Lower order terms in the 1-level density for families of holomorphic
  cuspidal newforms
LOWER ORDER TERMS IN THE 1-LEVEL DENSITY FOR FAMILIES OF
HOLOMORPHIC CUSPIDAL NEWFORMS
STEVEN J. MILLER
ABSTRACT. The Katz-Sarnak density conjecture states that, in the limit as the analytic conductors
tend to infinity, the behavior of normalized zeros near the central point of families of L-functions
agree with the N → ∞ scaling limits of eigenvalues near 1 of subgroups of U(N). Evidence for
this has been found for many families by studying the n-level densities; for suitably restricted test
functions the main terms agree with random matrix theory. In particular, all one-parameter families of
elliptic curves with rank r over Q(T ) and the same distribution of signs of functional equations have
the same universal limiting behavior for their main term. We break this universality and find family
dependent lower order correction terms in many cases; these lower order terms have applications
ranging from excess rank to modeling the behavior of zeros near the central point, and depend on the
arithmetic of the family. We derive an alternate form of the explicit formula for GL(2) L-functions
which simplifies comparisons, replacing sums over powers of Satake parameters by sums of the
moments of the Fourier coefficients λf (p). Our formula highlights the differences that we expect to
exist from families whose Fourier coefficients obey different laws (for example, we expect Sato-Tate
to hold only for non-CM families of elliptic curves). Further, by the work of Rosen and Silverman we
expect lower order biases to the Fourier coefficients in one-parameter families of elliptic curves with
rank over Q(T ); these biases can be seen in our expansions. We analyze several families of elliptic
curves and see different lower order corrections, depending on whether or not the family has complex
multiplication, a forced torsion point, or non-zero rank over Q(T ).
1. INTRODUCTION
Assuming the Generalized Riemann Hypothesis (GRH), the non-trivial zeros of any L-function
have real part equal to 1/2. Initial investigations studied spacing statistics of zeros far from the
central point, where numerical and theoretical results [Hej, Mon, Od1, Od2, RS] showed excellent
agreement with eigenvalues from the Gaussian Unitary Ensemble (GUE). Further agreement was
found in studying moments of L-functions [CF, CFKRS, KeSn1, KeSn2, KeSn3] as well as low-
lying zeros (zeros near the critical point).
In this paper we concentrate on low-lying zeros of L(s, f), where f ∈ H⋆k(N), the set of all
holomorphic cuspidal newforms of weight k and level N . Before stating our results, we briefly
review some notation and standard facts. Each f ∈ H⋆k(N) has a Fourier expansion
f(z) =
af(n)e(nz). (1.1)
Date: September 1, 2021.
2000 Mathematics Subject Classification. 11M26 (primary), 11G40, 11M41, 15A52 (secondary).
Key words and phrases. n-Level Density, Low Lying Zeros, Elliptic Curves.
The author would like to thank Walter Becker, Colin Deimer, Steven Finch, Dorian Goldfeld, Filip Paun and Matt
Young for useful discussions. Several of the formulas for expressions in this paper were first guessed by using Sloane’s
On-Line Encyclopedia of Integer Sequences [Sl]. The numerical computations were run on the Princeton Math Server,
and it is a pleasure to thank the staff there for their help. The author was partly supported by NSF grant DMS0600848.
http://arxiv.org/abs/0704.0924v4
2 STEVEN J. MILLER
Let λf (n) = af (n)n
−(k−1)/2. These coefficients satisfy multiplicative relations, and |λf(p)| ≤ 2.
The L-function associated to f is
L(s, f) =
λf(n)
1− λf(p)
χ0(p)
, (1.2)
where χ0 is the principal character with modulus N . We write
λf(p) = αf(p) + βf (p). (1.3)
For p |rN , αf (p)βf(p) = 1 and |αf(p)| = 1. If p|N we take αf(p) = λf(p) and βf (p) = 0. Letting
L∞(s, f) =
)1/2 (√
k − 1
k + 1
(1.4)
denote the local factor at infinity, the completed L-function is
Λ(s, f) = L∞(s)L(s, f) = ǫfΛ(1− s, f), ǫf = ±1. (1.5)
Therefore H⋆k(N) splits into two disjoint subsets, H
k (N) = {f ∈ H⋆k(N) : ǫf = +1} and
H−k (N) = {f ∈ H⋆k(N) : ǫf = −1}. Each L-function has a set of non-trivial zeros ρf,j = 12+ßγf,ℓ.
