Investigations of Zeros near the Central Point of
Elliptic Curve L-Functions
Steven J. Miller, with an appendix by Eduardo Duen˜ez
CONTENTS
1. Introduction
2. Random-Matrix Models for Families of Elliptic Curves
3. Theoretical Results
4. Experimental Results
5. Summary and Future Work
Appendix A. “Harder” Ensembles of Orthogonal Matrices
Acknowledgments
References
2000 AMS Subject Classification: Primary 11M26;
Secondary 11G05, 11G40, 11M26
Keywords: elliptic curves, low-lying zeros, n-level statistics,
random-matrix theory
We explore the effect of zeros at the central point on nearby
zeros of elliptic-curve L-functions, especially for one-parameter
families of rank r over Q. By the Birch and Swinnerton-Dyer
conjecture and Silverman’s specialization theorem, for t suffi-
ciently large the L-function of each curve Et in the family has r
zeros (called the family zeros) at the central point. We observe
experimentally a repulsion of the zeros near the central point,
and the repulsion increases with r. There is greater repulsion in
the subset of curves of rank r + 2 than in the subset of curves
of rank r in a rank-r family. For curves with comparable con-
ductors, the behavior of rank-2 curves in a rank-0 one-parameter
family over Q is statistically different from that of rank-2 curves
from a rank-2 family. In contrast to excess-rank calculations, the
repulsion decreases markedly as the conductors increase, and
we conjecture that the r family zeros do not repel in the limit.
Finally, the differences between adjacent normalized zeros near
the central point are statistically independent of the repulsion,
family rank, and rank of the curves in the subset. Specifically,
the differences between adjacent normalized zeros are statisti-
cally equal for all curves investigated with rank 0, 2, or 4 and
comparable conductors from one-parameter families of rank 0
or 2 over Q.
1. INTRODUCTION
Random-matrix theory has successfully modeled the be-
havior of the zeros and values of many L-functions; see,
for example, the excellent surveys [Keating and Snaith
03, Farmer xx]. The correspondence first appeared in
Montgomery’s analysis of the pair correlation1 of the ze-
ros of the Riemann zeta function as the zeros tend to
1If {αj}∞j=1 is an increasing sequence of numbers and B ⊂ R
n−1
is a compact box, the n-level correlations are
lim
N→∞
#
{(αj
1
− αj
2
, . . . , αj
n−1
− αj
n
) ∈ B, ji ≤ N, ji = jk}
N
.
One may replace the boxes with smooth test functions; see [Rudnick
and Sarnak 96] for details.
c
© A K Peters, Ltd.
1058-6458/2006 $ 0.50 per page
Experimental Mathematics 15:3, page 257

258 Experimental Mathematics, Vol. 15 (2006), No. 3
infinity [Montgomery 73]. Dyson noticed that Mont-
gomery’s answer, though limited to test functions sat-
isfying certain support restrictions, agrees with the pair
correlation of the eigenvalues from the Gaussian unitary
ensemble (GUE).2 Montgomery conjectured that his re-
sult holds for all correlations and all support. Again with
suitable restrictions and in the limit as the zeros tend to
infinity, Hejhal [Hejhal 94] showed that the triple correla-
tion of zeros of the Riemann zeta function agrees with the
GUE, and more generally, Rudnick and Sarnak [Rudnick
and Sarnak 96] showed that the n-level correlations of
the zeros of any principal L-function (the L-function at-
tached to a cuspidal automorphic representation of GLm
over Q) also agree with the GUE.
In this paper we explore another connection between
L-functions and random-matrix theory: the effect of mul-
tiple zeros at the central point on nearby zeros of an L-
function and the effect of multiple eigenvalues at 1 on
nearby eigenvalues in a classical compact group. Par-
ticularly interesting cases are families of elliptic-curve
L-functions. It is conjectured that zeros of primitive
L-functions are simple, except potentially at the cen-
tral point for arithmetic reasons. For an elliptic curve
E, the Birch and Swinnerton-Dyer conjecture [Birch
and Swinnerton-Dyer 63, Birch and Swinnerton-Dyer 65]
states that the rank of the Mordell–Weil group E(Q)
equals the order of vanishing of the L-function L(E, s)
at the central point s = 12 .
3 Let E be a one-parameter
family of elliptic curves over Q with (Mordell–Weil) rank
r:4
y2 = x3 + A(T )x + B(T ), A(T ), B(T ) ∈ Z[T ]. (1–1)
For all t sufficiently large, each curve Et in the family E
has rank at least r, by Silverman’s specialization theorem
[Si1verman 94]. Thus we expect each curve’s L-function
to have at least r zeros at the central point. We call the
r conjectured zeros from the Birch and Swinnerton-Dyer
conjecture the family zeros. Thus, at least conjecturally,
these families of elliptic curves offer an exciting and ac-
cessible laboratory in which we can explore the effect of
multiple zeros on nearby zeros.
The main tool for studying zeros near the central point
(the low-lying zeros) in a family is the n-level density.
2The GUE is the N →∞ scaling limit of N × N complex Her-
mitian matrices with entries independently chosen from Gaussians;
see [Mehta 91] for details.
3We normalize all L-functions to have functional equation s →
1− s, and thus the central point is at s = 1
2
.
4The group of rational function solutions (x(T ), y(T )) ∈ Q(T )2
to y2 = x3 +A(T )x+B(T ) is isomorphic to Zr ⊕T, where T is the
torsion part and r is the rank.
Let φ(x) =
∏n
i=1 φi(xi), where the φi are even Schwartz
functions whose Fourier transforms ̂φi are compactly sup-
ported. Following [Iwaniec et al. 00], we define the n-level
density for the zeros of an L-function L(s, f) by
Dn,f (φ)=
∑
j
1
,...,j
n
j
k

=±j


φ1
(
γf,j
1
log Cf
2π
)
· · ·φn
(
γf,j
n
log Cf
2π
)
,
(1–2)
where Cf is the analytic conductor of L(s, f), whose non-
trivial zeros are 12 +iγf,j . Under the generalized Riemann
hypothesis (GRH), the nontrivial zeros all lie on the criti-
cal line (s) = 12 , and thus γf,j ∈ R. Since φi is Schwartz,
note that most of the contribution is from zeros near the
central point. The analytic conductor of an L-function
normalizes the nontrivial zeros of the L-function so that
near the central point, the average spacing between nor-
malized zeros is 1; it is determined by analyzing the Γ-
factors in the functional equation of the L-function (see,
for example, [Iwaniec et al. 00]). For elliptic curves the
analytic conductor is the conductor of the elliptic curve
(the level of the corresponding weight-2 cuspidal new-
form from the modularity theorem of [Wiles 95, Taylor
and Wiles 95, Breuil et al. 01]).
We order a family F of L-functions by analytic con-
ductors. Let FN = {f ∈ F : Cf ≤ N}. The n-level
density for the family F with test function φ is
Dn,F (φ) = lim
N→∞
Dn,F
N
(φ), (1–3)
where
Dn,F
N
=
1
|FN |
∑
f∈F
N
Dn,f (φ). (1–4)
We can of course investigate other subsets. Other com-
mon choices are {f : Cf ∈ [N, 2N ]} and, for a one-
parameter family E of elliptic curves over Q, {Et ∈ E :
t ∈ [N, 2N ]}.
Let U(N) be the ensemble of N ×N unitary matrices
with Haar measure. The classical compact groups are
subensembles G(N) of U(N); the most frequently en-
countered ones are USp(2M), SO(2N), and SO(2N +1).
Katz and Sarnak’s density conjecture [Katz and Sarnak
99a, Katz and Sarnak 99b] states that as the conduc-
tors tend to infinity, the behavior of the normalized ze-
ros near the central point equals the N → ∞ scaling
limit of the normalized eigenvalues near 1 of a classical
compact group; see (1–7) for an exact statement. In the
function-field case, the corresponding classical compact
group can be identified from the monodromy group; in
the number-field case, however, the reason behind the

