cid:0) mn (cid:1) n s is modular, and hence it admits an analytic continuation to the whole of C and has an Euler productand a functional equation. It is therefore an L -function in the sense of Iwaniec and Kowalski, andthus [IK, Thm. 5.15] takes the form (cid:88) p e ≤ ye ≥ (cid:0) α E ( p ) e + β E ( p ) e (cid:1) log p (cid:18) mp e (cid:19) (cid:28) y (log y ) log(2 | m | N E y ) . The result follows by trivially bounding the contribution of prime powers. (cid:3)
Lemma 3.9.
Fix ε > and K ∈ Z ≥ , and assume ECRH. If σ = sup ( supp (cid:98) φ ) < ∞ , then we havethe bound S ,K := (cid:88) K ≤ k ≤
10 log X (cid:88) (cid:96) | N E µ ( (cid:96) ) (cid:96) (cid:88) p (cid:45) N E p>X − ε k +1 (cid:16) − (cid:96)p (cid:17) (cid:15) p λ E ( p ) log pp k (cid:98) φ (cid:18) log pL (cid:19) (cid:88) t ∈ Z (cid:18) tp (cid:19) (cid:98) w (cid:18) Xtp k (cid:96) (cid:19) (cid:28) ε,E X min( − K , − σ )+ ε . Proof.
We will show that for ≤ k ≤
10 log X , we have R := (cid:88) (cid:96) | N E µ ( (cid:96) ) (cid:96) (cid:88) t ∈ Z (cid:88) p>X − ε k +1 (cid:16) − t(cid:96)p (cid:17) (cid:15) p λ E ( p ) log pp k (cid:98) φ (cid:18) log pL (cid:19) (cid:98) w (cid:18) Xtp k (cid:96) (cid:19) (cid:28) ε,E X min( − k , − σ )+ ε , from which the lemma clearly follows. Notice that we have added back the primes dividing N E ,since by a calculation similar to (3.4), their contribution is (cid:28) (cid:88) (cid:96) | N E (cid:96) (cid:88) t ∈ Z (cid:88) p | N E log pp k (cid:12)(cid:12)(cid:12)(cid:12) (cid:98) w (cid:18) Xtp k (cid:96) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:28) (cid:88) (cid:96) | N E (cid:96) (cid:88) p | N E log pp k p k (cid:96)X (cid:28) E X − . We now apply Lemma 3.8. Note that (cid:15) p = i χ ( p ) + − i χ ( p ) , where χ and χ are respectivelythe trivial and the nontrivial character modulo . Using this fact and applying Lemma 3.8, we have S t,(cid:96) ( y ) := (cid:88) p ≤ y (cid:18) − t(cid:96)p (cid:19) (cid:15) p λ E ( p ) log p (cid:28) y (log y ) log(2 | t | (cid:96)yN E ) . Let m ( X ) := min( X k , ( c E X ) σ ) , with c E := N E / (2 πe ) . We first treat the terms in R for which X − ε k +1 < p ≤ m ( X ) . Denoting the sum of these terms by R , we have R = (cid:88) (cid:96) | N E µ ( (cid:96) ) (cid:96) (cid:88) (cid:54) = t ∈ Z P t,(cid:96) , where P t,(cid:96) := (cid:90) m ( X ) X − ε k +1 (cid:98) φ (cid:16) log xL (cid:17) (cid:98) w (cid:0) Xtx k (cid:96) (cid:1) x k d S t,(cid:96) ( x ) . In the case K = 0 , we adopt the convention that min( − K , − σ ) = − σ. Performing integration by parts, we obtain the bound P t,(cid:96) (cid:28) (cid:12)(cid:12) (cid:98) w (cid:0) X ε t(cid:96) (cid:1) (cid:12)(cid:12) X − ε (cid:12)(cid:12)(cid:12) S t,(cid:96) (cid:16) X − ε k +1 (cid:17)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12) (cid:98) w (cid:0) Xtm ( X ) k (cid:96) (cid:1)(cid:12)(cid:12) m ( X ) k |S t,(cid:96) ( m ( X )) | + ( k + 1) (cid:90) m ( X ) X − ε k +1 (cid:34) (cid:12)(cid:12) (cid:98) w (cid:0) Xtx k (cid:96) (cid:1) (cid:12)(cid:12) x k + (cid:12)(cid:12) (cid:98) w (cid:48) (cid:0) Xtx k (cid:96) (cid:1) (cid:12)(cid:12) X | t | x k (cid:96) (cid:35) x (cid:0) log(2 x | t | (cid:96)N E ) (cid:1) dx. (3.6)For any fixed M ≥ and x ∈ [ X − ε k +1 , m ( X )] , we have (cid:88) (cid:54) = t ∈ Z (cid:12)(cid:12)(cid:12)(cid:12) (cid:98) w (cid:18) X ε t(cid:96) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) log(2 | t | ) (cid:28) M (cid:96) M X − εM ; (cid:88) (cid:54) = t ∈ Z (cid:12)(cid:12)(cid:12)(cid:12) (cid:98) w (cid:18) Xtx k (cid:96) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:0) log(2 | t | ) (cid:1) (cid:28) x k (cid:96)X (cid:0) log(2 x k X(cid:96) ) (cid:1) ; (cid:88) (cid:54) = t ∈ Z X | t | (cid:12)(cid:12)(cid:12)(cid:12) (cid:98) w (cid:48) (cid:18) Xtx k (cid:96) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:0) log(2 | t | ) (cid:1) (cid:28) ( x k (cid:96) ) X (cid:0) log(2 x k X(cid:96) ) (cid:1) , from which we obtain R (cid:28) ε,E X − + X min( − k , − σ )+ ε + ( k + 1) (cid:88) (cid:96) | N E (cid:96) (cid:90) m ( X ) X − ε k +1 (cid:20) (cid:96)xX + (cid:96)xX (cid:21) x (cid:0) log(2 x k X(cid:96) ) log(2 x(cid:96) ) (cid:1) dx (cid:28) ε,E X min( − k , − σ )+ ε , since k ≤
10 log X .For those k for which σ < k , we have already covered all possible values of p (for X (cid:29) E ).For the remaining values of k , we apply the Pólya-Vinogradov inequality in the exact same manneras in the proof of Lemma 3.4. Thus the sum of the terms with p > X k is (cid:28) ε X − k + ε . The proof is complete since in this case we have that min( − k , − σ ) = − k . (cid:3) Proposition 3.10.
