MOMENTS OF THE RANK OF ELLIPTIC CURVES

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1 MOMENTS OF THE RANK OF ELLIPTIC CURVES STEVEN J MILLER AND SIMAN WONG Abstract. Fix an ellitic curve E/Q, and assume the Riemann Hyothesis for the L- function LE D, s for every quadratic twist E D of E by D Z. We combine Weil s exlicit formula with techniques of Heath-Brown to derive an asymtotic uer bound for the weighted moments of the analytic rank of E D. We derive from this an uer bound for the density of low-lying zeros of LE D, s which is comatible with the random matrix models of Katz and Sarnak. We also show that for any unbounded increasing function f on R, the analytic rank and assuming in addition the Birch and Swinnerton-Dyer conjecture the number of integral oints of E D are less than fd for almost all D.. Introduction Let E be an ellitic curve over Q. The Birch and Swinnerton-Dyer conjecture redicts that the geometric rank r mw E := is equal to the analytic rank r an E := This imlies in articular the Parity Conjecture: the rank of the Mordell-Weil grou of E/Q the order at s = of the L-function LE, s. we = rmwe, where we denotes the sign of the functional equation of LE, s. Nekovár [9] shows that this follows from the finiteness of the Tate-Shafarevich grou. Denote by N E the conductor of E/Q and by E D the quadratic twist of E by an integer D. If E/Q is given by y = x 3 +Ax+B, then an equation for E D is Dy = x 3 + Ax + B. If D is square-free and is rime to N E, we have the relation [5] we D = weχ D N E, where χ D denotes the quadratic character associated to Q D. Thus among the squarefree integers D rime to N E, the Parity Conjecture imlies that half of the twists E D have odd Mordell-Weil rank, and the other half, even. Early exerimental investigations see for instance [3, 0,, 37] suggested that a ositive ortion of families of ellitic curves including the family of all curves, curves with rime conductor, one-arameter families, quadratic and cubic twists, et cetera have rank. The numerical investigations can be misleading, though, as the convergence could be on the order of the logarithm of the conductor, which is still small for the families above. Recently Watkins [35] considered the family of cubic twists studied by Zagier-Kramarz [37], and went far enough to see the 00 Mathematics Subject Classification. Primary G05 ; Secondary G40. Key words and hrases. Ellitic curve, exlicit formula, integral oint, low-lying zeros, quadratic twist, rank.

2 ercentage of the higher ranks dro, with his data suggesting the roortion of rank and higher tends to zero in the limit. On the other hand, the random matrix models of Katz and Sarnak [0, 4 and 5], [8,. 9-0], which resuos the Riemann Hyothesis RH, redict that half of the twists should have analytic rank 0, and the other half, analytic rank, whence the average analytic rank over all twists should be /. In fact, function field analogues suggest that as the conductors tend to infinity, the limiting behavior of the normalized zeros near the central oint should agree with the scaling limit of eigenvalues near one of orthogonal grous if we slit by sign of the functional equation, the even sub-family should agree with SOeven and the odd with SOodd. See [5, 0, ] for general surveys on random matrix theory, and [, 6, 7, 8,, 3, 6, 7, 30, 36] for some of the many results on ranks in ellitic curve families, as well as agreements with scaling limits of random matrix ensembles. Goldfeld seems to have been the first erson to investigate the average rank of ellitic curves in a quadratic twist family. His main tool is Weil s exlicit formula. For the rest of this aer F denotes the triangle function. F x = max0, x. The exlicit formula says that the sum over owers of traces of Frobenius of E D, weighted by F, is essentially equal to a sum of the Mellin transform of F extended over the non-trivial zeros of LE D, s. Under RH, each term of this latter sum is non-negative. Since r an E D is the order of LE D, s at s =, to bound the average analytic rank we are led to study the average of the non-archimedean side of the twisted exlicit formula. In this way, Goldfeld [3] shows that under RH, for x E,ɛ we have. D <x r an E D ɛ He also oints out that any imrovement of the constant 3.5 to a number strictly less than would imly that a ositive ortion of the twists would have analytic rank 0, a statement which at resent has been roved unconditionally only for secial classes of E. Heath-Brown [4] makes a major breakthrough by imroving Goldfeld s constant, also under RH, from 3.5 to.5, and with D restricted to twists with the same root number. This imlies that under RH, a ositive ortion of the twists of E have rank 0 and, resectively. This imrovement is a result of better control over the non-archimedean side of the twisted exlicit formula, so Heath-Brown s uer bounds are in fact uer bounds for the average of the Archimedean side. For the rest of this aer, the constants involved in any O, o and exressions are with resect to the variable x only and deend only on those arameters rinted as subscrits next to these symbols. In articular, any unadorned O, o and constants are absolute. The ellitic curve L-functions are normalized to have functional equation s s, so s = corresonds to the central oint. D <x. Theorem.. Fix a ositive, thrice continuously differentiable function W comactly suorted on /, or, /. Fix an ellitic curve E/Q, and assume RH for every

3 LE D, s. For any ositive integer k = o E log log log x, as + iτ D runs through the nontrivial zeros of LE D, s with τ D 0 we have [r an E D + sinτd log x/ ] k D W τ D log x/ x k/ log k+ x D τ D 0 [ k + + k + k + 3 k ] D + oe,w W. 3 x k/ log k+ x We now investigate consequences of Theorem.. First, fixing a number R > 0 and setting k = [R/e]+, we get the following weighted uer bound on the density of large rank twists. Corollary.. Fix an ellitic curve E/Q, and assume RH for every LE D, s. Then for any fixed R > 0 and x R, we have D D W e O R/e + o E,W e/r R/e W. x x r ane D R D D Remark.3. For k =, Theorem. is essentially due to Heath-Brown [4]. More recisely, denote by E + and E the set of square-free integers D rime to N E for which LE D, s has root numbers + and, resectively. Then Heath-Brown shows that D 3.3 r an E D W E x + o D W. x It then follows that.4.5 D E ± D E + r ane D =0 D E r ane D = D W x D W x 4 + o E D E ± D E o E D E D W, x D W. x The general outline of the roof of Theorem. follows that of Heath-Brown; secifically, we make crucial use of his smooth averaging, resulting in a better asymtotic constant in the Theorem, cf. 4. Note that Heath-Brown considered a two arameter family of ellitic curves; in general, one obtains better results the larger the family is for examle, see the larger suort M. Young [36] obtains for the -level density or the better estimates on vanishing at the central oint for two-arameter families of ellitic curves than S. J. Miller [7] obtains for one-arameter families. Our main contribution is in the handling of certain truncated multivariable sums Proosition 4.3 and in the arithmetic alications Theorem.9 and the Corollaries. In articular, for k > Theorem. and hence Corollary. can also be refined to sum over D E ± only; we can even dro the condition D, N E =, at the cost of introducing tedious congruence argument on D in the roof of Theorem.. Additionally, in Heath-Brown s analysis he only needed to study moments of the rime sums, whereas for our alications towards bounding the analytic rank we must comute the moments of the full exlicit formula. 3