The Generalized Riemann Hypothesis asserts that all γf,ℓ ∈ R.
In studying the behavior of low-lying zeros, the arithmetic and analytic conductors determine the
appropriate scale. For f ∈ H⋆k(N), the arithmetic conductorNf is the integerN from the functional
equation, and the analytic conductor Qf is
(k+1)(k+3)
. The number of zeros within C units of the
central point (where C is any large, absolute constant) is of the order logQf . For us k will always
be fixed, so Nf and Qf will differ by a constant. Thus logQf ∼ logNf , and in the limit as the level
N tends to infinity, we may use either the analytic or arithmetic conductor to normalize the zeros
near the central point. See [Ha, ILS] for more details.
We rescale the zeros and study γf,ℓ
logQf
. We let F = ∪FN be a family of L-functions ordered by
conductor (our first example will be FN = H∗k(N); later we shall consider one-parameter families
of elliptic curves). The n-level density for the family is
Dn,F(φ) := lim
|FN |
ℓ1,...,ℓn
ℓi 6=±ℓk
γf,ℓ1
logQf
· · ·φn
γf,ℓn
logQf
, (1.6)
where the φi are even Schwartz test functions whose Fourier transforms have compact support and
+ßγf,ℓ runs through the non-trivial zeros of L(s, f). As the φi’s are even Schwartz functions, most
of the contribution to Dn,F(φ) arises from the zeros near the central point; thus this statistic is well-
suited to investigating the low-lying zeros. For some families, it is more convenient to incorporate
weights (for example, the harmonic weights facilitate applying the Petersson formula to families of
cuspidal newforms).
Katz and Sarnak [KaSa1, KaSa2] conjectured that, in the limit as the analytic conductors tend to
infinity, the behavior of the normalized zeros near the central point of a family F of L-functions
agrees with the N → ∞ scaling limit of the normalized eigenvalues near 1 of a subgroup of U(N):
Dn,F(φ) =
· · ·
φ1(x1) · · ·φn(xn)Wn,G(F)(x1, . . . , xn)dx1 · · · dxn, (1.7)
LOWER ORDER TERMS IN 1-LEVEL DENSITIES 3
where G(F) is the scaling limit of one of the following classical compact groups: N × N unitary,
symplectic or orthogonal matrices.1 Evidence towards this conjecture is provided by analyzing the
n-level densities of many families, such as all Dirichlet characters, quadratic Dirichlet characters,
L(s, ψ) with ψ a character of the ideal class group of the imaginary quadratic field Q(
families of elliptic curves, weight k level N cuspidal newforms, symmetric powers of GL(2) L-
functions, and certain families of GL(4) and GL(6) L-functions; see [DM1, FI, Gü, HR, HM, ILS,
KaSa2, Mil2, OS, RR1, Ro, Rub, Yo2].
Different classical compact groups exhibit a different local behavior of eigenvalues near 1, thus
breaking the global GUE symmetry. This correspondence allows us, at least conjecturally, to assign
a definite “symmetry type” to each family of primitive L-functions.2
Now that the main terms have been shown to agree with random matrix theory predictions (at
least for suitably restricted test functions), it is natural to study the lower order terms.3 In this paper
we see how various arithmetical properties of families of elliptic curves (complex multiplication,
torsion groups, and rank) affect the lower order terms. 4 For families of elliptic curves these lower
order terms have appeared in excess rank investigations [Mil3], and in a later paper [DHKMS] they
will play a role in explaining the repulsion observed in [Mil4] of the first normalized zero above the
central point in one-parameter families of elliptic curves.
We derive an alternate version of the explicit formula for a family F of GL(2) L-functions
of weight k which is more tractable for such investigations, which immediately yields a useful
expansion for the 1-level density for a family F of GL(2) cuspidal newforms. We should really
1For test functions φ̂ supported in (−1, 1), the one-level densities are
φ(u)W1,SO(even)(u)du = φ̂(u) +
φ(0)∫
φ(u)W1,SO(odd)(u)du = φ̂(u) +
φ(0)∫
φ(u)W1,O(u)du = φ̂(u) +
φ(0)∫
φ(u)W1,USp(u)du = φ̂(u)− 12φ(0)∫
φ(u)W1,U(u)du = φ̂(u).