Assume ECRH, fix m ∈ N and assume that m +1) ≤ σ = sup ( supp (cid:98) φ ) < m − . Then, for any fixed ε > , we have the bound S odd (cid:28) ε,E X min( − m , − σ )+ ε . Moreover, for ≤ σ < , we have S odd (cid:28) ε,E X − σ + ε . Proof.
The proof is similar to that of Proposition 3.6, except that we substitute Lemma 3.4 withLemma 3.9. (cid:3)
We summarize the findings of this section in the following theorem.
Theorem 3.11.
Fix ε > . Then, in the range σ = sup ( supp (cid:98) φ ) < , we have the following uncon-ditional bound: S odd (cid:28) ε,E X η ( σ )+ ε , where η ( σ ) = − max( m − m +1 , − σ ) for m +1) ≤ σ < m − , with m ∈ N . Moreover, if weassume ECRH, then, in the wider range σ = sup ( supp (cid:98) φ ) < , we have the improved bound S odd (cid:28) ε,E X θ ( σ )+ ε , where θ ( σ ) = (cid:40) − max(1 − m , − σ ) for m +1) ≤ σ < m − with m ∈ N , − σ for ≤ σ < . Note that the domain of this function is (0 , ) . OW-LYING ZEROS OF ELLIPTIC CURVE L -FUNCTIONS 21 Proof.
The unconditional bound follows directly from Proposition 3.6, and the conditional boundfollows from Proposition 3.10. (cid:3)
We are now ready to complete the proof of our main result.
Proof of Theorems 1.6 and 1.9.
By Corollary 2.12 and (3.1), we have D ( φ ; X ) = (cid:98) φ (0) LW ( X ) (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) log (cid:18) N E d (2 π ) (cid:19) − L (cid:90) ∞ (cid:32) (cid:98) φ ( x/L ) e − x − e − x − (cid:98) φ (0) e − x x (cid:33) dx + S odd + S even . Moreover, Theorem 3.11 shows that we have S odd (cid:28) ε X η ( σ )+ ε unconditionally, and that underECRH we have S odd (cid:28) ε X θ ( σ )+ ε . We conclude the proof by applying Lemma 2.5 and Remark 2.6to S even , yielding the estimate S even = − L ∞ (cid:88) (cid:96) =1 (cid:88) p (cid:0) α E ( p ) (cid:96) + β E ( p ) (cid:96) (cid:1) log pp (cid:96) (cid:98) φ (cid:18) (cid:96) log pL (cid:19) (cid:18) ψ N E ( p ) p (cid:19) − + O ε (cid:0) X − ε N εE (cid:1) . (cid:3) The prime sum in D ∗ ( φ ; X ) In this section, we study the prime sum appearing in (2.8), that is − LW ∗ ( X ) (cid:88) p,m ( α E ( p ) m + β E ( p ) m ) log pp m/ (cid:98) φ (cid:18) m log pL (cid:19) (cid:88) ∗ ( d,N E )=1 w (cid:18) dX (cid:19) (cid:18) dp m (cid:19) = S ∗ odd + S ∗ even , where again S ∗ odd and S ∗ even denote respectively the sum of the terms with m odd and even. Through-out, we do not indicate the dependence on φ and w of the implied constants in the error terms.We first give an estimate for S ∗ odd , showing that the terms with m ≥ are negligible. Lemma 4.1.
Fix ε > , and assume RH. Denoting by [ a, b ] the least common multiple of a and b ,we have S ∗ odd = − LW ∗ ( X ) (cid:88) (cid:96) | N E s ∈ N µ ( (cid:96) ) µ ( s ) (cid:88) p (cid:45) sN E λ E ( p ) log pp / (cid:98) φ (cid:18) log pL (cid:19) (cid:88) (cid:54) = d ∈ Z w (cid:18) d [ (cid:96), s ] X (cid:19) (cid:18) d [ (cid:96), s ] p (cid:19) + O ε (cid:0) N εE X − + ε (cid:1) . Proof.
We first see that (cid:16) dp m (cid:17) = (cid:16) dp (cid:17) , and so Lemma 2.10 implies that − LW ∗ ( X ) (cid:88) pm ≥ odd ( α E ( p ) m + β E ( p ) m ) log pp m/ (cid:98) φ (cid:18) m log pL (cid:19) (cid:88) ∗ ( d,N E )=1 w (cid:18) dX (cid:19) (cid:18) dp m (cid:19) (cid:28) ε N εE X − + ε . The same lemma also implies the bound − LW ∗ ( X ) (cid:88) p | N E λ E ( p ) log pp / (cid:98) φ (cid:18) log pL (cid:19) (cid:88) ∗ ( d,N E )=1 w (cid:18) dX (cid:19) (cid:18) dp (cid:19) (cid:28) ε N εE X − + ε . The claimed formula follows from using the identity µ ( d ) = (cid:80) s | d µ ( s ) and interchanging the orderof summation. (cid:3) We now follow the arguments of [KaS1].