Miller: Investigations of Zeros near the Central Point of Elliptic Curve L-Functions 259
identification is often a mystery (see [Duen˜ez and Miller
xx]). Since the eigenvalues of a unitary matrix are of the
form eiθ, we often talk about the eigenangles θ instead
of the eigenvalues eiθ, and the eigenangle 0 corresponds
to the eigenvalue 1.
Using the explicit formula we replace the sums over
zeros in (1–2) with sums over the Fourier coefficients at
prime powers. For example, if E : y2 = x3 +Ax+B is an
elliptic curve, then assuming GRH, the nontrivial zeros
of the associated L-function
L(E, s) =
∞
∑
n=1
λE(n)n−s (1–5)
(normalized to have functional equation s → 1 − s) are
1
2 + iγ, γ ∈ R. If φ is a Schwartz test function, then the
explicit formula for L(E, s) is
∑
γ
j
φ
(
γj
log CE
2π
)
(1–6)
= ̂φ(0) + φ(0) − 2
∑
p
log p
log CE
̂φ
(
log p
log CE
)
λE(p)
√
p
− 2
∑
p
log p
log CE
̂φ
(
2 log p
log CE
)
λ2E(p)
p
+ O
(
log log CE
log CE
)
;
see, for example, [Mestre 86, Mil1er 02]. Using appropri-
ate averaging formulas and combinatorics, the resulting
prime-power sums in the n-level densities can be evalu-
ated for ̂φi of suitably restricted support. The density
conjecture is that for each family of L-functions F , for
any Schwartz test function φ : Rn → Rn,
Dn,F (φ) =
∫
φ(x)Wn,G(x) dx =
∫
̂φ(u)̂Wn,G(u) du.
(1–7)
The density kernel Wn,G(x) is determined from the N →
∞ scaling limit of the associated classical compact group
G(N); the last equality follows by Plancherel. The most
frequently occurring answers are the scaling limits of uni-
tary, symplectic, and orthogonal ensembles. For n = 1
we have
̂W1,U(u) = δ(u),
̂W1,USp(u) = δ(u)−
1
2
I(u),
̂W1,SO(even)(u) = δ(u) +
1
2
I(u), (1–8)
̂W1,SO(odd)(u) = δ(u)−
1
2
I(u) + 1,
̂W1,O(u) = δ(u) +
1
2
,
where I(u) is the characteristic function of [−1, 1]. For
arbitrarily small support, unitary and symplectic are dis-
tinguishable from each other and the orthogonal groups;
however, for test functions ̂φ supported in (−1, 1), the
three orthogonal groups agree:
∫
̂φ(u)̂W1,U(u) du = ̂φ(u),
∫
̂φ(u)̂W1,USp(u) du = ̂φ(u) −
1
2
φ(0),
∫
̂φ(u)̂W1,SO(even)(u) du = ̂φ(u) +
1
2
φ(0), (1–9)
∫
̂φ(u)̂W1,SO(odd)(u) du = ̂φ(u) +
1
2
φ(0),
∫
̂φ(u)̂W1,O(u) du = ̂φ(u) +
1
2
φ(0).
Similar results hold for the n-level densities, but below
we need only the 1-level density; see [Conrey ??, Katz
and Sarnak 99a] for the derivations of the general n-level
densities, and Appendix 5 for the 1-level density for the
orthogonal groups.
For one-parameter families of elliptic curves, the re-
sults suggest that the correct models are orthogonal
groups (if all functional equations are even then the
answer is SO(even), while if all are odd the answer is
SO(odd)). Often, instead of normalizing each curve’s ze-
ros by the logarithm of its conductor (the local rescaling),
one instead uses the average log-conductor (the global
rescaling). If we are interested in only the average rank,
it suffices to study just the 1-level density from the global
rescaling. This is because we care only about the imag-
inary parts of the zeros at the central point, and both
scalings of the imaginary part of the central point are
zero; see, for example, [Brumer 92, Goldfeld 79, Heath-
Brown 04, Michel 95, Si1verman 98, Young xx]. To date,
all results have support in (−1, 1), where (1–9) shows that
the behavior of O, SO(even), and SO(odd) are indistin-
guishable. If we want to specify a unique corresponding
classical compact group, we study the 2-level density as
well, which for arbitrarily small support suffices to distin-
guish the three orthogonal candidates. Using the global
rescaling removes many complications in the 1-level sums
but not in the 2-level sums. In fact, for the 2-level inves-
tigations the global rescaling is as difficult as the local
rescaling; see [Miller 04] for details.
Our research was motivated by investigations into
the distribution of rank in families of elliptic curves as
the conductors grow. As we see below, for the ranges
of conductors studied there is poor agreement between
elliptic-curve-rank data and the N → ∞ scaling limits

260 Experimental Mathematics, Vol. 15 (2006), No. 3
of random-matrix theory. The purpose of this research
is to show that another statistic, the distribution of the
first few zeros above the central point, converges more
rapidly.
We briefly review the excess-rank phenomenon. A
generic one-parameter family of elliptic curves over Q
has half of its functional equations even and half odd (see
[Helfgott xx] for the precise conditions for a family to be
generic). Consider such a one-parameter family of ellip-
tic curves over Q, of rank r, and assume the Birch and
Swinnerton-Dyer conjecture. It is believed that the be-
havior of the nonfamily zeros is modeled by the N →∞
scaling limit of orthogonal matrices. Thus if the density
conjecture is correct, then at the central point in the limit
as the conductors tend to infinity, the L-functions have
exactly r zeros 50% of the time, and exactly r + 1 zeros
50% of the time. Thus in the limit, half the curves have
rank r and half have rank r + 1. In a variety of families,
however, one observes5 that 30% to 40% have rank r,
about 48% have rank r +1, 10% to 20% have rank r +2,
and about 2% have rank r+3; see, for example, [Brumer
and McGuinness 91, Fermigier 92, Fermigier 96, Zagier
and Kramarz 87].
We give a representative family below; see in particu-
lar [Fermigier 96] for more examples. Consider the one-
parameter family y2 = x3 + 16Tx + 32 of rank 0 over Q.
Each range in the table below has 2000 curves (the run
time in the last column is in hours):
T -range rank 0 rank 1 rank 2 rank 3 run time
[−1000, 1000) 39.4% 47.8% 12.3% 0.6% ¡1
[1000, 3000) 38.4% 47.3% 13.6% 0.6% ¡1
[4000, 6000) 37.4% 47.8% 13.7% 1.1% 1
[8000, 10000) 37.3% 48.8% 12.9% 1.0% 2.5
[24000, 26000) 35.1% 50.1% 13.9% 0.8% 6.8
[50000, 52000) 36.7% 48.3% 13.8% 1.2% 51.8
The relative stability of the percentage of curves in a
family with rank 2 above the family with rank r naturally
leads to the question whether this persists in the limit;
it cannot persist if the density conjecture (with orthog-
onal groups) is true for all support.6 Recently, Watkins
5Actually, this is not quite true. The analytic rank is estimated
by the location of the first nonzero term in the series expansion of
L(E, s) at the central point (see [Cremona 92] for the algorithms).
For example, if the zeroth through third coefficients are smaller
than 10−5 and the fourth is 1.701, then we say that the curve has
analytic rank 4, even though it is possible (though unlikely) that
one of the first four coefficients is really nonzero. It is difficult
to prove that an elliptic-curve L-function vanishes to order two or
greater. [Goldfeld 76] and [Gross and Zagier 87] give an effective
lower bound for the class number of imaginary quadratic fields by
an analysis of an elliptic-curve L-function that is proven to have
three zeros at the central point.
6Explicitly, if the large-conductor limits of the elliptic-curve
L-functions agree with the N → ∞ scaling limits of orthogonal
groups.
[Watkins ??] investigated the family x3 + y3 = m for
varying m, and in contrast to other families, his range
of m was large enough to see the percentage with rank
r+2 markedly decrease, providing support for the density
conjecture (with orthogonal groups).
In our example above, as well as in the other families
investigated, the logarithms of the conductors are quite
small. Even in our last set the log-conductors are under
40. An analysis of the error terms in the explicit formula
suggests that the rate of convergence of quantities related
to zeros of elliptic curves is like the logarithm of the con-
ductors. It is quite satisfying when we study the first
few normalized zeros above the central point to see that
in contrast to excess rank, we see a dramatic decrease in
repulsion with modest increases in conductor.
In Section 2 we study two random-matrix ensembles
that are natural candidates to model families of elliptic
curves with positive rank. Many natural questions con-
cerning the normalized eigenvalues for these models for
finite N lead to quantities that are expressed in terms
of eigenvalues of integral equations. Our hope is that
showing the possible connections between these models
and number theory will spur interest in studying these
models and analyzing these integral equations. We as-
sume the Birch and Swinnerton-Dyer conjecture, as well
as GRH. We calculate some properties of these ensembles
in Appendix 5.
In Section 3 we summarize the theoretical results of
previous investigations, which state the following:
• For one-parameter families of rank r over Q and suit-
ably restricted test functions, as the conductors tend
to infinity the 1-level densities imply that in this re-
stricted range, the r family zeros at the central point
are independent of the remaining zeros.
If this were to hold for all test functions, then as the
conductors tend to infinity the distribution of the first
few normalized zeros above the central point would be
independent of the r family zeros.
In Section 4 we numerically investigate the first few
normalized zeros above the central point for elliptic
curves from many families of different rank. Our main
observations are these:
• The first few normalized zeros are repelled from the
central point. The repulsion increases with the num-
ber of zeros at the central point, and even in the case
in which there are no zeros at the central point there
is repulsion from the large-conductor-limit theoret-
ical prediction. This is observed for the family of