4 Such refinements, however, do not imrove the lower bounds.4 and.5, so we will not ursue these issues here. From the roof of Theorem. we see that x k/ log k+ x can be relaced by x k/+ɛ for any ɛ > 0, rovided that we stiulate the o-term on the right side be deendent uon ɛ. We can then rewrite Theorem. in a more suggestive form:.6 [r an E D + sin τ D log T ] k+ɛ kw D τ D log T T D τ D 0 k+ɛ [ k + + k + ɛ + k + 3 k ] D + ɛ + oe,w,ɛ W. 3 T The factor k + ɛ in the τ D -sum is due to the fact that the asymtotic formula in.6 sums over D W x k/+ɛ. If we can rove a similar formula even just an uer bound by summing over D W x α for some fixed α, uniformly for infinitely many k, then we would be able to rove that almost all E D have analytic rank α +. The reason we need to take such a long sum is to ensure that the main term dominates the error term in.6. Now, our argument leading u to.6 is essentially otimal, excet in one ste where we estimate a difference of two terms by bounding each term; cf. Remark 4.5. Question. Can we imrove the main term in.6? Corollary. gives an uer bound for the weighted average of the multilicity of the otential zero at s = of LE D, s. This argument can be extended to count non-trivial zeros of bounded height. We begin with some notation. If E D is an even twist, then under RH the non-trivial zeros of LE D, s come in comlex conjugate airs + iτ ED,j with 0 τ ED, τ ED,. If E D is an odd twist, then LE D, s has a zero at s = ; we label the remaining zeros as + iτ ED,j with 0 τ ED, τ ED,. Finally, regardless of the arity of E D, define τ ED,j = τ ED,jlog N ED /π. Since sin x/ x is decreasing for 0 < x < π, for any fixed α > 0, if for some D E we have τ ED,3k < α/π, then for this D and for every j 3k, sinτed,jlog D / sinα/. > τ ED,jlog D / α/ We invoke Theorem., noting that D x since W is comactly suorted on /, or, /, and we get Corollary.4. Fix an ellitic curve E/Q, assume RH for every LE D, s, and let W be as in Theorem.. For any α 0, π, any integer k > 0 and x k, we have D W O + o E,W D x sin α / k W. α x D τ ED,3k<α/π D 4

5 Remark.5. To deduce the Corollary from Theorem., first note that since the summands in the left side of the theorem are non-negative, the contribution from the square-free D with xk/ log k+ x < D x k/ log k+ x < D is already included in the left side of the theorem. To run through all square-free D we can then aly a geometric series, and to remove the square-free condition, a simle sieve argument, all the while maintaining the denominator on the right side of the Corollary at the rice of scaling the numerator by a finite constant. To ut this result into context, recall that random matrix theory [9, 6.9, 7.5.5] furnishes a family of robability measures for the scaling limits of classical comact grous. For SOeven and SOodd we have the measures v+, j, v, j on R, j =,,..., with resect to which Katz and Sarnak formulate the following conjecture. Conjecture.6 Katz-Sarnak. For any integer j and any comactly suorted comlexvalue function h on R, h τ ED,j = + o E,h h dv+, j, R we D =+ we D =+ where D signifies that D runs through all square-free integers D. Similarly for v, j. As is ointed out in [0,. ], [8,. 0], this Conjecture imlies that almost all even res. odd twists of E have analytic rank 0 res.. By choosing h to be suorted on an arbitrarily small neighborhood of 0 R, this Conjecture imlies that for any fixed j and any ɛ > 0, there exists δ j ɛ > 0 so that δ j ɛ 0 as ɛ 0; and the set of square-free D for which τ ED,j < ɛ and we D = has density < δ j ɛ. In articular, for any ɛ > 0 the δ j ɛ if they exist form a non-increasing sequence that converges to 0. With resect to this formalism, Corollary.4 can be viewed as roving the existence of δ j α/π under RH instead of the full random matrix theory conjecture, such that δ j /π 0 as j. However, our resent argument does not allow us to relace sin α/α with an arbitrarily large constant by relacing α/π with an arbitrarily small number. Remark.7. If instead of sending ɛ to zero we ket ɛ fixed, we are asking about the number of normalized zeros in a given neighborhood. The answer here can also be redicted from the Katz-Sarnak conjectures; using the one-level density Goes and Miller [] have recently obtained exlicit results for one-arameter families; their calculations are similar to those by Mestre [8] and Hughes-Rudnick [5]. Remark.8. The method leading u to Theorem. and the Corollaries readily extends to twists by higher order Dirichlet characters; cf. Remark 4.4. We can also relace E by a cusidal newform of weight and trivial Nebentyus. As we mentioned before, Random Matrix Theory redicts that almost all twists of E D have analytic rank. Under RH alone we can show that the analytic rank grows slower 5

6 than any unbounded increasing function for almost all twists. This is significantly better than what can be shown for an arbitrary ellitic curve; due to Mestre [8] we know that the rank of an ellitic curve E with conductor N E is Olog N E / log log N E. Theorem.9. Let f be an unbounded increasing function on R. Fix an ellitic curve E/Q, and assume RH for every LE D, s. Then the set of integers D for which r an E D > fd has density zero. Proof. Let f be an unbounded, increasing function on R. Then r an D r an D ft/ # { T/ < D < T : r an D > fd }. D <T T/< D <T On the other hand, by Goldfeld s theorem. the left side is O E,W T. Since f is increasing and unbounded, the number of such D must be o E T, as desired. Remark.0. This roof of Corollary. is essentially due to Heath-Brown. Our original longer roof made use of the effective nature of Theorem. with resect to k. Conjectures of Lang and others giving height bounds for rational and integral oints on ellitic curves suggest that most ellitic curves have no integral oints. Thanks to Theorem.9 and the work of Silverman, we can make this recise for quadratic twist families. Let.7 E : y = x 3 + Ax + B be a quasi-minimal model for E/Q i.e. 4A 3 + 7B is minimal subject to A, B Z. Silverman [33, Theorem A] shows that there exists an absolute constant κ such that, if the j-invariant of E/Q is non-integral for δ rimes, then [ ] the number of S-integral oints.8 κ on the quasi-minimal model.7 +rmwe+δ+#s. Since.7 is quasi-minimal for E, u to a bounded ower of and 3 the Weierstrass equation.9 y = x 3 + AD x + BD 3 is quasi-minimal for E D if D is square-free. Since the j-invariant is constant in a quadratic twist family, Silverman s theorem lus Theorem.9 immediately yields the following conditional result which makes recise for quadratic twist families the heuristic above on integral oints. 3 Corollary.. Fix an ellitic curve E/Q, and assume RH and the Birch and Swinnerton- Dyer conjecture for every LE D, s. Then for any unbounded increasing function f on R, the set of integers D for which the Weierstrass equation.9 has more than fn ED integral oints has density zero. We would like to thank Joe Silverman for bringing this to our attention. 3 To see this, note that for any ellitic curve E /Q and for any rime, we have ord N E if > 3, ord 3 N E 5, and ord N E 0 [7,. 385]. 6