Lemma 4.2.
Fix ε > . Assume RH and ECRH, and suppose that σ := sup ( supp (cid:98) φ ) < ∞ . Then,for any S ≥ , we have S ∗ odd = − XLW ∗ ( X ) (cid:88) (cid:96) | N E s ≤ S µ ( (cid:96) ) µ ( s )[ (cid:96), s ] (cid:88) p (cid:45) sN E (cid:16) − [ (cid:96),s ] p (cid:17) (cid:15) p λ E ( p ) log pp (cid:98) φ (cid:18) log pL (cid:19) (cid:88) t ∈ Z (cid:18) tp (cid:19) (cid:98) w (cid:18) Xtp [ (cid:96), s ] (cid:19) + O ε (cid:0) N εE X ε (log S ) S − + N εE X − + ε (cid:1) . Proof.
The starting point is Lemma 4.1, in which we will bound the terms with s > S using ECRH.Applying Lemma 3.8 and a routine summation by parts, we obtain that LW ∗ ( X ) (cid:88) (cid:96) | N E s>S µ ( (cid:96) ) µ ( s ) (cid:88) (cid:54) = d ∈ Z w (cid:18) d [ (cid:96), s ] X (cid:19) (cid:88) p (cid:45) sN E (cid:18) d [ (cid:96), s ] p (cid:19) λ E ( p ) log pp / (cid:98) φ (cid:18) log pL (cid:19) (cid:28) W ∗ ( X ) (cid:88) (cid:96) | N E s>S (cid:88) (cid:54) = d ∈ Z w (cid:18) d [ (cid:96), s ] X (cid:19) (cid:0) log(2 | d | (cid:96)sN E X ) (cid:1) (cid:28) ε ( XN E ) ε (cid:88) s>S (log s ) s (cid:28) ε N εE X ε (log S ) S .
The rest of the proof is similar to that of Lemma 3.2, the main ingredient being Poisson Summation. (cid:3)
We now handle the terms with s ≤ S in S ∗ odd . Lemma 4.3.
Assume ECRH, fix ε > and suppose that σ := sup ( supp (cid:98) φ ) < ∞ . Then, for any ≤ S ≤ X , we have that (cid:88) (cid:96) | N E s ≤ S µ ( (cid:96) ) µ ( s )[ (cid:96), s ] (cid:88) t ∈ Z (cid:88) p (cid:45) sN E (cid:16) − [ (cid:96),s ] tp (cid:17) (cid:15) p λ E ( p ) log pp (cid:98) φ (cid:18) log pL (cid:19) (cid:98) w (cid:18) Xtp [ (cid:96), s ] (cid:19) (cid:28) ε,E SX σ − ε . Proof.
We first add back the primes dividing sN E , at the cost of an error term which is (cid:28) (cid:88) (cid:96) | N E s ≤ S (cid:96), s ] (cid:88) p | sN E log pp p [ (cid:96), s ] X (cid:28) ε N εE SX − log(2 N E S ) . We then follow the steps of Lemma 3.9. The sum we are interested in equals (cid:88) (cid:96) | N E s ≤ S µ ( (cid:96) ) µ ( s )[ (cid:96), s ] (cid:88) (cid:54) = t ∈ Z P (cid:48) t,(cid:96),s + O ε (cid:0) N εE SX − log(2 N E S ) (cid:1) , where (write c E := N E / (2 πe ) ) P (cid:48) t,(cid:96),s := (cid:90) ( c E X ) σ (cid:98) φ (cid:16) log xL (cid:17) (cid:98) w (cid:16) Xtx [ (cid:96),s ] (cid:17) x dS t,(cid:96),s ( x ) , This range can be replaced by ≤ S ≤ X M , for any fixed M ∈ N . However, the important range for our analysisis ≤ S ≤ X . OW-LYING ZEROS OF ELLIPTIC CURVE L -FUNCTIONS 23 with S t,(cid:96),s ( y ) = (cid:88) p ≤ y (cid:18) − [ (cid:96), s ] tp (cid:19) (cid:15) p λ E ( p ) log p (cid:28) y (log(2 y )) log(2 N E | t | s(cid:96)y ) . Performing integration by parts, we obtain the bound P (cid:48) t,(cid:96),s (cid:28) (cid:90) ( c E X ) σ (cid:12)(cid:12)(cid:12) (cid:98) w (cid:16) Xtx [ (cid:96),s ] (cid:17) (cid:12)(cid:12)(cid:12) x + (cid:12)(cid:12)(cid:12) (cid:98) w (cid:48) (cid:16) Xtx [ (cid:96),s ] (cid:17) (cid:12)(cid:12)(cid:12) X | t | x [ (cid:96), s ] x (cid:0) log(2 N E x | t | s(cid:96) ) (cid:1) dx. (4.1)Recall that (cid:88) (cid:54) = t ∈ Z (cid:12)(cid:12)(cid:12)(cid:12) (cid:98) w (cid:18) Xtx [ (cid:96), s ] (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:0) log(2 | t | ) (cid:1) (cid:28) x [ (cid:96), s ] X (cid:0) log(2 xX(cid:96)s ) (cid:1) ; (cid:88) (cid:54) = t ∈ Z X | t | (cid:12)(cid:12)(cid:12)(cid:12) (cid:98) w (cid:48) (cid:18) Xtx [ (cid:96), s ] (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:0) log(2 | t | ) (cid:1) (cid:28) ( x [ (cid:96), s ]) X (cid:0) log(2 xX(cid:96)s ) (cid:1) , from which we obtain the bound (cid:88) (cid:96) | N E s ≤ S µ ( (cid:96) ) µ ( s )[ (cid:96), s ] (cid:88) (cid:54) = t ∈ Z P (cid:48) t,(cid:96),s (cid:28) (cid:88) (cid:96) | N E s ≤ S (cid:96), s ] (cid:90) ( c E X ) σ (cid:20) [ (cid:96), s ] xX + [ (cid:96), s ] xX (cid:21) x (cid:0) log(2 xX(cid:96)s ) log(2 N E xs(cid:96) ) (cid:1) dx (cid:28) ε,E S (log(2 S )) X σ − ε . This concludes the proof. (cid:3)
We summarize the current section in the following theorem.