Miller: Investigations of Zeros near the Central Point of Elliptic Curve L-Functions 261
all elliptic curves, and for one-parameter families of
rank r over Q.
• There is greater repulsion in the first normalized zero
above the central point for subsets of curves of rank
2 from one-parameter families of rank 0 over Q than
for subsets of curves of rank 2 from one-parameter
families of rank 2 over Q. It is conjectured that as
the conductors tend to infinity, 0% of curves in a
family of rank r have rank r + 2 or greater. If this
is true, we are comparing a subset of zero relative
measure to one of positive measure. Since the first
set is (conjecturally) so small, it is not surprising
that to date there is no known theoretical agreement
with any random-matrix model for this case.
• In contrast to most excess-rank investigations, as the
conductors increase, the repulsion of the first few
normalized zeros markedly decreases. This supports
the conjecture that in the limit as the conductors
tend to infinity, the family zeros are independent of
the remaining normalized zeros (i.e., the repulsion
from the family zeros vanishes in the limit).
• The repulsion from additional zeros at the central
point cannot entirely be explained by collapsing
some zeros to the central point and leaving all the
other zeros alone. See in particular Remark 4.5.
• While the first few normalized zeros are repelled
from the central point, the differences between nor-
malized zeros near the central point are statistically
independent of the repulsion as well as the method
of construction. Specifically, the differences between
adjacent zeros near the central point from curves of
rank 0, 2, or 4 with comparable conductors from
one-parameter families of rank 0 or 2 over Q are
statistically equal. Thus the data suggest that the
effect of the repulsion is simply to shift all zeros by
approximately the same amount.
The numerical data are similar to those in excess-rank
investigations. While both seem to contradict the den-
sity conjecture, the density conjecture describes the lim-
iting behavior as the conductors tend to infinity. The
rate of convergence is expected to be on the order of the
logarithms of the conductors, which is under 40 for our
curves. Thus our experimental results are likely mislead-
ing as to the limiting behavior. It is quite interesting
that in contrast to most excess-rank investigations, we
can easily go far enough to see conductor-dependent be-
havior.
Thus our theoretical and numerical results, as well as
the Birch and Swinnerton-Dyer and density conjectures,
lead us to the following conjecture:
Conjecture 1.1. Consider one-parameter families of ellip-
tic curves of rank r over Q and their subfamilies of curves
with rank exactly r + k for k ∈ {0, 1, 2, . . . }. For each
subfamily there are r family zeros at the central point,
and these zeros repel the nearby normalized zeros. The
repulsion increases with r and decreases to zero as the
conductors tend to infinity, implying that in the limit the
r family zeros are independent of the remaining zeros.
If k ≥ 2, these additional nonfamily zeros at the central
point may influence nearby zeros, even in the limit as the
conductors tend to infinity. The spacings between adja-
cent normalized zeros above the central point are indepen-
dent of the repulsion; in particular, it does not depend on
r or k, but only on the conductors.
2. RANDOM-MATRIX MODELS FOR FAMILIES OF
ELLIPTIC CURVES
We want a random-matrix model for the behavior of ze-
ros from families of elliptic-curve L-functions with a pre-
scribed number of zeros at the central point. We concen-
trate on models for either one- or two-parameter families
over Q, and refer the reader to [Farmer xx] for more
on random-matrix models. Both of these families are ex-
pected to have orthogonal symmetries. Many researchers
(see, for example, [David et al. 04, Goldfeld 79, Gouve´a
and Mazur 91, Mai 93, Rubin and Silberberg 01, Rubin-
stein 01, Stewart and Top 95]) have studied families con-
structed by twisting a fixed elliptic curve by characters.
The general belief is that such twisting should lead to
unitary or symplectic families, depending on the orders
of the characters.
There are two natural models for the corresponding
situation in random-matrix theory of a prescribed num-
ber of eigenvalues at 1 in subensembles of orthogonal
groups. For ease of presentation we consider the case
of an even number of eigenvalues at 1; the odd case is
handled similarly.
Consider a matrix in SO(2N). It has 2N eigenvalues
in pairs e±iθj , with θj ∈ [0, π]. The joint probability
measure on Θ = (θ1, . . . , θN ) ∈ [0, π]N is
d0(Θ) = cN
∏
1≤j<k≤N
(cos θk − cos θj)2
∏
1≤j≤N
dθj , (2–1)
where cN is a normalization constant such that d0(Θ)
integrates to 1. From (2–1) we can derive all quantities

262 Experimental Mathematics, Vol. 15 (2006), No. 3
of interest on the random-matrix side; in particular, n-
level densities, distribution of first normalized eigenvalue
above 1 (or eigenangle above 0), and so forth.
We now consider two models for subensembles of
SO(2N) with 2r eigenvalues at 1, and the N → ∞ scaling
limit of each.
Independent Model: The subensemble of SO(2N)
with the upper left block a 2r × 2r identity matrix. The
joint probability density of the remaining N − r pairs is
given by
dε2r,Indep(Θ) = c2r,Indep,N
∏
1≤j<k≤N−r
(cos θk − cos θj)2
×
∏
1≤j≤N−r
dθj . (2–2)
Thus the ensemble is matrices of the form
{(
I2r×2r
g
)
: g ∈ SO(2N − 2r)
}
; (2–3)
the probabilities are equivalent to choosing g with re-
spect to Haar measure on SO(2N − 2r). We call this
the independent model, since the forced eigenvalues at 1
from the I2r×2r block do not interact with the eigenval-
ues of g. In particular, the distribution of the remaining
N −r pairs of eigenvalues is exactly that of SO(2N −2r);
this block’s N → ∞ scaling limit is just SO(even). See
[Conrey ??, Katz and Sarnak 99a] as well as Appendix 5.
Interaction Model: The subensemble of SO(2N) with
2r of the 2N eigenvalues equaling 1:
dε2r,Inter(Θ) = c2r,Inter,N
∏
1≤j<k≤N−r
(cos θk − cos θj)2
×
∏
1≤j≤N−r
(1 − cos θj)2rdθj . (2–4)
We call this the interaction model because the forced
eigenvalues at 1 do affect the behavior of the other eigen-
values near 1. Note here that we condition on all SO(2N)
matrices with at least 2r eigenvalues equal to 1. The
(1− cos θj)2r factor results in the forced eigenvalues at 1
repelling the nearby eigenvalues.
Remark 2.1. Since the calculations for the local statis-
tics near the eigenvalue at 1 in the interaction model
have not appeared in print, Appendix 5 (written by Ed-
uardo Duen˜ez) provides a derivation of formula (2–4) (see
especially Section A.2), as well as the relevant integral
(Bessel) kernels dictating such statistics. See also [Snaith
05] for the value distribution of the first nonzero deriva-
tive of the characteristic polynomials of this ensemble.
While both models have at least 2r eigenvalues equal
to 1, they are very different subensembles of SO(2N),
and they have distinct limiting behavior (see also Remark
3.1). We can see this by computing the 1-level density
for each, and comparing with (1–9). Letting ̂W1,SO(even)
(respectively ̂W1,SO(even),Indep,2r and ̂W1,SO(even),Inter,2r)
denote the Fourier transform of the kernel for the 1-level
density of SO(even) (respectively, of the independent
model for the subensemble of SO(even) with 2r eigenval-
ues at 1 and of the interaction model for the subensemble
of SO(even) with 2r eigenvalues at 1), we find in Ap-
pendix 5 that
̂W1,SO(even)(u) = δ(u) +
1
2
I(u),
̂W1,SO(even),Indep,2r(u) = δ(u) +
1
2
I(u) + 2, (2–5)
̂W1,SO(even),Inter,2r(u) = δ(u) +
1
2
I(u) + 2
+ 2(|u| − 1)I(u).
Since I(u) is positive for |u| < 1, note that the density
is smaller for |u| < 1 in the interaction model versus the
independent model. We can interpret this as a repulsion
of zeros, as the following heuristic shows (though see Ap-
pendix 5 for proofs). We compare the 1-level density of
zeros from curves with and without repulsion, and show
that for a positive decreasing test function, the 1-level
density is smaller when there is repulsion.
Consider two elliptic curves, E of rank 0 and conduc-
tor CE and E′ of rank r and conductor CE′ . Assume
CE ≈ CE′ ≈ C, and assume GRH for both L-functions.
If the curve E has rank 0 then we expect the first zero
above the central point, 12 + iγE,1, to have γE,1 ≈
1
log C .
For Er, if the r family zeros at the central point repel,
it is reasonable to posit a repulsion of size brlog C for some
br > 0. This is because the natural scale for the distance
between the low-lying zeros is 1log C , so we are merely
positing that the repulsion is proportional to the dis-
tance. We assume that all zeros are repelled equally;
evidence for this is provided in Section 4.6. Thus for E′
(the repulsion case) we assume γE′,j ≈ γE,j + brlog C . We
can detect this repulsion by comparing the 1-level densi-
ties of E and E′. Take a nonnegative decreasing Schwartz
test function φ. The difference between the contribution