7 Denote by E A,B the Weierstrass equation y = x 3 + Ax + B so that 4A 3 + 7B 0, and such that there exists no rime with 4 A and 6 B. The latter condition imlies that the discriminant of this equation differs from the minimial discriminant by at most 6. Also, as HE A,B := max A /3, B / goes to infinity we cature all ellitic curves over Q. Under RH for the L-functions of all ellitic curves over Q, Brumer [] shows that as x,.0 r an E A,B.3 + o. HE A,B x HE A,B x The same argument for Theorem.9 readily yields the following result. Young [36] has imroved this, relacing.3 with 5/4; as this constant is less than, under RH we find that a ositive ercentage of curves have rank 0 or. Corollary.. Assume RH for the L-function of every ellitic curve over Q. For any unbounded, increasing function f on R, the set of ellitic curves E A,B, as ordered by the height function HA, B, for which r an E > fha, B has density zero. Lang [4,. 40] conjectures that the number of integral oints on a quasi-minimal model of any E/Q should be bounded solely in terms of r mw E D. Silverman [3,. 5] conjectures that.8 should hold for all E with no δ-deendence. This conjecture lus Corollary. would imly an analog of Corollary. for the set of all ellitic curves over Q. Added in roof. Bhargava et al. recently announced an unconditional roof that the -Selmer rank of the quadratic twists of any ellitic curve over Q is bounded by.5. Acknowledgment. We are indebted to Professor Heath-Brown for sending a coy of his rerint [4], and for showing a simler roof of Theorem.9. We would like to thank Professors Hajir, Hoffstein, Mazur, Rosen and Silverman for many useful discussions and comments, and the referee for numerous helful and detailed comments on an earlier draft. The first named author was artially suorted by NSF Grants DMS and DMS , and the second named author was artially suorted by NSF grant DMS Exlicit formula Fix a modular ellitic curve E/Q of conductor N E. Denote by a n E the n-th coefficient of LE, s. For any rime N E, denote by α E and α E the eigenvalues of the Frobenius of E/F. Define. c n E = α E m + α E m if n = m > and N E ; a E m if n = m > and N E ; 0 otherwise. Note that c E = a E. For any λ > 0, define F λ x = F x/λ, where F is the triangle function given by. F x = max0, x. Following Weil s exlicit formula, we set Φ λ u = 7 F λ xe u x dx

8 which is closely related to the Lalace transform of F λ. Note that if s = + it with t R, then sinλt/ ;. Φ λ s = λ λt/ in fact, when Res = then Φ λ is essentially the Fourier transform of F λ. As ρ = β + iτ runs through the zeros of LE, s with 0 < β <, counted with multilicity, Weil s exlicit formula [8, II.] says that Φ λ ρ := lim Φ λ ρ z.3 ρ ρ <z = log N E c me log log m F log π m λ,m m 3 m > Note that c me m/. Since F, that means c me log log m F m λ,m m 3 log log n. m/ n 3/ n> 0 F t/λ e t dt. te t The integral in.3 is O/λ, so for λ, the exlicit formula now takes the form Φ λ ρ = log N E c E log log F c E log log F + O. λ λ ρ Next, we study how the exlicit formula behaves under quadratic twists. If N E D note that N E and D need not be corime and D need not be square-free, then D.4 c E D := a E, c E D := c E. Since F, N E D log log D F c E D a E λ N E D log log F c λ E D c E N E D N E D log, log. Since log /4, for D the right side of both exressions above are /4. N E D As the summands are decreasing and there are at most + logn E D terms, the sum is bounded by /4 E log 3/4 D. +logn E D 8

9 As ρ D runs through the zeros of LE D, s with 0 < Reρ D <, the exlicit formula becomes c E log D log Φ λ ρ D = log N ED F λ ρ D c E log log.5 F + Olog 3/4 D. λ Remark.. Though it does not matter for the uroses of this aer, we note that we can imrove the error term above, relacing Olog 3/4 D with log D /. Clearly log N E D log N E D, so it suffices to bound the latter sum. This sum has +logn E D terms, and the function log t t is decreasing for t 8. Thus log logn E D O + logn E D logn E D / E log D /, as claimed. log Lemma.. We have the estimates c E D log log F λ which imly.6 = λ/4 + o E λ, a E D log log F = λ / + o E λ, λ Φ λ ρ D = log N ED + λ ρ D c E D log log F + o E λ. λ Proof. If N E D, then by.4 c E D = c E and c E D = a E; as Ls, E is a cus form, we immediately obtain c E = a E. Thus N E D c E log s = N E D a E log s N E D log s. U to the bad rimes and a term holomorhic for Rs > 3/, the two sums on the right are times the logarithmic derivative of, resectively, the Rankin-Selberg L-function of the cus form associated to E with itself, and ζs. Each of the convolution L-function and ζs has a simle ole at s =. The Tauberian theorem and trivially estimating the bad rimes now immediately imlies that both <x a E log and <x c E log are x + o E x + Olog D /. The first two claims now follow from artial summation, and the third follows from substituting the first claim into.5. Set λ = log x and define.7 β = a E log F log log x, = x k/ log k+ x. 9

10 In what follows, we will take D so that D. From now on, assume 4.8 k = o E log log log x, whence O E log 3/4 D = o E log x. Combine all these and recall that N ED N E D, we now arrive at the final form of the exlicit formula for E D, obtained by combining the last definitions of λ and β and the bound O E log 3/4 D = o E log x:.9 Φ log x ρ D logd + log x ρ D > D β + o E log x. We emhasize again that D need not be corime to N E or square-free. 3. Moments of analytic rank Below we reduce the roof of Theorem. to a weighted sum of the β s defined in.7. Define fx, D = log D + log x, Rx, D = D β ; > we will aly this to our test function F λ with λ = log x which is why we have a log x/ term above. Let W be a ositive, thrice continuously differentiable function with comact suort on /, or, /. The k-th moment of the twisted exlicit formula, weighted by W, now becomes kw D Φ log x ρ D X D ρ k D k k + r r r= D k k k r k r + o E,k r i r= i= D log D + log x D fx, D k r Rx, D r W log i x D We begin by tackling the first of the three sums on the right. + o E log x k W D D fx, D k r i Rx, D r W. Lemma 3.. For l 0, we have lw D fx, D+o E log x D = k+/ log x+o E,W log x l [ D D ] W +o W. 4 We choose this o-bound for k to simlify the exosition. The otimal choice would be that which renders the O-term in Proosition 3. to be o E log x, but such refinements have no material imact on the arithmetic alications of the main theorem. 0