Theorem 4.4.
Assume RH and ECRH, and suppose that σ := sup ( supp (cid:98) φ ) < . Then, for any fixed ε > , we have the bound S ∗ odd (cid:28) ε,E X σ − + ε . Proof.
Take S = X − σ in Lemmas 4.2 and 4.3. (cid:3) Finally, we complete the proof of Theorem 1.10.
Proof of Theorem 1.10.
The desired result follows directly from Corollary 2.12, Lemma 2.10 andTheorem 4.4 (cf. the proof of Theorems 1.6 and 1.9). (cid:3)
Appendix A. The ratios conjecture’s prediction
The lower order terms in the -level density for the family of quadratic twists of a given ellipticcurve E with prime conductor and even sign of the functional equation was computed by Huynh,Keating and Snaith in [HKS] using the Ratios Conjecture techniques of [CFZ] and [CS]. In thisappendix we perform the corresponding calculations in the context of our weighted family of allquadratic twists coprime to the (not necessarily prime) conductor N E of the given elliptic curve E .Throughout this section we assume the Riemann Hypothesis for all L -functions that we encounter.As in Sections 3 and 4, every error term in this section is allowed to depend on φ and w , but wenow allow an additional dependence on E . Theorem A.1.
Fix ε > . Let E be an elliptic curve defined over Q with conductor N E . Let w bea nonnegative Schwartz function on R which is not identically zero and let φ be an even Schwartzfunction on R whose Fourier transform has compact support. Assuming GRH and Conjecture A.5 (the Ratios Conjecture for our family), the -level density for the zeros of the family of L -functionsattached to the quadratic twists of E coprime to N E is given by D ( φ ; X ) = (cid:98) φ (0) LW ( X ) (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) log (cid:18) N E d (2 π ) (cid:19) + 12 π (cid:90) R φ (cid:18) tL π (cid:19) (cid:20) Γ (cid:48) (1 + it )Γ(1 + it )+ Γ (cid:48) (1 − it )Γ(1 − it ) + 2 (cid:18) − ζ (cid:48) (1 + 2 it ) ζ (1 + 2 it ) + L (cid:48) (cid:0) it, Sym E (cid:1) L (cid:0) it, Sym E (cid:1) + A α,E ( it, it ) (cid:19) − it (cid:21) dt + φ (0)2 + O ε (cid:0) X − / ε (cid:1) , where ∗ indicates that we are summing over square-free d , the functions L , W and (cid:101) w are definedby (1.6) , (1.8) and (1.10) respectively, L (cid:0) s, Sym E (cid:1) is the symmetric square L -function of E (see (A.10) ), and the function A α,E is defined by (A.14) (see also (A.7) , (A.8) , (A.12) and (A.13) ). Rewriting the rather complicated expression for the function A α,E , we obtain the following alter-native formula for the sum of the second and third terms appearing in Theorem A.1. Theorem A.2.
Fix ε > . Let E be an elliptic curve defined over Q with conductor N E , and let φ be an even Schwartz function on R whose Fourier transform has compact support. We have thefollowing expression for the sum of the second and third terms appearing in Theorem A.1: π (cid:90) R φ (cid:18) tL π (cid:19) (cid:20) Γ (cid:48) (1 + it )Γ(1 + it ) + Γ (cid:48) (1 − it )Γ(1 − it ) + 2 (cid:18) − ζ (cid:48) (1 + 2 it ) ζ (1 + 2 it ) + L (cid:48) (cid:0) it, Sym E (cid:1) L (cid:0) it, Sym E (cid:1) + A α,E ( it, it ) (cid:19) − it (cid:21) dt + φ (0)2 = − L (cid:90) ∞ (cid:32) (cid:98) φ ( x/L ) e − x − e − x − (cid:98) φ (0) e − x x (cid:33) dx − L ∞ (cid:88) (cid:96) =1 (cid:88) p (cid:0) α E ( p ) (cid:96) + β E ( p ) (cid:96) (cid:1) log pp (cid:96) (cid:98) φ (cid:18) (cid:96) log pL (cid:19) (cid:18) ψ N E ( p ) p (cid:19) − + O (cid:15) (cid:0) X − ε (cid:1) , where ψ N E is the principal Dirichlet character modulo N E and the function L is defined by (1.6) . Remark A.3.
The proof of Theorem A.1 can, with only minimal changes, be turned into a proofof the corresponding result for the weighted family of all square-free quadratic twists coprime to theconductor N E of the given elliptic curve E . We record this result (combined with Theorem A.2)here for convenience:Let E , w , φ and ε be as in Theorem A.1. Assuming GRH and Conjecture A.5, the -level densityfor the zeros of the family of L -functions attached to the square-free quadratic twists of E coprimeto N E is given by D ∗ ( φ ; X ) = (cid:98) φ (0) LW ∗ ( X ) (cid:88) ∗ ( d,N E )=1 w (cid:18) dX (cid:19) log (cid:18) N E d (2 π ) (cid:19) − L (cid:90) ∞ (cid:32) (cid:98) φ ( x/L ) e − x − e − x − (cid:98) φ (0) e − x x (cid:33) dx − L ∞ (cid:88) (cid:96) =1 (cid:88) p (cid:0) α E ( p ) (cid:96) + β E ( p ) (cid:96) (cid:1) log pp (cid:96) (cid:98) φ (cid:18) (cid:96) log pL (cid:19) (cid:18) ψ N E ( p ) p (cid:19) − + O ε (cid:0) X − / ε (cid:1) , where the functions L and W ∗ are defined by (1.6) and (1.16) respectively. Remark A.4.