Miller: Investigations of Zeros near the Central Point of Elliptic Curve L-Functions 263
from the jth zero of each is
φ
(
γE′,j
log C
2π
)
− φ
(
γE,j
log C
2π
)
≈ φ
(
γE,j
log C
2π
+
br
2π
)
− φ
(
γE,j
log C
2π
)
(2–6)
≈ φ′
(
γE,j
log C
2π
)
·
br
2π
.
Since ̂φ is decreasing, its derivative is negative and thus
the above shows that the 1-level density for the zeros
from E′ (assuming repulsion) is smaller than the 1-level
density for zeros from E. Thus the lower 1-level density
in the interaction model versus the independent model
can be interpreted as a repulsion; however, this repulsion
can be shared among several zeros near the central point.
In fact, the observations in Section 4.6 suggest that the
repulsion shifts all normalized zeros near the central point
approximately equally.
3. THEORETICAL RESULTS
Consider a one-parameter family of elliptic curves of rank
r over Q. We summarize previous investigations of the
effect of the (conjectured) r family zeros on the other
zeros near the central point. For convenience we state
the results for the global rescaling, though similar results
hold for the local rescaling (under slightly more restric-
tive conditions; see [Miller 04] for details). For small sup-
port, the 1- and 2-level densities agree with the scaling
limits of
(
Ir×r
O(N)
)
,
(
Ir×r
SO(2N)
)
,
and
(
Ir×r
SO(2N + 1)
)
,
depending on whether the signs of the functional equa-
tion are equidistributed or all the signs are even or all the
signs are odd. The 1- and 2-level densities provide evi-
dence in support of the Katz–Sarnak density conjecture
for test functions whose Fourier transforms have small
support (the support is computable and depends on the
family). See [Mil1er 02] for the calculations with the
global rescaling, though the result for the 1-level density
is implicit in [Si1verman 98]. Similar results are observed
for two-parameter families of elliptic curves in [Mil1er
02, Young xx].
While the above results are consistent with the Birch
and Swinnerton-Dyer conjecture that each curve’s L-
function has at least r zeros at the central point, it is
not a proof (even in the limit) because our supports are
finite. For families with t ∈ [N, 2N ] the errors are of
size O( 1log N ) or O(
log log N
log N ). Thus for large N we cannot
distinguish a family with exactly r zeros at the central
point from a family in which each Et has exactly r zeros
at ±(log Ct)−2006.
For one-parameter families of elliptic curves over Q,
in the limit as the conductors tend to infinity the fam-
ily zeros (those arising from our belief in the Birch and
Swinnerton-Dyer conjecture) appear to be independent
of the other zeros. Equivalently, if we remove the contri-
butions from the r family zeros, for test functions with
suitably restricted support the spacing statistics of the
remaining zeros agree perfectly with the standard or-
thogonal groups O, SO(even), and SO(odd), and it is
conjectured that these results should hold for all sup-
port. Thus the n-level density arguments support the
independent over the interaction model when we study
all curves in a family; however, these theoretical argu-
ments do not apply if we study the subfamily of curves
of rank r + k (k ≥ 2) in a rank-r one-parameter family
over Q.
Remark 3.1. It is important to note that our theoretical
results are for the entire one-parameter family. Specif-
ically, consider the subset of curves of rank r + 2 from
a one-parameter family of rank r over Q. If the den-
sity conjecture (with orthogonal groups) is true, then in
the limit, 0% of curves are in this subfamily. Thus these
curves may behave differently without contradicting the
theoretical results for the entire family. Situations in
which the behavior of subensembles is different from that
of the entire ensemble are well known in random-matrix
theory. For example, to any simple graph we may attach
a real symmetric matrix, its adjacency matrix, for which
aij = 1 if there is an edge connecting vertices i and j, and
0 otherwise. The adjacency matrices of d-regular graphs
are a thin subensemble of real symmetric matrices with
entries independently chosen from {−1, 0, 1}. The densi-
ties of normalized eigenvalues in the two cases are quite
different, given by Kesten’s measure [McKay 81] for d-
regular graphs and Wigner’s semicircle law [Mehta 91]
for the real symmetric matrices.
It is an interesting question to determine the appro-
priate random-matrix model for rank-(r + 2) curves in a
rank-r one-parameter family over Q, both in the limit of
large conductors as well as for finite conductors. We ex-
plore this issue in greater detail in Sections 4.3 through
4.6, where we compare the behavior of rank-2 curves from

264 Experimental Mathematics, Vol. 15 (2006), No. 3
rank-0 one-parameter families over Q to that of rank-2
curves from rank-2 one-parameter families over Q.
4. EXPERIMENTAL RESULTS
We investigate the first few normalized zeros above the
central point. We used Michael Rubinstein’s L-function
calculator [Rubinstein ??] to determine the zeros. The
program does a contour integral to ensure that we have
found all the zeros in a region, which is essential in stud-
ies of the first zero! See [Rubinstein 98] for a descrip-
tion of the algorithms. The analytic ranks were found
(see Footnote 1) by determining the values of the L-
functions and their derivatives at the central point by
the standard series expansion; see [Cremona 92] for the
algorithms. Some of the programs and all of the data
(minimal model, conductor, discriminant, sign of the
functional equation, first nonzero Taylor coefficient from
the series expansion at the central point, and the first
three zeros above the central point) are available online
at http://www.math.brown.edu/∼sjmiller/repulsion.
We study several one-parameter families of elliptic
curves over Q. Since all of our families are rational sur-
faces,7 Rosen and Silverman’s result that the weighted
average of fibral Frobenius trace values determines the
rank over Q (see [Rosen and Silverman 98]) is applicable,
and evaluating simple Legendre sums suffices to deter-
mine the rank. We mostly use one-parameter families
from Fermigier’s tables [Fermigier 96], though see [Arms
et al. xx] for how to use the results of [Rosen and Silver-
man 98] to construct additional one-parameter families
with rank over Q.
We cannot obtain a decent number of curves with ap-
proximately equal log-conductors by considering a soli-
tary one-parameter family. The conductors in a family
typically grow polynomially in t. The number of Fourier
coefficients needed to study a value of L(s,Et) on the
critical line is of order
√
Ct log Ct (Ct is the conductor of
Et), and we must then additionally evaluate numerous
special functions. We can readily calculate the needed
quantities up to conductors of size 1011, which usually
translates to just a few curves in a family. We first stud-
ied all elliptic curves (parameterized with more than one
parameter), found the minimal models, and then sorted
by conductor. We then studied several one-parameter
7An elliptic surface y2 = x3 + A(T )x + B(T ), A(T ), B(T ) ∈
Z[T ], is a rational surface if and only if one of the following is true:
(a) 0 < max{3degA, 2degB} < 12; (b) 3degA = 2degB = 12 and
ordT=0T 12∆(T−1) = 0.
families, amalgamating data from different families if the
curves had the same rank and similar log-conductor.
Remark 4.1. Amalgamating data from different one-
parameter families warrants some discussion. We expect
that the behavior of zeros from curves with similar con-
ductors and the same number of zeros and family zeros
at the central point should be approximately equal. In
other words, we hope that curves with the same rank
and approximately equal conductors from different one-
parameter families of the same rank r over Q behave
similarly, and we may regard the different one-parameter
families of rank r over Q as different measurements of this
universal behavior. This is similar to numerical investi-
gations of the spacings of energy levels of heavy nuclei;
see, for example, [Harvey and Hughes 58, Haq et al. 82].
In studying the spacings of these energy levels, there were
very few (typically between 10 and 100) levels for each
nucleus. The belief is that nuclei with the same angular
momentum and parity should behave similarly. The re-
sulting amalgamations often have thousands of spacings
and excellent agreement with random-matrix predictions.
As with the excess-rank phenomenon, we found dis-
agreement between the experiments and the predicted
large-conductor limit; however, we believe that this dis-
agreement is due to the fact that the logarithms of
the conductors investigated are small. In Sections 4.2
through 4.5 we find that for curves with zeros at the
central point, the first normalized zero above the central
point is repelled, and the greater the number of zeros at
the central point, the greater the repulsion. However, the
repulsion decreases as the conductors increase. Thus the
repulsion is probably due to the small conductors, and
in the limit the independent model (which agrees with
the function-field analogue and the theoretical results of
Section 3) should correctly describe the first normalized
zero above the central point in curves of rank r in fam-
ilies of rank r over Q. It is not known what the correct
model is for curves of rank r + 2 in a family of rank r
over Q, though it is reasonable to conjecture that it is the
interaction model with the sizes of the matrices related
to the logarithms of the conductors. Keating and Snaith
[Keating and Snaith 00, Keating and Snaith 03] showed
that to study zeros at height T it is better to look at
N × N matrices, with N = log T , than to look at the
N → ∞ scaling limit. A fascinating question is to de-
termine the correct finite-conductor analogue for the two
different cases here. Interestingly, we see in Section 4.6
that the spacings between adjacent normalized zeros is