11 Proof. Since W x = 0 if x, the sum in the lemma extends over D only. Thus with X := x k/, from.8 we see that l D k + / log x + o E log x lw D fx, D + o E log x D >X W D >X l k + / log x + o E log x D >X W The condition D > X can be droed at the cost of introducing a term l D lx k + / log x + o E log x W k + / log x + o E log x k/, and the lemma follows. D X W The rest of the aer is devoted to roving the following result. The roof of Theorem. makes use of the conditional estimate only; we state the unconditional result for comarison. Proosition 3.. For r > 0, we have the estimate D fx, D i Rx, D r W D = log + log x i + o E,W log x 3 + o E r/ log r x D D W + O E,W 4 r r 3 x 3r log Xk + log x r+i /Xk if r is even, = O E,W 4 r r 3 x 3r log Xk + log x r+i+ /Xk if r is odd. If we assume RH for every LE D, s, then the O-term can be imroved to E,W c r Er r+3 x r/ log Xk + log x r+i for some constant c E deending on E only. Assuming the RH-estimate, we then see that kw D Φ log k log x ρ D k + / + o E k x X D ρ k D D k k + + o r E r/ k + / + o E,W k r / 3 r D r= r even +O E,W k 4+k c k Ex k/ log Xk + log x k. D W D W From.8 we have k = o E log log log x and = x k/ log x k+, which imlies that this O-term is o E,W. We relace r even gr with the equivalent all r + r gr. D.

12 Exanding the rest of the second line above accordingly and using. for Φ note we have chosen λ to equal log x, we find [r an E D + sinτd log x/ ] k D W τ D log x/ D τ D 0 [ k + + k + k + 3 k ] D 3. + oe,w W, 3 and Theorem. follows. 4. Poisson summation In this section we adat Heath-Brown s argument to reduce Proosition 3. to a multivariable rime number theorem for ellitic curves, to be roved in Section 6. We first set notation, and then rove an auxiliary result. We define the Fourier transform by ĝy = gxe πixy dx; this normalization of the Fourier transform facilitates alying the Poisson Summation formula later. Recall W is a ositive, thrice continuously differentiable function with comact suort on /, or, /. Denote by Ŵl the Fourier transform with resect to t of W l x, t, := logt Xk + log x l W t. Note that the integral defining Ŵl makes sense since W 0 = 0. Lemma 4.. There exists a constant γ W > 0 deending on W only, so that for > and integers l 0, m 0, as t, a W t < γ W, W l t < γ W llog Xk + log x l, and Ŵl < γ W l 3 log Xk + log x l min, t 3 for all l ; x r b Ŵl x, X km t dt, t t < γ W max, l 3 log Xk + log x l 3/ Xk m min / x, r. m Proof. For the rest of this roof, γ i denotes a constant deending on W only. Since W t is zero around an oen neighborhood of 0 and since W has comact suort, 3 t W lx, t, X 3 k < γ l 3 log Xk + log x l. Aly integration by arts three times and recall that W has comact suort. We get Ŵlx, t, < γ 3 t 3 y W lx, y, X 3 k dy < γ 3 l 3 log Xk + log x l min, t 3. D

13 The same argument yields the same estimate for tŵlx, t, with a different constant note / t < / t 3 as / < t <. Consequently, [Ŵl x, X km t ], t t x, X km Xk m, + tŵl t t 5/ Ŵl x, X km, t 3/ chain rule t { γ4 l log 3 Xk + log x l Xk m 3 m t t + Xk m 3t 3/ if X < 5/ k m/t, t γ 5 l 3 log Xk + log x l m t + 5/ t 3/ if Xk m/t < < γ 6 l 3 log Xk + log x l t min 3/, m. t So if m x r, the integral in the lemma becomes < γ 7 l 3 log Xk + log x l x r t 3/ t m dt < γ 8l 3 log X k + log x l x 3r/ m. On the other hand, if m x r, then slitting the integral as m + x r m gives < γ 9 l 3 log Xk + log x l x r m / + t 3/ dt m < γ 0 l 3 log Xk + log x l m /. Take γ W to be the maximum of the γ i and the lemma follows for l > 0. The argument for l = 0 is similar and simler. Recalling the definition of Rx, D r, we have 4. D = r D D fx, D i Rx, D r W D fx, D i W,..., r> D D β β r. r Note that the rimes,..., r in the inner-sum above need not be distinct. In articular, the roduct of the quadratic symbols is a non-trivial character recisely when r is not a square, and is zero if any j divides D. We roceed accordingly. Contribution to 4. from those,..., r whose roduct is a square In this case, every rime in the r-tule aears with even multilicity, which means i r is even, and ii the roduct of quadratic characters in 4. is if every i D, and is zero 3

14 otherwise. Thus the contribution in question is 4. r β β r/,..., r/ D 0 i = r β β r/ µδ,..., r/ δ π d D fx, D i W dδ fx, dδ i W, where π = r/ and µ is the Möbius function. 5 The terms in 4. with δ = sum to 4.3 d r β β r/ fx, d i W X,..., k r/ = r β r/ d d log d + log x i W. dxk By Lemma. and Lemma 3., this is = log + log x + o E log x i /3 + oe r/ log r x d d W + o W. where the factor of r was absorbed in the / r/ factor. We now bound the contribution from the terms in 4. with δ >. As W is suorted on either, / or /,, the d-sum below can be restricted to d /δ, and we find the contribution is bounded by 4.4 r β β r/,..., r/ δ π δ> W r log + log x + o E log x d dδ fx, dδ i W i β β r/,..., r/ W r log + log x i + o E log x β β r/,..., r/ δ π δ> d /δ /δ, where in the second line we use Lemma 4.a. Recall that π = r/ with < < r. Hence δ π,δ> = δ δ + r/ /. Thus 4.4 is 3r/ log + log x + o E log x i r/ /3 + o E r/ log + log x β q β q δ π δ> r/ + o E log x i log r x. 5 The Möbius function is multilicative, and µ r = r. 4