(1.8)
2For families of zeta orL-functions of curves or varieties over finite fields, the corresponding classical compact group
can be determined by the monodromy (or symmetry group) of the family and its scaling limit. No such identification
is known for number fields, though function field analogues often suggest what the symmetry type should be. See also
[DM2] for results about the symmetry group of the convolution of families, as well as determining the symmetry group
of a family by analyzing the second moment of the Satake parameters.
3Recently Conrey, Farmer and Zirnbauer [CFZ1, CFZ2] conjectured formulas for the averages over a family of
ratios of products of shifted L-functions. Their L-functions Ratios Conjecture predicts both the main and lower order
terms for many problems, ranging from n-level correlations and densities to mollifiers and moments to vanishing at
the central point (see [CS]). In [Mil6, Mil7] we verified the Ratios Conjecture’s predictions (up to error terms of size
O(X−1/2+ǫ)!) for the 1-level density of the family of quadratic Dirichlet characters and certain families of cuspidal
newforms for test functions of suitably small support. Khiem is currently calculating the predictions of the Ratios
Conjecture for certain families of elliptic curves.
4While the main terms for one-parameter families of elliptic curves of rank r over Q(T ) and given distribution of
signs of functional equations all agree with the scaling limit of the same orthogonal group, in [Mil1] potential lower
order corrections were observed (see [FI, RR2, Yo1] for additional examples, and [Mil3] for applications of lower order
terms to bounding the average order of vanishing at the central point in a family). The problem is that these terms are
of size 1/ logR, while trivially estimating terms in the explicit formula lead to errors of size log logR/ logR; here
logR is the average log-conductor of the family. These lower order terms are useful in refining the models of zeros
near the central point for small conductors. This is similar to modeling high zeros of ζ(s) at height T with matrices of
size N = log(T/2π) (and not the N → ∞ scaling limits) [KeSn1, KeSn2]; in fact, even better agreement is obtained
by a further adjustment of N arising from an analysis of the lower order terms (see [BBLM, DHKMS]).
4 STEVEN J. MILLER
write FN and RN below to emphasize that our calculations are being done for a fixed N , and then
take the limit as N → ∞. As there is no danger of confusion, we suppress the N in the FN and
LetNf be the level of f ∈ F and let φ be an even Schwartz function such that φ̂ has compact sup-
port, say supp(φ̂) ⊂ (−σ, σ). We weight each f ∈ F by non-negative weights wR(f), where logR
is the weighted average of the logarithms of the levels, and we rescale the zeros near the central
point by (logR)/2π (in all our families of interest, logR ∼ logN). Set WR(f) =
f∈F wR(f).
The 1-level density for the family F with weights wR(f) and test function φ is
D1,F(φ) =
WR(F)
wR(f)
f∈F wR(f)(A(k) + logNf)
WR(F) logR
φ̂(0)
WR(F)
wR(f)
αf(p)
m + βf(p)
log p
log p
log2R
f∈F wR(f)(A(k) + logNf)
WR(F) logR
φ̂(0) + S(F) +Ok
log2R
, (1.9)
with ψ(z) = Γ′(z)/Γ(z), A(k) = ψ(k/4) + ψ((k + 2)/4)− 2 log π, and
S(F) = − 2
WR(F)
wR(f)
αf(p)
m + βf (p)
log p
log p
. (1.10)
The above is a straightforward consequence of the explicit formula, and depends crucially on having
an Euler product for our L-functions; see [ILS] for a proof. As φ is a Schwartz function, most of
the contribution is due to the zeros near the central point. The error of size 1/ log2R arises from
simplifying some of the expressions involving the analytic conductors, and could be improved to
be of size 1/ log3R at the cost of additional analysis (see [Yo1] for details); as we are concerned
with lower order corrections due to arithmetic differences between the families, the above suffices
for our purposes.
