Secondary terms in the number of vanishings of quadratic twists of elliptic curve L-functions

We examine the number of vanishings of quadratic twists of the L-function associated to an elliptic curve. Applying a conjecture for the full asymptotics of the moments of critical L-values we obtain a conjecture for the first two terms in the ratio of the number of vanishings of twists sorted according to arithmetic progressions.


Introduction
Let E be an elliptic curve over Q with associated L-function given by Here, ∆ is the discriminant of E, and a p = p + 1− #E(F p ), with #E(F p ) the number of points, including the point at infinity, of E over F p . L E (s) has analytic continuation to C and satisfies a functional equation [12] [11] [1] of the form where Q is the conductor of the elliptic curve E and w E = ±1. Let 1 be the L-function of the elliptic curve E d , the quadratic twist of E by the fundamental discriminant d. If (d, Q) = 1, then L E (s, χ d ) satisfies the functional equation (5) In [5] and [6] conjectures, modeled after corresponding theorems in random matrix theory, are stated concerning the distribution of values of L E (1, χ d ) with an application made to counting the number of vanishings of L E (1, χ d ). We focus on the case w E χ d (−Q) = 1, since otherwise L E (1, χ d ) is trivially equal to zero. One quantity studied concerns the ratio of the number of vanishings sorted according to residue classes mod q for a fixed prime q ∤ Q. Let be the ratio of the number of vanishings of L E (1, χ d ) sorted according to whether χ d (q) = 1 or −1.
By looking at this ratio, certain elusive and mysterious quantities that appear in the asymptotics for both the numerator and denominator cancel each other out and one is left with a precise prediction for its limit. Let A conjecture from [5] asserts that, for q ∤ Q, It is believed that this continues to hold if the set of quadratic twists is restricted to subsets such as d < 0 or d > 0, or to |d| prime, though in the latter case we must be sure to rule out there being no vanishings at all due to arithmetic reasons [7]. Numerical evidence for three elliptic curves is presented in [5] and confirms this prediction. However, even taking X of size roughly 10 9 (and, in that paper, d < 0 and |d| prime), the numeric value of the ratio was found in that paper to agree with the predicted value to about two decimal places. In other cases, when a q of L E (S) in (1) equals 0, the numeric value of R q (X) compared to the predicted limit R q to three or more decimal places.
In this paper we examine secondary terms in the above conjecture applying new conjectures [4] for the full asymptotics of the moments of L E (1, χ d ). We obtain a conjectural formula for the next to leading term in the asymptotics for R q (X). It is of size O(1/ log(X)) and explains the slow convergence to the limit R q . We also explain in Section 3 the tighter fit when a q = 0.
While the main term, R q , in the above conjecture is robust and does not depend heavily on the set of d's considered, the secondary terms are more sensitive, for example, to the residue classes of d modulo the primes that divide Q. Therefore, for simplicity we focus on the following dense collection of fundamental discriminants d. Assume that Q is squarefree and let For curves of prime conductor Q we also consider the set of fundamental discriminants These sets of discriminants are also chosen because they allow us to efficiently compute L E (1, χ d ) using a relationship to the coefficients of certain modular forms of weight 3/2 that has been worked out explicitly for many examples by Tornaria and Rodiguez-Villegas [9] (see [6] for more details).
The sets S ± (X) restrict d according to certain residue classes mod Q in the case that Q is odd and squarefree, and 4Q in the case that Q is even and squarefree.

Moments of
be the kth moment of L E (1, χ d ).
The conjecture of Conrey-Farmer-Keating-Rubinstein-Snaith [4, 4.4] says here that, for k ≥ 1, k ∈ Z, as X → ∞, where Υ k is the polynomial of degree k(k − 1)/2 given by the where the contours above enclose the poles at z j = 0 and (14) A ± k , which depends on E, is the Euler product which is absolutely convergent for k j=1 |z j | < 1/2, with, for p ∤ Q, and, for p | Q, Because we are limiting ourselves to Q squarefree (Q prime in the S + case), we have a p = ±1 when p | Q and so The r.h.s. of (12) is [4] asymptotically, as X → ∞, where The leading asymptotics given above for the moments of L E (1, χ d ) was first made in [8] and [2], though the arithmetic factor was off for primes dividing Q. One nice thing about (19) is that it makes sense for complex values of k and in [8] was conjectured to hold for ℜk > −1/2. In [5] it is shown how the conjectured asymptotics for moments can be used to obtain information concerning the distribution of values of L E (1, χ d ). That paper discusses the importance of the first pole of the r.h.s. of (21) at k = −1/2 in analyzing the number of vanishings of L E (1, χ d ).