15 Keeing in mind that D W D/ W, we see that if r is even, then the terms in 4. coming from those,..., r whose roduct is a square, is log + log x i + o E log x /3 + oe r/ d log r x + O W r/ log r x W. d Contribution to 4. from those,..., r whose roduct is not a square Set 4.5 π = r, π 0 = largest erfect square divisor of π such that π 0, π/π 0 =, π = the roduct of the distinct rime divisors of π 0, so π = π 0, π = the roduct of the distinct rime divisors of π/π 0, so π = π/π 0. For examle, if π = = , then π 0 = 3 7 4, π = 3 7 and π = 5. We write D as j + mπ π with m Z and j {0,,..., π π }. Note,..., r π > D π = j mod π π D π π π = j π π Then the contribution in question to 4. is equal to j j r β β r π,..., r π > j mod π π m= Set ez = exπiz. Alying Poisson summation gives j j r β β r π 4.6 = r,..., r π > β β r π π m= We break the analysis into cases. π m= = Ŵ i x, m π π, j j π π. j + fx, j + mπ π i mπ π W. Ŵ i x, X km Xk, e mj π π π π π π j mod π π Lemma 4.. The contribution to 4.6 from m divisible by π π is 4.7 O 3r i 3 log + log x r+i x r/. X k j π j e mj. π π π Proof. As π π m, e mj/π π =. If we didn t have the j π factor which is the case if π = then the sum over j would be zero. In general it is resent, and the j-sum is bounded by the number of numbers at most π π that share a divisor with π, which we may trivially bound by π π. 5

16 As β = β log by ae log F log log x, we see each rime is at most x, and for x we have by Hasse s bound a E. We use Lemma 4.a to trivially bound Ŵi Ŵ i x, m π π, i 3 log Xk + log x i 3 log X k + log x i π π min, i X 3 k m3, 3 m 3 where we write m as π π m. The sum over m converges, and we are left with r i 3 log Xk + log x i r log log r. r,...,r x π,π > Ignoring now the restrictions on the rimes, our contribution is bounded by r X k i3 log Xk + log x i r log. x log r x / r and thus the contri- Since k = olog log log x and r k, we have 6 x bution to 4.6 from m with π π m is bounded by r X k i3 log Xk + log x X 3 k i x / r 3r i 3 log X k + log x i x r/ /X k. The error term from Lemma 4. is significantly smaller than the other error terms which arise below. We may now assume that π π m; in articular, m 0. For l =,, set δ l = π l, m, π l = δ l π l, m = δ l n l. 6 By artial summation, log / x x log + x u log du. x / u 3/ Using u log = u + Ou/ log u see [9], there is a c such that log x/ + c / log x x in bounding the contribution of the integral, it s convenient to slit it to [, x /8 ] and [x /8, x], where on the r log x latter interval we relace / log u with 8/ log x. As r k = olog log log x, + c log x + c log x r e c log, and thus x x / r. 6

17 Since π, π =, by the Chinese remainder theorem we may write j as j = j π + j π with j {0,..., π } and j {0,..., π }. The j-sum in 4.6 is thus [ j π e mj ][ j π e mj ] π j π π π π j π π [ j = e mj ][ l e n l j ] 4.8, π π π π π δ j π l π j π j l π where we used the fact that π and π are relatively rime to relace π π with. Note that the j -sum in 4.8 is zero unless δ =. As j π = j, π we see that we have the rincial character, and the j -sum becomes a Ramanujan sum, as the Ramanujan sums are defined by Sπ, m := d π,m µ π d = e d j mod π j,π = mj While we have a negative sign in the exonential s argument, this does not matter as its resence is equivalent to taking the comlex conjugate of the Ramanujan sum; as the Ramanujan sum is real valued, we may add or remove the minus sign. Note π = δ π and δ = π, m. As π is square-free, we must have δ and π relatively rime. In articular, if d δ then d does not divide π, so in this case µπ δ /d = µπ µδ /d. Thus the j -sum is just d δ µ π δ d d = µπ d δ µ δ d. d As δ is a roduct of a subset of the r rimes, we may write δ = ν νl. Using the multilicativity of the Ramanujan sums, we find the d-sum equals ν νl = ϕδ where ϕ is Euler s totient function. Using the above, 4.8 simlifies to π = µπ π ϕδ j e nδ j π π jπ =,..., r π > π π µπ ϕδ π + i π. nδ i, π π by the standard quadratic Gauss sum calculation. Note π π µπ ϕδ δ. In the analysis below, remember m = n δ 0. Thus the terms in 4.6 where π π does not divide m contribute r β β r Ŵ i x, δ n ±n δ, δ π π π π π 4.9 r n =0,..., r π > δ π n =0 ±n δ β β r π δ π 7 π Ŵ i x, X kn., X π π π k

18 There is no contribution from any n divisible by a j which divides π because of the resence of the factor n δ π. We now estimate 4.9 in two ways, first unconditionally and then assuming RH. Unconditional Estimate Since F, we have β log 4.9 as and β = 0 if > x. Letting u = π π, we rewrite 4.0 r Ŵ i x, n u, n =0 u u,..., r π > r=u Q,..., r, n, where ±n δ Q,..., r, n := β β r. π π δ π As δ is a roduct of at most r distinct rimes, there are at most r terms in the δ -sum in Q,..., r, n. Since F vanishes outside,, we have β = 0 if > x. We use Lemma 4.a to bound Ŵi and we see that 4.9 W r n =0,..., r<x W 4 r i 3 log + log x r log log r r 3 r Xk 3 n i 3 log X 3 k + log x i i x 3r /Xk, where we trivially bounded x log with x x log x3. RH Estimate Note that if r x r, then Q,..., r, n = 0 for any n because one of the β terms vanish. In articular, the u-sum in 4.0 is a finite sum. To evaluate this u-sum we roceed by artial summation. That calls for the following estimate, to be roved in Sections 5 and 6. Proosition 4.3. Assume RH for every LE D, s, and let U x r with x 0. Then there exists a constant c E deending only on E so that, for any integers m, r > 0, as,..., r run through all rime numbers, 4. r U π > Q,..., r, n c E r r [log N E + log n + log x] r log r+ x. 8

19 Assuming this, the u-sum in 4.0 is [ = Q,..., r, n ]Ŵi x, X kn, X x r k x r 4. r x r π > x r [ r t π > ] Q,..., r, n tŵi x, n t, t dt ] r W c E r [log r N E + log n + log x log r+ x i 3 log + log x i [ min x, r 3 +, x r n Xk n min x r 3 ] n ] r E,W r r+3 c r Elog + log x [log i log r+ x n + log x, Xk n min x r 3. n Consequently, 4.0 becomes 4.3 E,W r r+3 c r Elog + log x i log n + log x r Xk, n min x r 3. n n =0 Thus the contribution to the n-sum from those n x r / is 4.4 E,W r r+3 c r Elog + log x i log n + log x r x r 3 n n n x r / E,W r r+3 c r Elog + log x i x r 3/ We now roceed to analyze 4.4. log x r+ /x r. log n + log xr n n x r /X k We claim the n -sum is bounded by O log + We slit the n -sum into two cases, n and n >, where = x k/ log k+ x. In the first case, we relace log n + log x r with log + log x r. The resulting n sum is dominated by n x r / /n, which is O /x r. Consider now n, and remember 0 < r k = olog log log x. The claim is trivial if r = k =. We may thus assume k, which imlies x so n x. We have log n + log x r r log r n, and r log log x. We are left with bounding Note n maxx,x r / log r n. n log r n n r log log n / log n, and the exonent is decreasing with increasing n. 9.