Miller: Investigations of Zeros near the Central Point of Elliptic Curve L-Functions 269
Family Median µ˜ Mean µ StDev σ
µ
log(conductor) No.
1: [0, 1, 1, 1, T ] 1.28 1.33 0.26 [3.93, 9.66] 7
2: [1, 0, 0, 1, T ] 1.39 1.40 0.29 [4.66, 9.94] 11
3: [1, 0, 0, 2, T ] 1.40 1.41 0.33 [5.37, 9.97] 11
4: [1, 0, 0,−1, T ] 1.50 1.42 0.37 [4.70, 9.98] 20
5: [1, 0, 0,−2, T ] 1.40 1.48 0.32 [4.95, 9.85] 11
6: [1, 0, 0, T, 0] 1.35 1.37 0.30 [4.74, 9.97] 44
7: [1, 0, 1,−2, T ] 1.25 1.34 0.42 [4.04, 9.46] 10
8: [1, 0, 2, 1, T ] 1.40 1.41 0.33 [5.37, 9.97] 11
9: [1, 0,−1, 1, T ] 1.39 1.32 0.25 [7.45, 9.96] 9
10: [1, 0,−2, 1, T ] 1.34 1.34 0.42 [3.26, 9.56] 9
11: [1, 1,−2, 1, T ] 1.21 1.19 0.41 [5.73, 9.92] 6
12: [1, 1,−3, 1, T ] 1.32 1.32 0.32 [5.04, 9.98] 11
13: [1,−2, 0, T, 0] 1.31 1.29 0.37 [4.73, 9.91] 39
14: [−1, 1,−3, 1, T ] 1.45 1.45 0.31 [5.76, 9.92] 10
All Curves 1.35 1.36 0.33 [3.26, 9.98] 209
Distinct Curves 1.35 1.36 0.33 [3.26, 9.98] 196
TABLE 1. First normalized zero above the central point for 14 one-parameter families of elliptic curves of rank 0 over Q
(smaller conductors).
Family Median µ˜ Mean µ StDev σ
µ
log(conductor) No.
1: [0, 1, 1, 1, T ] 0.80 0.86 0.23 [15.02, 15.97] 49
2: [1, 0, 0, 1, T ] 0.91 0.93 0.29 [15.00, 15.99] 58
3: [1, 0, 0, 2, T ] 0.90 0.94 0.30 [15.00, 16.00] 55
4: [1, 0, 0,−1, T ] 0.80 0.90 0.29 [15.02, 16.00] 59
5: [1, 0, 0,−2, T ] 0.75 0.77 0.25 [15.04, 15.98] 53
6: [1, 0, 0, T, 0] 0.75 0.82 0.27 [15.00, 16.00] 130
7: [1, 0, 1,−2, T ] 0.84 0.84 0.25 [15.04, 15.99] 63
8: [1, 0, 2, 1, T ] 0.90 0.94 0.30 [15.00, 16.00] 55
9: [1, 0,−1, 1, T ] 0.86 0.89 0.27 [15.02, 15.98] 57
10: [1, 0,−2, 1, T ] 0.86 0.91 0.30 [15.03, 15.97] 59
11: [1, 1,−2, 1, T ] 0.73 0.79 0.27 [15.00, 16.00] 124
12: [1, 1,−3, 1, T ] 0.98 0.99 0.36 [15.01, 16.00] 66
13: [1,−2, 0, T, 0] 0.72 0.76 0.27 [15.00, 16.00] 120
14: [−1, 1,−3, 1, T ] 0.90 0.91 0.24 [15.00, 15.99] 48
All Curves 0.81 0.86 0.29 [15.00, 16.00] 996
Distinct Curves 0.81 0.86 0.28 [15.00, 16.00] 863
TABLE 2. First normalized zero above the central point for 14 one-parameter families of elliptic curves of rank 0 over Q
(larger conductors).
pothesis, the number of minus signs is a random variable
from a binomial distribution with N = 21 and θ = 12 .
The probability of observing four or fewer minus signs
is about 3.6%, supporting the claim of decreasing repul-
sion with increasing conductor. For the medians there
are seven minus signs out of twenty-one; the probabil-
ity of seven or fewer minus signs is about 9.4%. Every
time the smaller-conductor set had the lesser mean, it
also had the lesser median; the mean and median tests
are not independent.
4.4 One-Parameter Families of Rank 2
over the Rationals
4.4.1 The Family y2 = x3 − T 2x + T 2. We study
the first normalized zero above the central point for
69 rank-2 elliptic curves from the one-parameter fam-
ily y2 = x3 − T 2x + T 2 of rank 2 over Q. There are
35 curves with log(cond) ∈ [7.8, 16.1] in Figure 9 and 34
with log(cond) ∈ [16.2, 23.3] in Figure 10. Unlike the
previous examples in which we chose many curves of the
same rank from different families, here we have just one
family. Since the conductors grow rapidly, we have far
fewer data points, and the range of the log-conductors is
much greater. However, even for such a small sample, the
repulsion decreases with increasing conductors, and the
shape begins to approach the conjectured distribution.
The pooled and unpooled two-sample t-procedures give
t-statistics over 5 with over 60 degrees of freedom, and
we may use the central limit theorem. Since the proba-
bility of a z-value of 5 or more is less than 10−4%, the
data do not support the null hypothesis (i.e., the data
support our conjecture that the repulsion decreases as
the conductors increase).
4.4.2 Rank-2 Curves. We consider 21 one-parameter
families of rank 2 over Q, and investigate curves of rank
2 in these families. The families are from [Fermigier 96].
We again amalgamated the different families, and sum-
marize the results in Table 4.
The difference between these experiments and those of
Section 4.3.2 is that while both deal with one-parameter
families over Q, here we study curves of rank 2 from
families of rank 2 over Q; earlier we studied curves of rank

Miller: Investigations of Zeros near the Central Point of Elliptic Curve L-Functions 271
Family Mean Standard Deviation log(conductor) No.
1: [1, T, 0,−3− 2T, 1] 1.91 0.25 [15.74, 16.00] 2
2: [1, T,−19,−T − 1, 0] 1.57 0.36 [15.17, 15.63] 4
3: [1, T, 2,−T − 1, 0] 1.29 [15.47, 15.47] 1
4: [1, T,−16,−T − 1, 0] 1.75 0.19 [15.07, 15.86] 4
5: [1, T, 13,−T − 1, 0] 1.53 0.25 [15.08, 15.91] 3
6: [1, T,−14,−T − 1, 0] 1.69 0.32 [15.06, 15.22] 3
7: [1, T, 10,−T − 1, 0] 1.62 0.28 [15.70, 15.89] 3
8: [0, T, 11,−T − 1, 0] 1.98 [15.87, 15.87] 1
9: [1, T,−11,−T − 1, 0]
10: [0, T, 7,−T − 1, 0] 1.54 0.17 [15.08, 15.90] 7
11: [1, T,−8,−T − 1, 0] 1.58 0.18 [15.23, 25.95] 6
12: [1, T, 19,−T − 1, 0]
13: [0, T, 3,−T − 1, 0] 1.96 0.25 [15.23, 15.66] 3
14: [0, T, 19,−T − 1, 0]
15: [1, T, 17,−T − 1, 0] 1.64 0.23 [15.09, 15.98] 4
16: [0, T, 9,−T − 1, 0] 1.59 0.29 [15.01, 15.85] 5
17: [0, T, 1,−T − 1, 0] 1.51 [15.99, 15.99] 1
18: [1, T,−7,−T − 1, 0] 1.45 0.23 [15.14, 15.43] 4
19: [1, T, 8,−T − 1, 0] 1.53 0.24 [15.02, 15.89] 10
20: [1, T,−2,−T − 1, 0] 1.60 [15.98, 15.98] 1
21: [0, T, 13,−T − 1, 0] 1.67 0.01 [15.01, 15.92] 2
All Curves 1.61 0.25 [15.01, 16.00] 64
TABLE 4. First normalized zero above the central point for 21 one-parameter families of rank 2 over Q with log(cond) ∈
[15, 16] and t ∈ [0, 120]. The median of the first normalized zero of the 64 curves is 1.64.
Curves / Family Median Mean Std. Dev. No.
Rank 2 / Rank 0 over Q 1.926 1.936 0.388 701
Rank 2 / Rank 2 over Q 1.642 1.610 0.247 64
TABLE 5. First normalized zero above the central point.
The first family is the 7014 rank-2 curves from the 21 one-
parameter families of rank 0 over Q from Table 3 with
log(cond) ∈ [15, 16]; the second family is the 64 rank-2
curves from the 21 one-parameter families of rank 2 over
Q from Table 4 with log(cond) ∈ [15, 16].
range) depends on how we choose the curves. For the
range of conductors studied, rank-2 curves from rank-0
one-parameter families over Q do not behave the same as
rank-2 curves from rank-2 one-parameter families over Q.
4.6 Spacings between Normalized Zeros
For finite conductors, even when there are no zeros at the
central point, the first normalized zero above the central
point is repelled from the predicted N →∞ scaling lim-
its. The repulsion increases with the number of zeros at
the central point and decreases with increasing conduc-
tor. For an elliptic curve E, let z1, z2, z3, . . . denote the
imaginary parts of the normalized zeros above the cen-
tral point. We investigate whether zj+1 − zj depends on
the repulsion from the central point.
We consider the following two sets of curves in Table 6:
• the 863 distinct rank-0 curves with log(cond) ∈
[15, 16] from the 14 one-parameter families of rank 0
over Q from Table 2;
• the 701 distinct rank-2 curves with log(cond) ∈
[15, 16] from the 21 one-parameter families of rank 0
over Q from Table 3.
863
Rank-0
Curves
701
Rank-2
Curves
t-Statistic
Median z
2
− z
1
1.28 1.30
Mean z
2
− z
1
1.30 1.34 −1.60
StDev z
2
− z
1
0.49 0.51
Median z
3
− z
2
1.22 1.19
Mean z
3
− z
2
1.24 1.22 0.80
StDev z
3
− z
2
0.52 0.47
Median z
3
− z
1
2.54 2.56
Mean z
3
− z
1
2.55 2.56 −0.38
StDev z
3
− z
1
0.52 0.52
TABLE 6. Spacings between normalized zeros. All curves
have log(cond) ∈ [15, 16], and zj is the imaginary part of
the jth normalized zero above the central point. The 863
rank-0 curves are from the 14 one-parameter families of rank
0 over Q from Table 2; the 701 rank-2 curves are from the
21 one-parameter families of rank 0 over Q from Table 3.
In Table 6 we calculate the median and mean for z2− z1,
z3−z2, and z3−z1. The last statistic involves the sum of
differences between two adjacent normalized zeros, and
allows the possibility of some effects being averaged out.
Although the normalized zeros are repelled from the cen-
tral point (and by different amounts for the two sets), the
differences between the normalized zeros are statistically
independent of this repulsion. We performed a t-test on
the means in the three cases. In each case, the t-statistic
was less than 2, strongly supporting the null hypothesis
that the differences are independent of the repulsion.
We have consistently observed that the greater the
number of zeros at the central point, the greater the
repulsion. One possible explanation is as follows: For
rank-2 curves in a rank-0 one-parameter family over Q,
the first zero above the central point collapses down to
the central point, and the other zeros are left alone. Since