We prove Theorems A.1 and A.2 for Schwartz test functions φ for which the Fouriertransforms have compact support. This is a more restricted class of test functions than is typicallyused in results based on the Ratios Conjecture. However, this class is more than sufficient for ourpurposes in the present paper. Let us also point out that even though we could, with more work,prove Theorem A.1 for a larger class of test functions, we are at present not aware of any proof OW-LYING ZEROS OF ELLIPTIC CURVE L -FUNCTIONS 25 of Theorem A.2 which avoids the assumption that the test functions φ have compactly supportedFourier transforms.A.1. Proof of Theorem A.1.
To begin, we derive the appropriate version of the Ratios Conjec-ture. Thus we consider the sum R ( α, γ ) := 1 W ( X ) (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) L (cid:0) + α, E d (cid:1) L (cid:0) + γ, E d (cid:1) . (A.1)In order to rewrite the expression for R ( α, γ ) we recall two well-known formulas. The first formulais L ( s, E d ) = ∞ (cid:88) n =1 µ E ( n ) χ d ( n ) n s , (A.2)where µ E is the multiplicative function given by µ E ( p k ) = − λ E ( p ) if k = 1 ,ψ N E ( p ) if k = 2 , if k > , and ψ N E is the principal Dirichlet character modulo N E . The second formula is the approximatefunctional equation for L ( s, E d ) : L ( s, E d ) = (cid:88) n
Let ε > and let w be a nonnegative Schwartz function on R which is notidentically zero. Let δ > and suppose that the complex numbers α and γ satisfy − δ < (cid:60) ( α ) < , X (cid:28) (cid:60) ( γ ) < and (cid:61) ( α ) , (cid:61) ( γ ) (cid:28) X − ε . Then we have that W ( X ) (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) L ( + α, E d ) L ( + γ, E d ) = Y E ( α, γ ) A E ( α, γ ) + O ε (cid:0) X − / ε (cid:1) , where Y E ( α, γ ) is defined in (A.12) and A E ( α, γ ) is defined in (A.13) . We require the family average of the logarithmic derivative of the L -functions L ( s, E d ) in ourcalculation of the -level density. Thus we differentiate the result of Conjecture A.5 with respect to α . First we define A α,E ( r, r ) := ∂∂α A E ( α, γ ) (cid:12)(cid:12)(cid:12)(cid:12) α = γ = r . (A.14) Lemma A.6.
Let ε > and let w be a nonnegative Schwartz function on R which is not identicallyzero. Suppose that r ∈ C satisfies X (cid:28) (cid:60) ( r ) < and (cid:61) ( r ) (cid:28) X − ε . Then, assuming ECRH andConjecture A.5, we have that W ( X ) (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) L (cid:48) ( + r, E d ) L ( + r, E d )= − ζ (cid:48) (1 + 2 r ) ζ (1 + 2 r ) + L (cid:48) (cid:0) r, Sym E (cid:1) L (cid:0) r, Sym E (cid:1) + A α,E ( r, r ) + O ε (cid:0) X − / ε (cid:1) . (A.15) Proof.
We have that ∂∂α Y E ( α, γ ) A E ( α, γ ) (cid:12)(cid:12)(cid:12)(cid:12) α = γ = r = − ζ (cid:48) (1 + 2 r ) ζ (1 + 2 r ) + L (cid:48) (cid:0) r, Sym E (cid:1) L (cid:0) r, Sym E (cid:1) + A α,E ( r, r ) , which gives the main term in (A.15). The fact that the error term remains the same under dif-ferentiation follows immediately from a standard argument based on Cauchy’s integral formula forderivatives. (cid:3) We stress that the error term O ε ( X − / ε ) is part of the statement of the Ratios Conjecture. Let us also pointout that the condition on the imaginary parts of α and γ is not used in the derivation of Conjecture A.5, but isincluded as a plausible (and by now standard) condition under which conjectures produced by the Ratios Conjecturerecipe are expected to hold. Proof of Theorem A.1.