20 Assume first x r / 0. We thus have 4.5 log r n n n x r /X k n +r log logx r / logx r / n x r / x r = x r x r x r +r log logx r / logx r / ex logx r / r log logxr / logx r / logxr / r rr log r x x r log + log x r k k ; however, k k log x to see this, taking logarithms leads us to comare k log k and log log x, and log log x is clearly larger since k = olog log log x. Combining the above with the factor of r log log x log + log x roves the claim in the case x r / 0. If instead x r / 0, we argue similarly excet now the n sum starts at x instead of x r /, and we may add the factor of /x r to our bound as it is bounded below by /0. We use the above analysis to finish bounding the contribution to the n -sum from n x r /. Substituting into 4.4 yields the contribution is bounded by E,W r r+3 c r Elog + log x i+r+ x r E,W r r+3 c r Elog + log x r+i+ x r/. 3/ x r On the other hand, the contribution from those n < x r / is E,W r r+3 c r Elog + log x i r log n + log x n 0< n <x r / We argue as before. As n x r, log n + log x r r + r log r x, and from above we know r + r log log x. We thus find the contribution from these n is bounded by This comletes the roof of Proosition 3.. E,W r r+3 c r Elog + log x r+i+ x r/. Remark 4.4. The argument in this section readily extends to twists by Dirichlet characters of fixed order n >. The main difference is that the argument now roceeds according to whether r is a erfect n-th ower or not. Also, if n > then βn converges. The effect of this is that our family is now exected to agree with the scaling limits of unitary matrices, and not orthogonal matrices see [3]. The rest of the argument, including Proosition 4.3, extends with no change. Going through the whole roof, we see that for twists by characters of order n > that Theorem. holds with the new asymtotic constant k + / + o E,W k. 0

21 In terms of arithmetic, given an ellitic curve E/Q and a number field K/Q with an Abelian Galois grou of order n, the L-function of EK is equal to the roduct of all the twisted L-functions Ls, E, χ, where χ runs through all non-necessarily rimitive Dirichlet characters of orders dividing [K : Q] and of conductors dividing the Artin conductor of K/Q. So the analog of Theorem. for twists by Dirichlet characters of order n would rovide information about the average analytic ranks for a fixed ellitic curve over Q as we vary over Abelian extensions of degree n over Q. Remark 4.5. While Proosition 4.3 gives an essentially otimal bound for the size of the Q- sum, we have no control over the sign of this Q-sum as u varies. Because of that, to estimate 4. using Proosition 4.3 we are forced to ut absolute value signs everywhere. This is essentially the only lace in the roof of Theorem. where we might lose information the in 4.9 does not have any material imact on the rest of the roof. 5. A comlex rime number theorem The results in this section are ellitic curves analogs of classical estimates. As is customary, given a comlex number s we denote by σ and t its real and imaginary art, resectively. Lemma 5.. Assume the Riemann hyothesis for LE, s. Then for σ + / log x and t, we have the estimate L E, s/le, s log N E + log s + log x. For a roof, see for examle Theorem 5.7 of [6]. Lemma 5.. Assume the Riemann Hyothesis for LE, s. For 0 j log x, x E and + / log x σ, we have the estimate a E log +j 5. log j log N x s E + log s + + log x log x. <x The roof is standard, and is given in Aendix A for comleteness. From the definition of F and Lemma 5., we obtain immediately Corollary 5.3. Assume the Riemann hyothesis for LE, s. Then for + / log x σ, we have the estimate a E log log F log N s E + log s + log x. log x <x 6. Proof of Proosition 4.3 When r =, Brumer [,.3] deduces Proosition 4.3 from the exlicit formula in conjunction with an estimate of a weighted sum of zeros of LE D, s. Another essentially equivalent way is to aly the Perron formula as in the roof of the rime number theorem to the logarithmic derivative of LE, s. The exlicit formula aroach does not seem to

22 generalize to r >, but the aroach via the Perron formula does, with the key analytic estimate rovided by Corollary 5.3. We rove Proosition 4.3 in several stes. We first give an overview of the stes leading to the roof of the roosition. The difficulty in the rime sums there is that we have factors such as ±nδ π ; in other words, only some of the rimes are involved in the Legendre symbols. We want to exloit cancelation from the Legendre symbols. We are able to do that for the rimes dividing π, but not for the rimes dividing π. Fortunately the contribution from rimes dividing π is small. The reason is that these rimes occur at least twice, and <x βk = Olog k x. We break the roof into four stes, which are given below. We assume x 0 below; as we are only interested in the limit as x, this assumtion is harmless. Remember also that U x r. Ste I. Define L x E, s = <x a E log log F. s log x As F has comact suort, this is a finite sum and hence it is holomorhic for all s. A standard alication of Perron s formula see for instance [6] or [34] gives log x +i x L x E, s + r U s s ds 6. β E β r E log x. log x i x r U Corollary 5.3 shows that there is a constant, which we denote c E, /5, such that the integral is 6. c E, /5 r U / log x log r x log x +i x log x i x log N E + log s + + log x r ds s. For x 0 we have s + x; further, U / log x e r as U x r. Trivially estimating the integrand gives that 6. is c E, e r log N E + log x r log r+ x, where we gained a log x from the integral. Using this in 6., we find β E β r E c E, e r log N E + log x r 6.3 log r+ x. r U Ste II. Fix an integer m 0. With π defined as in 4.5, we claim that there is a constant c E, such that m β β r c E, r log NE + log m + log x r 6.4 log r+ x. r r U π > Remember β log if x and 0 otherwise. We first show that we may dro the condition π > at a negligible cost. To say that π = means that r is even and π = r/. While this suggests that the π = term would have a large contribution, as the character m r = m π which is if m, π =, these terms give a small contribution because each