The difficult (and interesting) piece in the 1-level density is S(F). Our main result is an alternate
version of the explicit formula for this piece. We first set the notation. For each f ∈ F , let
S(p) = {f ∈ F : p |rNf}. (1.11)
Thus for f /∈ S(p), αf (p)m + βf (p)m = λf (p)m. Let
Ar,F(p) =
WR(F)
f∈S(p)
wR(f)λf(p)
r, A′r,F(p) =
WR(F)
f /∈S(p)
wR(f)λf(p)
r; (1.12)
we use the convention that 00 = 1; thus A0,F (p) equals the cardinality of S(p).
LOWER ORDER TERMS IN 1-LEVEL DENSITIES 5
Theorem 1.1 (Expansion for S(F) in terms of moments of λf(p)). Let logR be the average log-
conductor of a finite family of L-functions F , and let S(F) be as in (1.10). We have
S(F) = − 2
A′m,F(p)
log p
log p
−2φ̂(0)
2A0,F(p) log p
p(p+ 1) logR
2A0,F(p) log p
p logR
log p
A1,F(p)
log p
log p
+ 2φ̂(0)
A1,F(p)(3p+ 1)
p1/2(p+ 1)2
log p
A2,F(p) log p
p logR
log p
+ 2φ̂(0)
A2,F(p)(4p
2 + 3p+ 1) log p
p(p+ 1)3 logR
−2φ̂(0)
Ar,F(p)p
r/2(p− 1) log p
(p+ 1)r+1 logR
log3R
= SA′(F) + S0(F) + S1(F) + S2(F) + SA(F) +O
log3R
. (1.13)
If we let
ÃF (p) =
WR(F)
f∈S(p)
wR(f)
λf(p)
p+ 1− λf(p)
, (1.14)
then by the geometric series formula we may replace SA(F) with SÃ(F), where
SÃ(F) = −2φ̂(0)
ÃF (p)p
3/2(p− 1) log p
(p+ 1)3 logR
. (1.15)
Remark 1.2. For a general one-parameter family of elliptic curves, we are unable to obtain exact,
closed formulas for the rth moment terms Ar,F(p); for sufficiently nice families we can find exact
formulas for r ≤ 2 (see [ALM, Mil3] for some examples, with applications towards constructing
families with moderate rank over Q(T ) and the excess rank question). Thus we are forced to
numerically approximate the Ar,F(p) terms when r ≥ 3.5
We prove Theorem 1.1 by using the geometric series formula for
m≥3(αf (p)/
p)m (and sim-
ilarly for the sum involving βf(p)
m) and properties of the Satake parameters. We find terms like
λf(p)
3 − 3λf(p)
p+ 1− λf(p)
λf(p)
2 − 2
p+ 1− λf (p)
. (1.16)
While the above formula leads to tractable expressions for computations, the disadvantage is that
the zeroth, first and second moments of λf(p) are now weighted by 1/(p + 1 − λf (p)
p). For
many families (especially those of elliptic curves) we can calculate the zeroth, first and second
moments exactly up to errors of size 1/N ǫ; this is not the case if we introduce these weights in the
5This greatly hinders comparison with the L-Functions Ratios Conjecture, which gives useful interpretations for
the lower order terms. In [CS] the lower order terms are computed for a symplectic family of quadratic Dirichlet L-
functions. The (conjectured) expansions there show a remarkable relation between the lower order terms and the zeros
of the Riemann zeta function; for test functions with suitably restricted support, the number theory calculations are
tractable and in [Mil6] are shown to agree with the Ratios Conjecture.
6 STEVEN J. MILLER
denominator. We therefore apply the geometric series formula again to expand 1/(p+1−λf (p)
and collect terms.
An alternate proof involves replacing each αf (p)
m + βf (p)
m for p ∈ S(p) with a polynomial∑m
r=0 cm,rλf(p)
m, and then interchanging the order of summation (which requires some work, as
the resulting sum is only conditionally convergent). The sum over r collapses to a linear combina-
tion of polylogarithm functions, and the proof is completed by deriving an identity expressing these
sums as a simple rational function.6
Remark 1.3. An advantage of the explicit formula in Theorem 1.1 is that the answer is expressed
as a weighted sum of moments of the Fourier coefficients. Often much is known (either theoret-
ically or conjecturally) for the distribution of the Fourier coefficients, and this formula facilitates
comparisons with conjectures. In fact, often the r-sum can be collapsed by using the generating
function for the moments of λf(p). Moreover, there are many situations where the Fourier coef-
ficients are easier to compute than the Satake parameters; for elliptic curves we find the Fourier
coefficients by evaluating sums of Legendre symbols, and then pass to the Satake parameters by
solving aE(p) = 2
p cos θE(p). Thus it is convenient to have the formulas in terms of the Fourier
coefficients. As ÃF(p) = O(1/p)), these sums converge at a reasonable rate, and we can evaluate
the lower order terms of size 1/ logR to any specified accuracy by simply calculating moments and
modified moments of the Fourier coefficients at the primes.
We now summarize the lower order terms for several different families of GL(2) L-functions;
many other families can be computed through these techniques. The first example is analyzed in
§3, the others in §5. Below we merely state the final answer of the size of the 1/ logR term to a few
digits accuracy; see the relevant sections for expressions of these constants in terms of prime sums
with weights depending on the family. For sufficiently small support, the main term in the 1-level
density of each family has previously been shown to agree with the three orthogonal groups (we can
determine which by calculating the 2-level density and splitting by sign); however, the lower order
terms are different for each family, showing how the arithmetic of the family enters as corrections
to the main term. For most of our applications we have weight 2 cuspidal newforms, and thus the
conductor-dependent terms in the lower order terms are the same for all families. Therefore below
we shall only describe the family-dependent corrections.
• All holomorphic cusp forms (Theorem 3.4): Let Fk,N be either the family of even weight
k and prime level N cuspidal newforms, or just the forms with even (or odd) functional
equation. Up to O(log−3R), for test functions φ with supp(φ̂) ⊂ (−4/3, 4/3), as N → ∞
6The polylogarithm function is Lis(x) =
k=1 k
−sxk . If s is a negative integer, say s = −r, then the polylogarithm
function converges for |x| < 1 and equals
〉xr−j
(1 − x)r+1, where the 〈 r
〉 are the Eulerian numbers (the
number of permutations of {1, . . . , r} with j permutation ascents). In [Mil5] we show that if aℓ,i is the coefficient of
ki in
j=0(k
2 − j2), and bℓ,i is the coefficient of ki in (2k + 1)
j=0(k − j)(k + 1 + j), then for |x| < 1 and ℓ ≥ 1
we have
aℓ,2ℓLi−2ℓ(x) + · · ·+ aℓ,0Li0(x) =
(2ℓ)!
xℓ(1 + x)
(1− x)2ℓ+1
bℓ,2ℓ+1Li−2ℓ−1(x) + · · ·+ bℓ,0Li0(x) = (2ℓ+ 1)!
xℓ(1 + x)
(1− x)2ℓ+2
. (1.17)
Another application of this identity is to deduce relations among the Eulerian numbers.
LOWER ORDER TERMS IN 1-LEVEL DENSITIES 7
the (non-conductor) lower order term is
− 1.33258 · 2φ̂(0)/ logR. (1.18)
Note the lower order corrections are independent of the distribution of the signs of the func-
tional equations.
• CM example, with or without forced torsion (Theorem 5.6): Consider the one-parameter
families y2 = x3 + B(6T + 1)κ over Q(T ), with B ∈ {1, 2, 3, 6} and κ ∈ {1, 2}; these
families have complex multiplication, and thus the distribution of their Fourier coefficients
does not follow Sato-Tate. We sieve so that (6T + 1) is (6/κ)-power free. If κ = 1 then all
values ofB have the same behavior, which is very close to what we would get if the average
of the Fourier coefficients immediately converged to the correct limiting behavior.7 If κ = 2
the four values ofB have different lower order corrections; in particular, ifB = 1 then there
is a forced torsion point of order three, (0, 6T + 1). Up to errors of size O(log−3R), the
(non-conductor) lower order terms are approximately
B = 1, κ = 1 : −2.124 · 2φ̂(0)/ logR,
B = 1, κ = 2 : −2.201 · 2φ̂(0)/ logR,
B = 2, κ = 2 : −2.347 · 2φ̂(0)/ logR
B = 3, κ = 2 : −1.921 · 2φ̂(0)/ logR
B = 6, κ = 2 : −2.042 · 2φ̂(0)/ logR. (1.19)
• CM example, with or without rank (see §5.2): Consider the one-parameter families
y2 = x3 − B(36T + 6)(36T + 5)x over Q(T ), with B ∈ {1, 2}. If B = 1 the family
has rank 1, while if B = 2 the family has rank 0; in both cases the family has complex
multiplication. We sieve so that (36T + 6)(36T + 5) is cube-free. The most important
difference between these two families is the contribution from the S eA(F) terms, where the
B = 1 family is approximately −.11 · 2φ̂(0)/ logR, while the B = 2 family is approxi-
mately .63 · 2φ̂(0)/ logR. This large difference is due to biases of size −r in the Fourier
coefficients at(p) in a one-parameter family of rank r over Q(T ). Thus, while the main
term of the average moments of the pth Fourier coefficients are given by the complex mul-
tiplication analogue of Sato-Tate in the limit, for each p there are lower order correction
terms which depend on the rank. This is in line with other results. Rosen and Silverman
[RoSi] prove
t mod p at(p) is related to the negative of the rank of the family over Q(T );
see Theorem 5.8 for an exact statement.