272 Experimental Mathematics, Vol. 15 (2006), No. 3
the zeros are symmetric about the central point, the ef-
fect of one zero above the central point collapsing is to
increase the number of zeros at the central point by 2.
For our 14 one-parameter families of elliptic curves of
rank 0 over Q and log-conductors in [15, 16], we studied
the second and third normalized zeros above the central
point. The mean of the second normalized zero is 2.16
with a standard deviation of .39, while the third normal-
ized zero has a mean of 3.41 and a standard deviation
of .41. These numbers differ statistically11 from the first
and second normalized zeros of the rank-2 curves from
our 21 one-parameter families of rank 0 over Q with log-
conductor in [15, 16], where the means were respectively
1.93 (with a standard deviation of .39) and 3.27 (with a
standard deviation of .39). Thus while for a given range
of log-conductors the average second normalized zero of a
rank-0 curve is close to the average first normalized zero
of a rank-2 curve, they are not equal, and the additional
repulsion from extra zeros at the central point cannot be
entirely explained only by collapsing the first zero to the
central point while leaving the other zeros alone.
Remark 4.5. Since the second (respectively third) nor-
malized zero for rank-0 curves in rank-0 families over Q
is 2.16 (3.41) while the first (second) normalized zero for
rank-2 curves in rank-0 families over Q is 1.93 (3.27),
one can interpret the effect of the additional zeros at the
central point as an attraction. Specifically, for curves of
rank 2 in a rank-0 family over Q, by symmetry, two zeros
collapse to the central point, and the remaining zeros are
then attracted to the central point, being closer than the
corresponding zeros from rank-0 curves. As remarked in
[Farmer xx, Section 3.5], the term “lowest zero” is not
well defined when there are multiple zeros at the central
point. We can mean either the first zero above the cen-
tral point, or one of the many zeros at the central point.
In all cases, for finite conductors there is repulsion from
the N →∞ scaling limits of random-matrix theory; how-
ever, “attraction” might be a better term for the effect
of additional zeros at the central point, though the cur-
rent terminology is to talk about repulsion of zeros at the
central point.
We now study the differences between normalized ze-
ros coming from one-parameter families of rank 2 over
Q. Table 7 shows that while the normalized zeros are
repelled from the central point, the differences between
11The pooled and unpooled t-statistics in both experiments are
greater than 6, providing evidence against the null hypothesis that
the two means are equal.
64
Rank-2
Curves
23
Rank-4
Curves
t-Statistic
Median z
2
− z
1
1.26 1.27
Mean z
2
− z
1
1.36 1.29 0.59
StDev z
2
− z
1
0.50 0.42
Median z
3
− z
2
1.22 1.08
Mean z
3
− z
2
1.29 1.14 1.35
StDev z
3
− z
2
0.49 0.35
Median z
3
− z
1
2.66 2.46
Mean z
3
− z
1
2.65 2.43 2.05
StDev z
3
− z
1
0.44 0.42
TABLE 7. Spacings between normalized zeros. All curves
have log(cond) ∈ [15, 16], and zj is the imaginary part of
the jth normalized zero above the central point. The 64
rank-2 curves are the 21 one-parameter families of rank 2
over Q from Table 4; the 23 rank-4 curves are the 21 one-
parameter families of rank 2 over Q from Table 4.
the normalized zeros are statistically independent of the
repulsion. We performed a t-test for the means in the
three cases studied. For two of the three cases the t-
statistic was less than 2 (and in the third it was only
2.05), supporting the null hypothesis that the differences
are independent of the repulsion.
701
Rank-2
Curves
64
Rank-2
Curves
t-Statistic
Median z
2
− z
1
1.30 1.26
Mean z
2
− z
1
1.34 1.36 0.69
StDev z
2
− z
1
0.51 0.50
Median z
3
− z
2
1.19 1.22
Mean z
3
− z
2
1.22 1.29 1.39
StDev z
3
− z
2
0.47 0.49
Median z
3
− z
1
2.56 2.66
Mean z
3
− z
1
2.56 2.65 1.93
StDev z
3
− z
1
0.52 0.44
TABLE 8. Spacings between normalized zeros. All curves
have log(cond) ∈ [15, 16], and zj is the imaginary part of
the jth normalized zero above the central point. The 701
rank-2 curves are the 21 one-parameter families of rank 0
over Q from Table 4, and the 64 rank-2 curves are the 21
one-parameter families of rank 2 over Q from Table 4.
We performed one last experiment on the differences
between normalized zeros. In Table 8 we compare two
sets of rank-2 curves: the first are the 21 one-parameter
families of rank 0 over Q from Table 3, while the sec-
ond are the 21 one-parameter families of rank 2 over Q
from Table 4. While the first normalized zero is repelled
differently in the two cases, the differences are statisti-
cally independent of the nature of the zeros at the cen-

Miller: Investigations of Zeros near the Central Point of Elliptic Curve L-Functions 273
tral point, as indicated by all t-statistics being less than
2. This suggests that the spacings between adjacent nor-
malized zeros above the central point are independent
of the repulsion at the central point; in particular, this
quantity does not depend on how we construct our family
of rank-2 curves.
5. SUMMARY AND FUTURE WORK
As the conductors tend to infinity, theoretical results sup-
port the validity of the N → ∞ scaling limit of the inde-
pendent model for all curves in one-parameter families of
elliptic curves of rank r over Q; however, it is unknown
what the correct model is for the subfamily of curves
of rank r + 2. The experimental evidence suggests that
the first normalized zero, for small and finite conduc-
tors, is repelled by zeros at the central point. Further,
the greater the number of zeros at the central point, the
greater the repulsion; however, the repulsion decreases
as the conductors increase, and the difference between
adjacent normalized zeros is statistically independent of
the repulsion and the rank of the curves.
At present, we can calculate the first normalized zero
for log-conductors of size about 25. While we can use
more powerful computers to study larger conductors, it
is unlikely that these conductors will be large enough to
exhibit the predicted limiting behavior. It is interesting
that in contrast to the excess-rank investigations, here
we see noticeable convergence to the limiting theoretical
results as we increase the conductors.
An interesting project is to determine a theoretical
model to explain the behavior for finite conductors. In
the large-conductor limit, analogies with the function
field and calculations with the explicit formula lead us to
the independent model for curves of rank r from families
of rank r over Q, and theoretical results in the number-
field case support this. It is not unreasonable to posit
that in the finite-conductor analogues the size of the ma-
trices should be a function of the log-conductors. Un-
fortunately, the statistics for the finite N × N random-
matrix ensembles are expressed in terms of eigenvalues
of integral equations, and are usually plotted only in the
N → ∞ scaling limit. This makes comparison with the
experimental data difficult, and a future project is to an-
alyze the finite-N cases using the finite-N kernels. Such
an analysis will facilitate comparing the finite-N limits
of the independent and interaction models for curves of
rank r + 2 from families of rank r over Q.
APPENDIX A. “HARDER” ENSEMBLES OF
ORTHOGONAL MATRICES
(written by Eduardo Due~nez)
In this appendix we derive the conditional (interaction)
eigenvalue probability measure (2–4) and illustrate how
it affects eigenvalue statistics near the central point 1,
in particular through repulsion (observed via the 1-level
density). We also explain the relation to the classical
Bessel kernels of random-matrix theory, and to other
central-point statistics.
A.1 Full Orthogonal Ensembles
In view of our intended application, we will be concerned
exclusively with random-matrix ensembles of orthogonal
matrices in what follows. If we write the eigenvalues (in
no particular order) of a special12 orthogonal matrix of
size 2N (respectively 2N + 1) as {±eiθj}N1 (respectively
{+1}∪{±eiθj}N1 ) with 0 ≤ θj ≤ π, then the N -tuple Θ =
(θ1, . . . , θN ) parameterizes the eigenvalues. In terms of
the angles θj , the probability measure of the eigenval-
ues induced from normalized Haar measure on SO(2N)
(respectively on SO(2N + 1) upon discarding one forced
eigenvalue of +1) can be identified with a measure on
[0, π]N ,
dε0(Θ) = C˜
(0)
N
∏
1≤j<k≤N
(cos θk − cos θj)2
∏
1≤j≤N
dθj ,
(A–1)
dε1(Θ) = C˜
(1)
N
∏
1≤j<k≤N
(cos θk − cos θj)2
×
∏
1≤j≤N
sin2
(
θj
2
)
dθj (A–2)
in the 2N and 2N + 1 cases, respectively, as shown in
[Conrey ??, Katz and Sarnak 99a]; the normalization con-
stants C˜(m)N ensure that the measures on the right-hand
sides are probability measures. Note that formulas (A–1)
and (A–2) are symmetric upon permuting the θj ’s, so is-
sues related to a choice of a particular ordering of the
eigenvalues of the matrix are irrelevant. More impor-
tantly, observe the quadratic exponent of the differences
of the cosines.
The statistical behavior of the eigenvalues near +1 is
closely related to the order of vanishing of the measures
above at θ = 0. We change variables and replace the
eigenvalues e±iθ by the levels
x = cos θ (A–3)
12That is, of determinant one.