We recall from (2.10) that D ( φ ; X ) = 1 W ( X ) (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) (cid:88) γ d φ (cid:18) γ d L π (cid:19) . Hence, by the argument principle, we have that D ( φ ; X ) = 1 W ( X ) (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) πi (cid:32)(cid:90) ( c ) − (cid:90) (1 − c ) (cid:33) L (cid:48) ( s, E d ) L ( s, E d ) φ (cid:18) − iL π (cid:18) s − (cid:19)(cid:19) ds (A.16)with + X < c < .For the integral on the line with real part − c , making the change of variable s → − s andrecalling that φ is even, we find that it equals W ( X ) (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) πi (cid:90) ( c ) L (cid:48) (1 − s, E d ) L (1 − s, E d ) φ (cid:18) − iL π (cid:18) s − (cid:19)(cid:19) ds. (A.17)Also, using the functional equation (1.5) and (A.6), we obtain L (cid:48) ( s, E d ) L ( s, E d ) = X (cid:48) E d ( s ) X E d ( s ) − L (cid:48) (1 − s, E d ) L (1 − s, E d ) . (A.18)Hence, from (A.17) and (A.18) and making the change of variable s = 1 / r , we have that (A.16)becomes D ( φ ; X ) = 1 W ( X ) (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) πi (cid:90) ( c − / (cid:20) L (cid:48) (1 / r, E d ) L (1 / r, E d ) − X (cid:48) E d (1 / r ) X E d (1 / r ) (cid:21) φ (cid:18) iLr π (cid:19) dr. (A.19)We bring the summation inside the integral and substitute W ( X ) (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) L (cid:48) (1 / r, E d ) L (1 / r, E d ) with the right-hand side of (A.15). Note that this substitution is a priori valid only for r with (cid:61) ( r ) < X − ε . However, since (cid:98) φ is assumed to have compact support on R , it is clear that φ (cid:0) iLr π (cid:1) is rapidly decaying as |(cid:61) ( r ) | → ∞ . This fact, together with standard estimates of the logarithmicderivative of L -functions in the half-plane (cid:60) ( s ) > (see, e.g., [IK, Thm. 5.17]), make it possibleto bound the tail of the integral in (A.19) (where we cannot apply Lemma A.6) by O ε (cid:0) X − ε (cid:1) .Furthermore, applying the same tools to bound also the tail of the integral in (A.20), we arrive at D ( φ ; X ) = 1 W ( X ) (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) πi (cid:90) ( c − / (cid:20) − ζ (cid:48) (1 + 2 r ) ζ (1 + 2 r ) + 2 L (cid:48) (cid:0) r, Sym E (cid:1) L (cid:0) r, Sym E (cid:1) +2 A α,E ( r, r ) − X (cid:48) E d (1 / r ) X E d (1 / r ) (cid:21) φ (cid:18) iLr π (cid:19) dr + O ε (cid:0) X − / ε (cid:1) . (A.20)We now move the contour of integration from (cid:60) ( r ) = c − / c (cid:48) to (cid:60) ( r ) = 0 . However, thefunction (cid:32) − ζ (cid:48) (1 + 2 r ) ζ (1 + 2 r ) + L (cid:48) (cid:0) r, Sym E (cid:1) L (cid:0) r, Sym E (cid:1) + A α,E ( r, r ) (cid:33) − X (cid:48) E d (1 / r ) X E d (1 / r ) OW-LYING ZEROS OF ELLIPTIC CURVE L -FUNCTIONS 29 has a pole at r = 0 with residue 1. Thus, by Cauchy’s Theorem, we have that D ( φ ; X ) = 1 W ( X ) (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) π (cid:90) R (cid:20) − ζ (cid:48) (1 + 2 it ) ζ (1 + 2 it ) + 2 L (cid:48) (cid:0) it, Sym E (cid:1) L (cid:0) it, Sym E (cid:1) + 2 A α,E ( it, it ) + log (cid:18) N E d (2 π ) (cid:19) + Γ (cid:48) (1 − it )Γ(1 − it ) + Γ (cid:48) (1 + it )Γ(1 + it ) − it (cid:21) φ (cid:18) tL π (cid:19) dt + 1 W ( X ) (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) πi (cid:90) ( c (cid:48) ) φ (cid:0) iLr π (cid:1) r dr + O ε (cid:0) X − / ε (cid:1) . Finally, we note that φ (0) = 12 πi (cid:90) ( c (cid:48) ) φ (cid:0) iLr π (cid:1) r dr − πi (cid:90) ( − c (cid:48) ) φ (cid:0) iLr π (cid:1) r dr = 22 πi (cid:90) ( c (cid:48) ) φ (cid:0) iLr π (cid:1) r dr, which completes the proof. (cid:3) Remark A.7.
Note that, using [Me, Lemme I.2.1], we also get the following useful formula as acorollary of equation (A.20), D ( φ ; X ) = (cid:98) φ (0) LW ( X ) (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) log (cid:18) N E d (2 π ) (cid:19) − L (cid:90) ∞ (cid:32) (cid:98) φ ( x/L ) e − x − e − x − (cid:98) φ (0) e − x x (cid:33) dx + 1 πi (cid:90) ( c (cid:48) ) (cid:20) − ζ (cid:48) (1 + 2 r ) ζ (1 + 2 r ) + L (cid:48) (cid:0) r, Sym E (cid:1) L (cid:0) r, Sym E (cid:1) + A α,E ( r, r ) (cid:21) φ (cid:18) iLr π (cid:19) dr + O ε (cid:0) X − / ε (cid:1) . A.2.
Proof of Theorem A.2.