• Non-CM Example (see Theorem 5.14): Consider the one-parameter family y2 = x3 −
3x + 12T over Q(T ). Up to O(log−3R), the (non-conductor) lower order correction is
approximately
− 2.703 · 2φ̂(0)/ logR, (1.20)
7In practice, it is only as p → ∞ that the average moments converge to the complex multiplication distribution;
for finite p the lower order terms to these moments mean that the answer for families of elliptic curves with complex
multiplication is not the same as what we would obtain by replacing these averages with the moments of the complex
multiplication distribution.
8 STEVEN J. MILLER
which is very different than the family of weight 2 cuspidal newforms of prime level N .
Remark 1.4. While the main terms of the 1-level density in these families depend only weakly on
the family,8 we see that the lower order correction terms depend on finer arithmetical properties
of the family. In particular, we see differences depending on whether or not there is complex
multiplication, a forced torsion point, or rank. Further, the lower order correction terms are more
negative for families of elliptic curves with forced additive reduction at 2 and 3 than for all cuspidal
newforms of prime level N → ∞. This is similar to Young’s results [Yo1], where he considered
two-parameter families and noticed that the number of primes dividing the conductor is negatively
correlated to the number of low-lying zeros. A better comparison would perhaps be to square-free
N with the number of factors tending to infinity, arguing as in [ILS] to handle the necessary sieving.
Remark 1.5. The proof of the Central Limit Theorem provides a useful analogy for our results. If
X1, . . . , XN are ‘nice’ independent, identically distributed random variables with mean µ and vari-
ance σ2, then as N → ∞ we have (X1+ · · ·+XN −Nµ)/σ
N converges to the standard normal.
The universality is that, properly normalized, the main term is independent of the initial distribu-
tion; however, the rate of convergence to the standard normal depends on the higher moments of the
distribution. We observe a similar phenomenon with the 1-level density. We see universal answers
(agreeing with random matrix theory) as the conductors tend to infinity in the main terms; how-
ever, the rate of convergence (the lower order terms) depends on the higher moments of the Fourier
coefficients.
The paper is organized as follows. In §2 we review the standard explicit formula and then prove
our alternate version (replacing averages of Satake parameters with averages of the Fourier coef-
ficients). We analyze all cuspidal newforms in §3. After some preliminary expansions for elliptic
curve families in §4, we analyze several one-parameter families in §5.