23 rime occurs at least twice, leading to significantly smaller rime sums. Exlicitly, instead of having sums such as <x we have <x, so r U π = log m β β r r log r/ U β β r/ r/ β since β = 0 if x <x 4 log x r/, which suffices for our uroses as it is dominated by the claimed error in 6.4. Thus it suffices to study the sum in 6.4 without the additional condition π >. If N E m then a E m = a E m. If we didn t have to worry about the N E m condition, we could estimate the sum by a generalization of 6.3 the only difference being that now the conductor is N E m and not N E. We therefore relace β β m r r with β E m β r E m and control the error. We bound the error by the number of rimes dividing N E m as follows: we label the rimes so that,..., j are all the rimes dividing N E m, with j r this is the error term, and at least one rime in our list divides N E m. We denote the remaining rimes by q,..., q r j to emhasize the fact that they are relatively rime to N E m. For these rimes, we still have the character m q q r j, and we bound the contribution from these rimes sums by using 6.4 and induction. There is no harm in doing so; even though we are trying to rove 6.4, we are only using it with fewer rimes, and thus we are fine by induction note the base case is j = r, which leads to a vacuous sum. Using the above and β log, the left side of 6.4, without the π condition, is r U [ r O β E m β r E m + j=,..., j l N E m log log j j q q r j U/ j m ] β q E β qr j E. q q r j We estimate the first sum above by using a straightforward generalization of 6.3. Secifically, the bound in 6.3 deends on the conductor of the ellitic curve, N E ; as we are twisting by m, we must relace log N E with logn E m = log N E + log m. We now estimate each of the inner q-sum by using Ste I alied to the ellitic curve E m. The only change in the bound is that N E is relaced by N E m. All that remains is to bound the j-sum. We have 6.5 r j=,..., j l N E m log log j j r + j= N E m log The worst case is when N E m is a rimorial; if max denotes the largest rime, we would have max = N E m, which imlies max log = logn E m, so max logn E m. 3.

24 Using the Prime Number Theorem and Partial Summation, we have log y y tdt + y + y 4 y. y t 3/ y All together, this yields c E, e r[ log N E + log m + log x ] r log r+ x +8 r c E, e r[ log N E + log m + log x ] r log r+ x. c r E,[ log NE + log m + log x ] r log r+ x with c E, = 9ec E,, which comletes the analysis of Ste II. Ste III. 6.6 Fix an integer A 0,. We claim that there is a constant c E,3 such that r U j,a= π > m β β r r c r E,3 log NE + log m + log A + log x r log r+ x. We roceed by induction. We first consider the base case when r =. We extend the sum to be over all rimes at most U with π > which we can handle by Ste II, and bound the error from rimes dividing A. We have U,A= π > β m c E, log NE + log m + log x log 3 x + A log c E, log NE + log m + log x log 3 x + log A c E,3 log NE + log m + log A + log x log 3 x, where c E,3 = max, c E,. This gives the case r =. In general, r U j,a= π > m β β r r r +O j= j U l A = r U π > log log j j m β β r r q q r j U/ j q l,a= π > m β q β qr j. q q r j Ste III now follows from a similar analysis as in Ste II, the main difference being instead of N E m we now have A. 4

25 Ste IV. Finally we come to the roof of Proosition 4.3. We must show there is a constant c E such that r Q,..., r, n c E r [log r N E + log n + log x] log r+ x, where r U π > ±n δ Q,..., r, n = β β r. π We roceed by induction on r, the case r = being automatic in that case π = and π =. When π = which forces δ =, every rime occurs to an odd ower, and in articular m r = m π. Thus by 6.4, the sum of the Q,..., r, n terms with π = is 6.7 δ π c E, r log N E + log n + log x r log r+ x c E,3 r log N E + log n + log x r log r+ x from ulling out the and noting c E, c E,3. It remains to account for terms with π >. That haens recisely when π is exactly divisible by an even rime ower. In a slight abuse of notation, let us write r as λ q q r λ. We do not assume the different s are relatively rime to each other, nor do we assume the different q s are relatively rime to each other; however, it is imortant that the s are relatively rime to the q s, and no q rime occurs an even number of times. Then the contribution from these terms is therefore equal to z is the largest integer at most z 6.8 r/ λ= λ U β β λ q q r λ U/ λ q j, λ = π > π ±n δ β q β qr λ, π π where π and π above are defined with resect to the r-tule,,..., λ, λ, q,..., q r λ ; in articular, π, π and δ are indeendent of the q s. We switch the order of the δ and q-sums. Therefore 6.8 is bounded by r/ 6.9 β β ±n δ λ β q β qr λ. λ= λ U δ π q q r λ U/ λ q j, λ = π > We droed the / π factor as it only marginally imroves the final bound at a cost of more involved calculations, and the estimate without it suffices. We now aly 6.6 from Ste III to the q-sum. We use r λ and U/ λ for r and U in Ste III, take A = λ U = x λ/ x r/, m is ±n δ, and note δ π A U. Thus 6.9 is bounded by r/ λ= λ U β β λ c r λ E,3 δ π δ π log NE + log n + log x r/ + log x r log r λ+ x. 5 π

26 There are at most r/ choices for δ as δ π and π is the roduct of at most r/ distinct rimes; thus we may relace the δ sum with the harmless factor r/. We now turn to the sums over λ and the rimes,..., λ. Our definition of β forces each rime to be at most x. Using x / 4 log x and r r, we see that our sum is clearly dominated by r/ λ= x 4 log λ r/ c r λ E,3 log NE + log n + log x r/ + log x r log r λ+ x 8c E,3 r log N E + log n + log x r/ + log x r/ r log r+ x r/ 6c E,3 r r log N E + log n + log x r log r+ x r 64c E,3r r log N E + log n + log x r log r+ x 8c E,3 r r log N E + log n + log x r log r+ ; λ= λ= 4 λ log λ x log λ x log λ x x λ the claim follows by taking c E = 8c E,3. Aendix A. Proof of Lemma 5. Proof of Lemma 5.. We first rove the j = 0 case, and then show how arbitrary j follows by artial summation. In the arguments below σ +, and thus log x x σ e. For j = 0, we mimic the roof of the rime number theorem under the Riemann hyothesis. Note that because of our normalization for the ellitic curve L-functions that the central oint is s =, the functional equation relates s to s, and the coefficients c E see. are on the order of. Set c = / + / log x. Write s = σ + it with σ >. We extend the sum to all rime owers; as the squared and higher owers lead to convergent series, the cost of relacing < x with k < x is absorbed by the error term. Alying the Perron formula see for instance Chater 7 of [4], taking the T there to be x, a standard 6