274 Experimental Mathematics, Vol. 15 (2006), No. 3
so the measures above become measures on [−1,+1]N :
dε0(X) = C
(0)
N
∏
1≤j<k≤N
(xk − xj)2
×
N
∏
j=1
(1 − xj)−1/2(1 + xj)−1/2dxj , (A–4)
dε1(X) = C
(1)
N
∏
j<k
(xk − xj)2
×
N
∏
j=1
(1 − xj)1/2(1 + xj)−1/2dxj , (A–5)
where X = (x1, . . . , xN ) and C
(m)
N are suitable normal-
ization constants. Here we observe the appearance of the
weight functions on [−1, 1]:
w(x) = (1− x)a(1 + x)−1/2, (A–6)
a =
{
−
1
2 for SO(2N),
+ 12 for SO(2N + 1).
(A–7)
By the Gaudin–Mehta theory (see, for example, [Mehta
91]), and in view of the quadratic exponent of the differ-
ences of the “levels” xj , the study of eigenvalue statis-
tics using classical methods is intimately related to the
sequence of orthogonal polynomials with respect to the
weight w(x).13
In classical random-matrix-theory terminology (espe-
cially in the context of the Laguerre and Jacobi ensem-
bles) the endpoints −1,+1 are called the “hard edges”
of the spectrum because the probability measure, con-
sidered on RN , vanishes outside [−1,+1]N . We will keep
calling θ = 0, π the “central points” (endpoints of the
diameter with respect to which the spectrum is symmet-
ric). Phenomena about central points and hard edges
are equivalent in view of the change of variables (A–3).
Perhaps not surprisingly, the parameter a, which dictates
the order of vanishing of the weight function w(x) at the
hard edge +1, suffices to characterize the mutually differ-
ent statistics near the central point in each of SO(even)
and SO(odd). However, the importance of this parameter
is best understood in the context of certain subensembles
of SO as described below.
A.2 Conditional (“Harder”) Orthogonal Ensembles
The conditional eigenvalue measure for the subensemble
SO(2r)(2N) of SO(2N) consisting of matrices for which
some 2r of the 2N eigenvalues are equal to +1 can easily
13The inner product being 〈f, g〉 =
∫
1
0
f(x)g(x)w(x) dx.
be obtained from (A–4). Let
f(x1, . . . , xN ) = C
(m)
N
∏
1≤j<k≤N
(xk − xj)2
∏
1≤j≤N
w(xj)
(A–8)
be the normalized probability density function of the lev-
els for SO(2N), where w(x) is as in (A–6) with a = − 12
and m = 0. Now let t1, . . . , tr be chosen such that
0 < tk < 1, let K =
∏
j [1 − tj , 1], and let I = J × K
for some box J ⊂ [−1, 1]N−r. This means that we are
constraining r pairs of levels to lie in a neighborhood of
x = 1 (or equivalently that we are construing r pairs of
eigenvalues to lie in circular sectors about the point 1 on
the unit circle). Thus, the conditional probability that
the remaining N − r pairs of eigenvalues lie in J is given
by
F (T ;J) =
∫
J×K
f(x) dx
∫
[−1,1]N−r×K
f(x) dx
, (A–9)
where T = (t1, . . . , tr). The conditional probability mea-
sure of the eigenvalues for the subensemble SO(2r)(2N)
is the limit as all tk → 0+ of F (t;J), as a function of the
box J ; call it G(J). Applying L’Hoˆpital’s rule r times to
the quotient (A–9) (once on each variable tk) and using
the fundamental theorem of calculus we get
G(J) (A–10)
= limT→0+
∫
J
(V (X))2(M(X,T ))2w(X)dX·(V (T ))2w(T )
∫
[0,1]
N−r
(V (X))2(M(X,T ))2w(X)dX·(V (T ))2w(T )
,
where X = (x1, . . . , xN−r) and
V (X) =
∏
1≤j<k≤N−r
(xk − xj),
V (T ) =
∏
1≤j<k≤r
(tk − tj),
w(X) =
∏
1≤j≤N−r
w(xj),
w(T ) =
∏
1≤k≤r
w(1 − tk),
M(X,T ) =
∏
1≤j≤N−r
1≤k≤r
(1 − tk − xj).
Naturally, the factors of V (T ) and w(T ) cancel in equa-
tion (A–10). Since M(X,T ) is bounded, the integrands
in equation (A–10) are uniformly dominated by an inte-
grable function, ensuring that we can let all tj → 0 in
the integrands of (A–10) to obtain
G(J) =
∫
J (V (X))
2(M(X, 0))2w(X) dX
∫
[0,1]N−r (V (X))
2(M(X, 0))2w(X) dX
. (A–11)

Miller: Investigations of Zeros near the Central Point of Elliptic Curve L-Functions 275
Now observe that (M(X, 0))2w(X) =
∏
1≤j≤r w˜(xj),
where w˜(x) is given by equation (A–6) with a replaced by
a˜ = a + 2, so the probability measure of the eigenvalues
for SO(2r)(2N) is obtained from that of SO(2(N − m))
simply by changing the weight function w → w˜. Ex-
plicitly, the probability measure of the eigenvalues of the
ensemble SO(2r)(2N) is given by
dεm(X) = C
(m)
N−m
∏
j<k
(xk − xj)2
∏
j
(1− xj)m−1/2
∏
j
dxj ,
(A–12)
where m = 2r, X = (x1, . . . , xN−r), the indices j, k range
from 1 to N − r, and C(m)N−m are suitable normalization
constants (equal to the reciprocal of the denominator of
the right-hand side of equation (A–11)).
The same argument shows that the subensemble
SO(2r+1)(2N +1) of SO(2N +1) consisting of matrices for
which 2r + 1 eigenvalues are equal to +1 has the same
eigenvalue measure (A–12) with m = 2r + 1. Because
the density of the measure vanishes to a higher order
near the edge +1 the larger m is, we will say that the
edge becomes harder when m is larger (whence the title
of this section), and call m its hardness.
A.3 Independent Model
It is important to observe that the presence of the m-
multiple eigenvalues at the central point in these harder
subensembles of orthogonal matrices has a strong re-
pelling effect due to the extra factor (1 − x)m multi-
plied by the weight (1 − x)−1/2(1 + x)−1/2 of SO(even).
For comparison purposes consider the following situation,
first in the SO(even) case. The number of eigenvalues
equal to +1 of any SO(2N) matrix is always an even num-
ber 2r, and one may consider the subensemble A2N,2r of
SO(2N),
A2N,2r =
{(
I2r×2r
g
)
: g ∈ SO(2N − 2r)
}
, (A–13)
which is just SO(2N − 2r) in disguise. This is certainly
a subensemble of SO(2N) consisting of matrices with at
least 2r eigenvalues equal to +1, albeit quite a different
one from the 2r-hard subensemble of SO(2N) described
before. For example, the eigenvalue measure (apart from
the point masses at the last 2r eigenvalues) for A2N,2r is
dε0(x1, . . . , xN−r), (A–14)
and not dε2r(x1, . . . , xN−r). The same observation ap-
plies in the SO(2N + 1) case: with the obvious notation,
the eigenvalue measure for A2N+1,2r is14
dε1(θ1, . . . , θN−r), (A–15)
and not dε2r+1(θ1, . . . , θN−r).
A.4 1-Level Density: Full Orthogonal
Before dealing with the harder subensembles, we make
some comments about the hard edges of the full SO(even)
and SO(odd). The local statistics near the point +1 are
dictated by the even “+” (respectively odd “−”) sine
kernels
S
±
(ξ, η) = S(ξ, η) ± S(ξ,−η) (A–16)
in the case of SO(even) (respectively SO(odd)); see [Katz
and Sarnak 99a, Katz and Sarnak 99b]. Here ξ, η are
rescaled variables centered at the value 0, namely related
to the original variables by15
x = cos
( π
N
ξ
)
, (A–17)
and S(x, y) = sin(πx)/(πx) is the sine kernel, which has
the universal property of describing the local statistics
at any bulk point of any ensemble with local quadratic
local level repulsion [McLaughlin et al. 99]. However,
it is not the sine kernel but its even (odd) counterparts
that dictate the local statistics near the central point.
For example, the central one-level density is given by the
diagonal values at x = y of the respective kernel (see
Figure 11):
ρ+(x) = 1 +
sin 2πx
2πx
, for SO(even), (A–18)
ρ
−
(x) = 1 +
sin 2πx
2πx
+ δ(x), for SO(odd). (A–19)
(In the SO(odd) case the Dirac delta reflects the fact that
any such matrix has an eigenvalue at the central point.)
Observe that ρ
−
vanishes to second order, whereas ρ+
does not vanish at the central point x = 0.16
A.5 1-Level Density: Harder Orthogonal
We return to the more general case of m-hard ensem-
bles of orthogonal matrices. Because the classical Ja-
cobi polynomials
{
P (a,b)n
}
∞
0 are orthogonal with respect
14Observe that a matrix in A
2N+1,2r has 2r+1 eigenvalues equal
to +1 and N − r other pairs of eigenvalues.
15This is justified by the fact N/π is the average (angular) asymp-
totic density of the eigenangles θj of a random orthogonal matrix;
hence asymptotic equidistribution holds—away from the central
points!
16If the central point were not atypical, then the local density
would be dictated by the diagonal values S(x, x) ≡ 1 of the sine
kernel.