Vanishings of L E (1, χ d ) in progressions
We fix a prime q ∤ Q and restrict d further according to residue classes mod q as follows. For λ = ±1 we set denote the number of ratio of the number of vanishings of L E (1, χ d ), with d ∈ S ± , sorted according to residue classes mod q.
To study this ratio we need to look at the moments: The conjecture in [4] then gives where Υ ± k,q,λ (x) is given by the same formula as in (13) but with a slight but important modification: the local factor corresponding to the prime q, F ± k,q , gets replaced by Similarly, in (20), the local factor at the prime q gets replaced by From this we immediately surmise several things. First, R ± q (X) which is conjectured to be, asymptotically, equal to the ratio of the residues of the two moments (25), corresponding to λ = 1 and −1, at the pole k = −1/2 should thus equal, up to leading order, Second, when a q = 0, the complete asymptotic expansion for both moments are identical up to the conjectured error of size O(X −1/2+ǫ ). The reason for this is that, in (26), if a q = 0, there is no dependence on λ. Indulging in conjectural bravado, we predict that when a q = 0 and similarly for R q (X) in (6). This fits well with our numeric data. See section 6 and also Table 1 in [5] . Third, from this formula for the moments we are able to work out, in principle, arbitrarily many terms in the asymptotic expansion of R ± q (X). Below, we describe the next to leading term in detail. It is of size O(1/ log(X)). The lower terms in the asymptotics of R ± q (X) do depend on whether we are looking at S + (X) as opposed to S − (X). This arises from the fact that the local factors F ± k,p for p | Q in equation (18) depend on whether we are looking at S + or S − . While this does not affect the main term R q , it does show up in the secondary terms.
4 Evaluating the first two terms of M ± E (X, k; q, λ) To evaluate the residue that defines Υ ± k,q,λ we need to examine the multiple Laurent series about z j = 0 of the corresponding integrand. In the numerator, we must evaluate the coefficient of factors of the Vandermonde. Therefore, in computing the residue, we only need to take terms from the series for e x k j=1 z j up to degree k 2 . From this we see that Υ ± k,q,λ (x) is a polynomial in x of degree k 2 . To obtain the leading two terms of Υ ± k,q,λ (x), i.e. those of degree k 2 and k 2 − 1 in x, we need to evaluate the constant and linear terms in the multiple Maclaurin series of the function Here A ± k (z 1 , . . . , z k ; q, λ) is the same as the function A ± k (z 1 , . . . , z k ) but with the local factor F ± k,q replaced by F ± k,q,λ . For example, the term involving x k(k−1)/2 of Υ ± k,q,λ (x) is equal to It is shown in [3] that the above equals where We also have h ± k (0; q, λ) = A ± k (0, . . . , 0; q, λ).
To compute the leading two terms of the moments we prefer to write and evaluate the constant and linear terms of Notice that the linear terms all share the same coefficient because h ± k (z; q, λ) is symmetric in the z j 's.
The constant term can be pulled out of the integral as e α ± k (q,λ) = h ± k (0; q, λ). The linear terms can be absorbed into the exp(x k j=1 z j ). Dropping the terms of degree two or higher in log h ± k (z; q, λ) we can evaluate the residue using (33): and thus find that (39) Inserting (39) into (25) and integrating, we obtain and hence Therefore, the remaining work is to compute above the coefficient β ± k (q, λ). To do so we evaluate individually the linear terms in the Maclaurin expansions of: and log A ± k (z 1 , . . . , z k ; q, λ).
First, log Γ(1 + z) = −γz + π 2 12 z 2 + . . . hence and so (42) equals (−γ + log(Q 1/2 /(2π))) z j + . . . . Next, Therefore, (43) equals We now turn to (44). The function A ± k (z 1 , . . . , z k ; q, λ) is given by (15) except that the local factor at p = q, namely F ± k,q , gets replaced by (26). To find the coefficient of z j in the Maclaurin series for we can, because the above is symmetric in the z j 's, differentiate with respect to z 1 and set all z j equal to 0. We thus find that the coefficient of z j equals 9 Next we consider the contribution from the local factor when p = q: Differentiating w.r.t. z 1 and setting all z j = 0 we find that the coefficient of z j in the Maclaurin series for log F ± k,q,λ equals log q(λa q − 2) Finally, we consider the local factor when p = q. If p | Q, we have, on taking the logarithm of (18), differentiating w.r.t. z 1 , setting all z j = 0, that the coefficient of z j in the series for log F ± k,p equals If p ∤ Q, taking the logarithm of (16), differentiating w.r.t. z 1 , and letting z j = 0, we get the coefficient of where f 1 (p) = 1 − a p /p + 1/p f 2 (p) = 1 + a p /p + 1/p.
Hence, adding all the coefficients of z j we find that β ± k (q, λ) in (37), and hence in (39), equals (58) Notice that the only dependence in β ± k (q, λ) on q is in the term 10 5 Conjecture for the first two terms in R ± q (X) Dividing M ± E (X, k; q, 1) by M ± E (X, k; q, −1), using equation (41) M ± E (X, k; q, 1) The first factor (61) Interpolating to k = −1/2 gives our conjecture: where β ± − 1 2 (q, λ) is given explicitly by equation (57). The implied constant in the remainder term depends on E and q, and thus also on a q .