We determine the contribution of A α,E to Theorem A.1 and RemarkA.7 by first obtaining a useful expansion for it. Note that A E ( r, r ) = 1 and hence, following [HMM],from (A.12) and (A.13) we have that A α,E ( r, r ) = (cid:88) p | N E log p λ E ( p ) p r − λ E ( p ) p r − p r − p r − ∞ (cid:88) e =1 λ E ( p e ) p e (1+2 r ) + (cid:88) p (cid:45) N E log p (cid:34) λ E ( p ) p r − λ E ( p ) p r ) + p r ) − λ E ( p ) p r + λ E ( p ) p r ) − p r ) − p r − p r − ∞ (cid:88) e =0 λ E ( p e +2 ) − λ E ( p e ) p ( e +1)(1+2 r ) + 1 p + 1 ∞ (cid:88) e =0 λ E ( p e +2 ) − λ E ( p e ) p ( e +1)(1+2 r ) (cid:35) . (A.21)We can express the logarithmic derivative of ζ ( s ) as ζ (cid:48) (1 + 2 r ) ζ (1 + 2 r ) = − (cid:88) p log p (cid:34) p r − p r (cid:35) . (A.22)As for that of L ( s, Sym E ) , by (A.10) and (A.11), we obtain L (cid:48) (cid:0) r, Sym E (cid:1) L (cid:0) r, Sym E (cid:1) = − (cid:88) p | N E log p λ E ( p ) p r − λ E ( p ) p r − (cid:88) p (cid:45) N E log p λ E ( p ) p r − λ E ( p ) p r ) + p r ) − λ E ( p ) p r + λ E ( p ) p r ) − p r ) = − (cid:88) p | N E log p ∞ (cid:88) (cid:96) =1 α E ( p ) (cid:96) p (cid:96) (1+2 r ) − (cid:88) p (cid:45) N E ∞ (cid:88) (cid:96) =1 (cid:0) α E ( p ) (cid:96) + β E ( p ) (cid:96) + 1 (cid:1) log pp (cid:96) (1+2 r ) . (A.23) Thus, we have that (A.21) equals A α,E ( r, r ) = ζ (cid:48) (1 + 2 r ) ζ (1 + 2 r ) − L (cid:48) (cid:0) r, Sym E (cid:1) L (cid:0) r, Sym E (cid:1) − (cid:88) p | N E log p ∞ (cid:88) e =1 λ E ( p e ) p e (1+2 r ) + (cid:88) p (cid:45) N E log p (cid:34) − ∞ (cid:88) e =0 λ E ( p e +2 ) − λ E ( p e ) p ( e +1)(1+2 r ) + 1 p + 1 ∞ (cid:88) e =0 λ E ( p e +2 ) − λ E ( p e ) p ( e +1)(1+2 r ) (cid:35) . (A.24)Recall that when p | N E , we have α E ( p ) = λ E ( p ) . Hence we find, using (A.22), (A.23) and theidentity λ E ( p e ) − λ E ( p e − ) = α E ( p ) e + β E ( p ) e (for p (cid:45) N E ), that (A.24) becomes A α,E ( r, r ) = − (cid:88) p | N E ∞ (cid:88) (cid:96) =1 log pp (cid:96) (1+2 r ) + (cid:88) p (cid:45) N E log pp + 1 ∞ (cid:88) (cid:96) =1 α E ( p ) (cid:96) + β E ( p ) (cid:96) p (cid:96) (1+2 r ) . Combining this with the proof of Theorem A.1 and Remark A.7, we obtain π (cid:90) R φ (cid:18) tL π (cid:19) (cid:20) Γ (cid:48) (1 + it )Γ(1 + it ) + Γ (cid:48) (1 − it )Γ(1 − it ) + 2 (cid:18) − ζ (cid:48) (1 + 2 it ) ζ (1 + 2 it ) + L (cid:48) (cid:0) it, Sym E (cid:1) L (cid:0) it, Sym E (cid:1) + A α,E ( it, it ) (cid:19) − it (cid:21) dt + φ (0)2 = − L (cid:90) ∞ (cid:32) (cid:98) φ ( x/L ) e − x − e − x − (cid:98) φ (0) e − x x (cid:33) dx + 1 πi (cid:90) ( c (cid:48) ) (cid:20) − ζ (cid:48) (1 + 2 r ) ζ (1 + 2 r ) + L (cid:48) (cid:0) r, Sym E (cid:1) L (cid:0) r, Sym E (cid:1) − (cid:88) p | N E ∞ (cid:88) (cid:96) =1 log pp (cid:96) (1+2 r ) + (cid:88) p (cid:45) N E log pp + 1 ∞ (cid:88) (cid:96) =1 α E ( p ) (cid:96) + β E ( p ) (cid:96) p (cid:96) (1+2 r ) (cid:21) φ (cid:18) iLr π (cid:19) dr. (A.25)Next we consider the term πi (cid:90) ( c (cid:48) ) (cid:20) − ζ (cid:48) (1 + 2 r ) ζ (1 + 2 r ) + L (cid:48) (cid:0) r, Sym E (cid:1) L (cid:0) r, Sym E (cid:1) − (cid:88) p | N E ∞ (cid:88) (cid:96) =1 log pp (cid:96) (1+2 r ) + (cid:88) p (cid:45) N E log pp + 1 ∞ (cid:88) (cid:96) =1 α E ( p ) (cid:96) + β E ( p ) (cid:96) p (cid:96) (1+2 r ) (cid:21) φ (cid:18) iLr π (cid:19) dr (A.26)appearing in (A.25). It follows from (A.22) and (A.23) that we can rewrite (A.26) as πi (cid:90) ( c (cid:48) ) (cid:20) − (cid:88) p log p ∞ (cid:88) (cid:96) =1 α E ( p ) (cid:96) + β E ( p ) (cid:96) p (cid:96) (1+2 r ) + (cid:88) p (cid:45) N E log pp + 1 ∞ (cid:88) (cid:96) =1 α E ( p ) (cid:96) + β E ( p ) (cid:96) p (cid:96) (1+2 r ) (cid:21) φ (cid:18) iLr π (cid:19) dr. Furthermore, making the substitution u = − iLr π , we obtain L (cid:90) C (cid:48) (cid:20) − (cid:88) p log p ∞ (cid:88) (cid:96) =1 α E ( p ) (cid:96) + β E ( p ) (cid:96) p (cid:96) e − (log p ) ( πiu(cid:96)L )+ (cid:88) p (cid:45) N E log pp + 1 ∞ (cid:88) (cid:96) =1 α E ( p ) (cid:96) + β E ( p ) (cid:96) p (cid:96) e − (log p ) ( πiu(cid:96)L ) (cid:21) φ ( u ) du, (A.27)where C (cid:48) denotes the horizontal line (cid:61) ( u ) = − Lc (cid:48) π . On C (cid:48) the summations inside the integral in(A.27) converge absolutely and uniformly on compact subsets. Hence we can interchange the order OW-LYING ZEROS OF ELLIPTIC CURVE L -FUNCTIONS 31 of integration and summation and we have that (A.27) becomes − L (cid:88) p log p ∞ (cid:88) (cid:96) =1 α E ( p ) (cid:96) + β E ( p ) (cid:96) p (cid:96) (cid:90) C (cid:48) φ ( u ) e − πiu ( (cid:96) log pL ) du + 2 L (cid:88) p (cid:45) N E log pp + 1 ∞ (cid:88) (cid:96) =1 α E ( p ) (cid:96) + β E ( p ) (cid:96) p (cid:96) (cid:90) C (cid:48) φ ( u ) e − πiu ( (cid:96) log pL ) du. Finally, we change the contour of integration from C (cid:48) to the line (cid:61) ( u ) = 0 . This is possible since weare assuming that (cid:98) φ has compact support on R and since the entire function φ ( z ) := (cid:82) R (cid:98) φ ( x ) e πixz dx satisfies the estimate | φ ( T + it ) | ≤ π | T | (cid:90) R (cid:12)(cid:12) (cid:98) φ (cid:48) ( x ) (cid:12)(cid:12) max(1 , e xLc (cid:48) ) dx, uniformly for − Lc (cid:48) π ≤ t ≤ , as T → ±∞ . We conclude that (A.26) equals − L (cid:88) p log p ∞ (cid:88) (cid:96) =1 α E ( p ) (cid:96) + β E ( p ) (cid:96) p (cid:96) (cid:98) φ (cid:18) (cid:96) log pL (cid:19) + 2 L (cid:88) p (cid:45) N E log pp + 1 ∞ (cid:88) (cid:96) =1 α E ( p ) (cid:96) + β E ( p ) (cid:96) p (cid:96) (cid:98) φ (cid:18) (cid:96) log pL (cid:19) . (A.28) Lemma A.8.
Let ε > and let p be a fixed prime. Then we have that (cid:88) ∗ ( d,N E )=1 p | d (cid:101) w (cid:18) dX (cid:19) = (cid:40) W ( X ) p +1 + O ε (cid:0) ( p XN E ) ε (cid:1) if p (cid:45) N E , . (A.29) Proof.
It follows from Lemma 2.5 (with n = p ) and Remark 2.6 that if p (cid:45) N E , then (cid:88) ∗ ( d,N E )=1 p | d (cid:101) w (cid:18) dX (cid:19) = (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) − (cid:88) ∗ ( d,N E )=1 p (cid:45) d (cid:101) w (cid:18) dX (cid:19) = W ( X ) p + 1 + O ε (cid:0) ( p XN E ) ε (cid:1) . (cid:3) Combining (A.28) and Lemma A.8, we have that (A.26) becomes − L (cid:88) p log p ∞ (cid:88) (cid:96) =1 α E ( p ) (cid:96) + β E ( p ) (cid:96) p (cid:96) (cid:98) φ (cid:18) (cid:96) log pL (cid:19) + 2 LW ( X ) ∞ (cid:88) (cid:96) =1 (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) (cid:88) p | d (cid:0) α E ( p ) (cid:96) + β E ( p ) (cid:96) (cid:1) log pp (cid:96) (cid:98) φ (cid:18) (cid:96) log pL (cid:19) + O ε ( XN E ) ε LW ( X ) (cid:88) p (cid:45) N E ∞ (cid:88) (cid:96) =1 (cid:0) α E ( p ) (cid:96) + β E ( p ) (cid:96) (cid:1) log pp (cid:96) − ε (cid:98) φ (cid:18) (cid:96) log pL (cid:19) . (A.30)From the bounds | α E ( p ) | , | β E ( p ) | ≤ and W ( X ) (cid:29) X (together with the assumption that (cid:98) φ hascompact support), we have that the error term in (A.30) is at most O ε ( X − ε ) . Finally, notingthat (cid:18) dp (cid:96) (cid:19) = (cid:40) p (cid:45) d, p | d, we find that (A.30) equals − LW ( X ) ∞ (cid:88) (cid:96) =1 (cid:88) p (cid:0) α E ( p ) (cid:96) + β E ( p ) (cid:96) (cid:1) log pp (cid:96) (cid:98) φ (cid:18) (cid:96) log pL (cid:19) (cid:88) ∗ ( d,N E )=1 (cid:101) w (cid:18) dX (cid:19) (cid:18) dp (cid:96) (cid:19) + O ε ( X − ε ) , which together with (A.25) and Lemma 2.5 and Remark 2.6 (as in the proof of Theorem 1.6)concludes the proof of Theorem A.2. References [BCDT] C. Breuil, B. Conrad, F. Diamond, R. Taylor, On the modularity of elliptic curves over Q: wild 3-adicexercises, J. Amer. Math. Soc. (2001), no. 4, 843–939.[B] A. Brumer, The average rank of elliptic curves I, Invent. Math. (1992), no. 3, 445–472.[CFZ] J. B. Conrey, D. W. Farmer, M. R. Zirnbauer, Autocorrelation of ratios of L-functions, Commun. NumberTheory Phys. (2008), no. 3, 593–636.[CS] J. B. Conrey, N. C. Snaith, Applications of the L-functions ratios conjectures, Proc. Lond. Math. Soc. (3) (2007), no. 3, 594–646.[C] J. E. Cremona, Algorithms for modular elliptic curves , second edition, Cambridge University Press, Cambridge,1997.[D] H. Davenport,
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