27 argument yields 7 A. c+i x c i x n= n x L E, σ + it + ξ x ξ LE, σ + it + ξ ξ dξ n<x Λn x c min, n σ / n x log n x c n EΛn n σ+it + Λx xx σ /, where Λ denotes the usual von Mongoldt function, and the last term on the right side of A. is resent only if x is a rime ower though there is no harm in always including it as it is dominated by other error terms. Unlike [4], we have the factor n σ / in the denominator. The n σ is due to the fact that we re integrating a shifted L-function, while the n / which is really n in the numerator is due to c n E n / for n a rime ower. If n 5x or if n 3x then log x has a ositive lower bound. Thus the contribution of 4 4 n such n to the right side of A. is recall that σ > and c = / + / log x n Λn n + / log x +/ log x ζ ζ + / log x log x. We assume now that x is not a rime ower; if it is, we may relace x by x at the cost of losing at most one term in 5., and the contribution from that term may be absorbed by our error term. As x is not a rime ower, the argument in [4,. 07] shows that the contribution from those n such that 3x < n < 5x is 4 4 log x x min, x x x + log x, where x is the distance from x to the nearest rime ower. Putting everything together gives c+i x L E, σ + it + ξ x ξ c i x LE, σ + it + ξ ξ dξ c n E log n log x + log x x min,. n σ+it n<x x x x Our next ste is to estimate the integral. We are evaluating L /L at σ + it + ξ, which has real art σ + c = σ + +, which is greater than 3 + as σ >. We shift the contour log x log x and evaluate the integrand at arguments with real art +, which means shifting ξ to log x having real art σ +. The ξ-rectangle has vertices log x c ± i x, σ + log x ± i x. 7 We briefly comment on the modifications needed to Davenort s argument. First, we need to change the integration from c i, c + i to c i x, c + i x. This is easily done through contours, as the imaginary art is large where the two differ. We can thus look at the two vertical segments, each of which we shift to a vertical segment to the right this adds a horizontal segment, but by the same arguments as below the contribution here is negligible. Using Hasse s bound that a E, we see LE, s converges absolutely for Rs > 3/. As Rσ + it + ξ > 3/, we ass through no zeros or oles when shifting the contour, and are left with two vertical integrals in a region where the series exansion for L /L converges absolutely. We can interchange the integral and the sum, and argue as in Davenort to obtain an error subsumed in the error term below. 7

28 Under the Riemann Hyothesis, there are no zeros or oles inside or on this rectangle. Thus it remains to estimate the integral along the other three edges of this rectangle. The integral along the to edge is recall < σ σ+/ log x c σ+/ log x c L E, σ + it + ξ + i x LE, σ + it + ξ + i x x ξ+i x ξ + i x dξ log NE + log ξ + σ + it + i x + log x e x x dξ by Lemma 5. log N E + log s + + log x log x log N E + log s + log x. The same bound holds for the integral along the bottom edge. As for the vertical edge with real art of ξ equal to σ + with σ >, using Lemma 5. again and noting log x σ + so log x x σ+/ log x yields x x x x x 0 L E, + + it + iτ log x LE, + log N E + log x σ+/ log x + it + iτ + iτ idτ log x log x + + s + τ + log x log x log NE + log s + + log x log x dτ + τ log x x [ e log N E + log s + + log x log x log x dτ + 0 e dτ + τ log x dτ τ ] log N E + log s + + log x log x log x. Putting everything together, we find that for σ + / log x, n<x c n E log n n σ+it log x + log x x + log N E + log s + + log x log x. Since σ >, the contribution to the sum on the left side from non-rime n is m< x, which comletes the roof when j = 0. The case of general j follows immediately by artial summation. Set log m m 3/ Sx := x a E log σ+it C E,s log x, 8

29 where C E,s = log N E + log s +. Thus a E log +j log j x σ+it log j x x x logj x log j x Sx log j u dsu j x log j u log j x u C E,s log x + j log j x C E,s = C E,s log x + j log j x C E,s C E,s log x. x Sudu log j u log udu u x log j+ u j + References [] B. Bektemirov, B. Mazur, W Stein and M. Watkins, Average ranks of ellitic curves: Tension between data and conjecture. Bull. Amer. Math. Soc , [] A. Brumer, The average rank of ellitic curves I. Invent. Math [3] A. Brumer and O. McGuinness, The behavior of the Mordell-Weil grou of ellitic curves. Bull. A.M.S [4] H. Davenort, Multilicative number theory, 3rd edition, Sringer-Verlag, 000. [5] J. B. Conrey, L-Functions and random matrices. Pages in Mathematics unlimited 00 and Beyond, Sringer-Verlag, Berlin, 00. [6] J. B. Conrey, A. Pokharel, M. O. Rubinstein and M. Watkins, Secondary terms in the number of vanishings of quadratic twists of ellitic curve L-functions. In Ranks of ellitic curves and random matrix theory, ages 5 3, London Math. Soc. Lecture Note Ser. 34, Cambridge Univ. Press, Cambridge, 007. [7] C. David, J. Fearnley and H. Kisilevsky, On the vanishing of twisted L-functions of ellitic curves. Exeriment. Math , no., [8] E. Dueñez and S. J. Miller, The effect of convolving families of L-functions on the underlying grou symmetries. Proceedings of the London Mathematical Society 009, doi: 0./lms/d08. [9] P. Dusart, The k th rime is greater than kln k + ln ln k for k. Math. Comutat , no. 5, [0] S. Fermigier, Zéros des fonctions L de courbes ellitiques. Exerimental Math., 99, [] S. Fermigier, Étude exérimentale du rang de familles de courbes ellitiques sur Q. Exerimental Math. 5, 996, [] J. Goes and S. J. Miller, Towards an average version of the Birch and Swinnerton-Dyer conjecture, Journal of Number Theory 30 00, no. 0, [3] D. Goldfeld, Conjectures on ellitic curves over quadratic fields. In Number Theory, Carbondale. LNM 75, Sringer-Verlag, 979. [4] D. R. Heath-Brown, The average rank of ellitic curves IV. Duke Math. J. 004, no. 3, [5] C. Hughes and Z. Rudnick, Linear Statistics of Low-Lying Zeros of L-functions. Quart. J. Math. Oxford , [6] H. Iwaniec and E. Kowalski, Analytic Number Theory, AMS Colloquium Publications, Vol. 53, AMS, Providence, RI, 004. [7] J. H. Silverman, Advanced Toics in the Arithmetic of Ellitic Curves. Sringer-Verlag, 994. [8] N. Katz, Twisted L-functions and monodromy. Princeton University Press, 00. [9] N. Katz and P. Sarnak, Random matrices, Frobenius eigenvalues, and monodromy. AMS Colloquium Publ. 45,