Numerical Data
We verify the conjecture described above for over two thousand elliptic curves and the sets S ± E (X), with X = 10 8 . Altogether we have 2398 datasets. The curves in question and the method for computing L E (1, χ d ) are detailed in [6]. Tables of L-values can be obtained from [10].
We first depict in Figure 1 the distribution of the remainder in comparing R ± q (X) to the conjectured first and second order approximations. More precisely, for our 2398 datasets, we examine the distribution of values of and of with X = 10 8 , q ≤ 3571. We break up the horizontal axis into small bins of size .0002 and count how often the values fall within a given bin. The difference in (64) has smaller variance reflecting an overall better fit of the second order approximation compared with the first. These distributions are not Gaussian. There are yet further lower terms and these are given by complicated sums involving the Dirichlet coefficients of L E (s), and q. In the first plot of Figure 2 we depict, for one hundred of our datasets, the raw data for the values given by equation (63). The horizontal axis is q. For each q on the horizontal axis there are 100 points corresponding to the 100 values, one for each dataset, of R ± q (X) − R q , with X = 10 8 . We see the values fluctuating about zero, most of the time agreeing to within about .02. The convergence in X is predicted from the secondary term to be logarithmically slow and one gets a better fit by including the second order term. This is depicted in the second plot of Figure 2 which shows the difference given in (64). again with X = 10 8 , and the same one hundred elliptic curves E. We see an improvement to the first plot which uses just the main term. We only depict data for 100 datasets in these plots since otherwise there would be too many data points leading to a thick black mess.
Finally, a sequence of plots shows the dependence of the remainder term in the first and second order approximations on q and a q . Given an integer n, we display, in Figure 3 q v.s. R ± q (10 8 )−R q for the subset of our elliptic curves satisfying a q = n. For each of n = −20, −9, −6, −4, −3, −2, −1, 0, 1,2,3,4,6,9,20 there is one plot. Figure 4 does the same but for the values given by equation (64).
We notice several things. Overall, the plots in the Figure 4 are more symmetric about the horizontal axis reflecting a tighter fit by including the second order term. For smaller q however, incorporating the second order term leads to a correction that tends to overshoot. Compare for example the fourth plot in Figures 3 and 4. Presumably, the third and further order terms, while of size O(log(X) −2 ) can have relatively large constants for smaller q requiring one to take X larger than 10 8 to see an improvement from the second order term. This is also reflected in Tables 1-2     Values of R ± q (X) − R q , with X = 10 8 , 2 ≤ q < 500, for the subset of our elliptic curves satisfying a q = n. The blank white area on the left of the plots for larger n reflects Hasse's theorem that |a q | < 2q 1/2 which restricts how small q can be given a q = n.   Table 1: The values of R ± q (10 8 ), for the elliptic curve 11 A of conductor 11 given by y 2 + y = x 3 − x 2 − 10x − 20, compared to the conjectured first order approximation (63) and second order approximation (64).