276 Experimental Mathematics, Vol. 15 (2006), No. 3
0.5 1 1.5 2 2.5 3
0.25
0.5
0.75
1
1.25
1.5
1.75
2
FIGURE 11. The central 1-level densities ρ
+
(dotted) and
ρ
−
(dash-dotted) versus the “bulk” 1-level density ρ ≡ 1
observed away from the central points.
to the weight w(x) = (1 − x)a(1 + x)b on [−1, 1], the
local statistics near the central point x = +1 are de-
rived from the asymptotic behavior of these polynomi-
als at the right edge of the interval [−1,+1].17 More
specifically, the relevant kernel that takes the place of
the (even or odd) sine kernel is the “edge limit” as
N → ∞ of the Christoffel–Darboux/Szego˝ projection
kernel K(a,b)N (x, y) onto polynomials of degree less than N
in L2([−1, 1], (1− x)a(1 + x)bdx) (via the change of vari-
ables (A–17)). For the edge +1, the limit depends only
on the parameter a and is equal to the Bessel kernel18
B(a)(ξ, η) =
√
ξη
ξ2 − η2
[πξJa+1(πξ)Ja(πη) (A–20)
− Ja(πξ)πηJa+1(πη)],
B(a)(ξ, ξ) =
π
2
(πξ)[J2a(πξ) − Ja−1(πξ)Ja+1(πξ)],
(A–21)
where Jν stands for the Bessel function of the first kind
[Nagao and Wadati 91, Duen˜ez 04].
It is a little more natural for our purposes to use the
hardness m, rather than a = m− 12 , as the parameter, so
we define
K(m)(x, y) = B(m−1/2)(x, y), (A–22)
k(m)(x, y) =
1
π
K(m)(x/π, y/π). (A–23)
Using the recursion formula for Bessel functions we ob-
tain an alternative formula to (A–21) for the diagonal
17This “edge limit” and the ensuing Bessel kernels are also ob-
served in the somewhat simpler context of the so-called (unitary)
Laguerre ensemble.
18In fact, the even sine kernel is given by S
+
= B(−1/2), whereas
the odd sine kernel is given by S
−
= B(1/2).
1 2 3 4
0.25
0.5
0.75
1
1.25
1.5
1.75
2
FIGURE 12. The m-hard 1-level edge densities
for m = 0, 1, . . . , 5.
values of the kernel:
k(m)(x, x) =
x
2
[Jm+ 1
2
(x)2 + Jm− 1
2
(x)2] (A–24)
−
(
m −
1
2
)
Jm+ 1
2
(x)Jm− 1
2
(x).
Except for m times a point mass at x = 0, the central
m-hard 1-level density is given by
ρm(x) = K(m)(x, x). (A–25)
See Figure 12.
A.6 Spacing Measures
In this section we state some well-known formulas giving
the spacing measures or “gap probabilities” at the central
point. Their derivation is standard and depends only
on knowledge of the edge-limiting kernels K(m) (see, for
instance, [Katz and Sarnak 99a, Mehta 91, Tracy and
Widom 98]). Let E(m)(k; s) be the limit as N → ∞
of the probability that exactly k of the ξj ’s lie on the
interval (0, s), where ξj is related to xj via (A–17). Also
let p(m)(k; s)ds be the conditional probability that the
(k + 1)st of the ξj ’s to the right of ξ = 0 lies in the
interval [s, s + ds), in the limit N → ∞.
Abusing notation, let K(m)|s denote the integral op-
erator on L2([0, s], dx) with kernel K(m)(x, y):
K(m)|sf(·) =
∫ s
0
K(m)(·, y)f(y)dy. (A–26)
If I denotes the identity operator, then the following for-
mulas hold:
E(m)(k; s) =
1
k!
∂k
∂T k
det(I + TK(m)|s)
∣
∣
∣
∣
T=−1
, (A–27)
p(m)(k; s) = −
d
ds
k
∑
j=0
E(m)(j; s). (A–28)

Miller: Investigations of Zeros near the Central Point of Elliptic Curve L-Functions 277
On the right-hand side of (A–27), “det” is the Fredholm
determinant: for an operator with kernel K,
det(I+K) = 1+
∞
∑
n=1
1
n!
∫
· · ·
∫
R
n
det
n×n
(K(xj , xk)) dxn · · · dx1.
(A–29)
Identical formulas hold even for finite N , provided that
the limiting kernel K(m) is replaced by the Christoffel–
Darboux/Szego˝ projection kernel K
(m− 1
2
,− 1
2
)
N−m associated
with the weight w(x) of (A–6) with a = m− 12 , acting on
L2([−1, 1], w(x) dx). In this case, the corresponding op-
erator is of finite rank, the Fredholm determinant agrees
with the usual determinant, and the series (A–29) is fi-
nite.
A.7 Explicit Kernels
In view of the relation between Bessel functions of the
first kind of half-integral parameter and trigonometric
functions, it is possible to write the kernels K(m) in terms
of elementary functions.
A.7.1 m = 0 : The Even Sine Kernel.
K0(x, y) = S+(x, y) =
sin π(x− y)
π(x− y)
+
sin π(x + y)
π(x + y)
.
(A–30)
The one-level density is
ρ+(x) = S+(x, x) = 1 +
sin 2πx
2πx
. (A–31)
The Fourier transform of the one-level density is
ρˆ+(u) = δ(u) +
1
2
I(u), (A–32)
where I(u) is the characteristic function of the interval
[−1, 1].
A.7.2 m =1: The Odd Sine Kernel.
K1(x, y) = S−(x, y) =
sinπ(x− y)
π(x− y)
−
sinπ(x + y)
π(x + y)
.
(A–33)
The one-level density is
ρ
−
(x) = S
−
(x, x) = δ(x) + 1−
sin 2πx
2πx
. (A–34)
The Fourier transform of the one-level density is
ρˆ
−
(u) = δ(u) + 1−
1
2
I(u). (A–35)
A.7.3 m = 2: The “Doubly Hard” Kernel.
K2(x, y) =
sinπ(x− y)
π(x− y)
+
sinπ(x + y)
π(x + y)
− 2
sin πx
πx
sinπy
πy
.
(A–36)
The one-level density is
ρ2(x) = 2δ(x) + 1 +
sin 2πx
2πx
− 2
(
sinπx
πx
)2
. (A–37)
The Fourier transform of the one-level density is
ρˆ2(u) = δ(u) + 2 +
(
2|u| −
3
2
)
I(u). (A–38)
A.7.4 m = 3: The “Triply Hard” Kernel.
K3(x, y) = K1(x, y) +
18
π2xy
(
1 +
5
π2xy
)
K0(x, y)
− 6
(
cos πx
πx
cos πy
πy
+
sinπx
(πx)2
sin πy
(πy)2
)
.
(A–39)
ACKNOWLEDGMENTS
The program used to calculate the order of vanishing of
elliptic-curve L-functions at the central point was written by
Jon Hsu, Leo Goldmakher, Stephen Lu, and the author; the
program to calculate the first few zeros above the central
point in families of elliptic curves is due to Adam O’Brien,
Aaron Lint, Atul Pokharel, Michael Rubinstein, and the au-
thor. I would like to thank Eduardo Duen˜ez, Frank Firk,
Michael Rosen, Peter Sarnak, Joe Silverman, and Nina Snaith
for many enlightening conversations, the referees for a very
thorough reading of the paper and suggestions that improved
the presentation, and the information technology managers
at the Mathematics Departments at Princeton, Ohio State,
and Brown Universities for help in getting all the programs
to run compatibly.
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Steven J. Miller, Department of Mathematics, Brown University, 151 Thayer Street, Providence, RI 02912
(sjmiller@math.brown.edu)
Eduardo Duen˜ez, Department of Applied Mathematics, University of Texas at San Antonio, 6900 N Loop 1604 W,
San Antonio, TX 78249 (eduenez@math.utsa.edu)
Received August 8, 2005; accepted in revised form November 23, 2005.