2. EXPLICIT FORMULAS
2.1. Standard Explicit Formula. Let φ be an even Schwartz test function whose Fourier trans-
form has compact support, say supp(φ̂) ⊂ (−σ, σ). Let f be a weight k cuspidal newform of level
N ; see (1.1) through (1.5) for a review of notation. The explicit formula relates sums of φ over
the zeros of Λ(s, f) to sums of φ̂ and the Fourier coefficients over prime powers. We have (see for
example Equations (4.11)–(4.13) of [ILS]) that
Ak,N(φ)
αf(p)
m + βf(p)
log p
log p
(2.1)
8All that matters are the first two moments of the Fourier coefficients. All families have the same main term in
the second moments; the main term in the first moment is just the rank of the family. See [Mil2] for details for one-
parameter families of elliptic curves
LOWER ORDER TERMS IN 1-LEVEL DENSITIES 9
where
Ak,N(φ) = 2φ̂(0) log
Ak,N ;j(φ),
Ak,N ;j(φ) =
φ(x)dx,
(2.2)
with ψ(z) = Γ′(z)/Γ(z), α1 =
and α2 =
In this paper we concentrate on the first order correction terms to the 1-level density. Thus
we are isolating terms of size 1/ logR, and ignoring terms that are O(1/ log2R). While a more
careful analysis (as in [Yo1]) would allow us to analyze these conductor terms up to an error of size
O(log−3R), these additional terms are independent of the family and thus not as interesting for our
purposes. We use (8.363.3) of [GR] (which says ψ(a + bß) + ψ(a − bß) = 2ψ(a) + O(b2/a2) for
a, b real and a > 0), and find
Ak,N ;j(φ) = φ̂(0)ψ
(αj + 1)2 log
. (2.3)
This implies that
Ak,N(φ) = φ̂(0) logN + φ̂(0)
k + 2
− 2 log π
(αj + 1)2 log
. (2.4)
As we shall consider the case of k fixed and N → ∞, the above expansion suffices for our purposes
and we write
Ak,N(φ) = φ̂(0) logN + φ̂(0)A(k) +Ok
log2R
. (2.5)
We now average (2.1) over all f in our family F . We allow ourselves the flexibility to introduce
slowly varying non-negative weights wR(f), as well as allowing the levels of the f ∈ F to vary.
This yields the expansion for the 1-level density for the family, which is given by (1.9).
We have freedom to choose the weights wR(f) and the scaling parameter R. For families of
elliptic curves we often take the weights to be 1 for t ∈ [N, 2N ] such that the irreducible polynomial
factors of the discriminant are square or cube-free, and zero otherwise (equivalently, so that the
specialization Et yields a global minimal Weierstrass equation); logR is often the average log-
conductor (or a close approximation to it). For families of cuspidal newforms of weight k and
square-free levelN tending to infinity, we might takewR(f) to be the harmonic weights (to simplify
applying the Petersson formula) and R around k2N (i.e., approximately the analytic conductor).
The interesting piece in (1.9) is
S(F) = − 2
WR(F)
wR(f)
αf(p)
m + βf(p)
log p
log p
. (2.6)
10 STEVEN J. MILLER
We rewrite the expansion above in terms of the moments of the Fourier coefficients λf (p). If p|Nf
then αf(p)
m + βf(p)
m = λf(p)
m. Thus
S(F) = − 2
WR(F)
wR(f)
λf(p)
log p
log p
WR(F)
p |rNf
wR(f)
αf(p)
m + βf(p)
log p
log p
(2.7)
In the explicit formula we have terms such as φ̂(m log p/ logR). As φ̂ is an even function, Taylor
expanding gives
log p
= φ̂(0) +O
log p
. (2.8)
As we are isolating lower order correction terms of size 1/ logR in S(F), we ignore any term
which is o(1/ logR). We therefore may replace φ̂(m log p/ logR) with φ̂(log p/ logR) at a cost of
O(1/ log3R) for all m ≥ 3,9 which yields
S(F) = − 2
WR(F)
wR(f)
λf(p)
log p
log p
WR(F)
p |rNf
wR(f)
λf(p)
log p
log p
WR(F)
p |rNf
wR(f)
λf(p)
2 − 2
log p
log p
WR(F)
p |rNf
wR(f)
αf(p)
m + βf (p)
log p
log p
log3R
(2.9)
We have isolated them = 1 and 2 terms from p|rNf as these can contribute main terms (and not just
lower order terms). We used for p|rNf that αf(p)+βf (p) = λf(p) and αf (p)2+βf (p)2 = λf(p)2−2.
2.2. The Alternate Explicit Formula.
9As φ̂ has compact support, the only m that contribute are m ≪ logR, and thus we do not need to worry about the
m-dependence in this approximation because these terms are hit by a p−m/2.
LOWER ORDER TERMS IN 1-LEVEL DENSITIES 11
Proof of Theorem 1.1. We use the geometric series formula for the m ≥ 3 terms in (2.9). We have
M3(p) :=
αf (p)√
βf(p)√
αf(p)
p− αf(p))
βf(p)
p− βf(p))
(αf (p)
3 + βf(p)
p− (αf (p)2 + βf (p)2)
p(p+ 1− λf(p)
λf(p)
3√p− λf (p)2 − 3λf(p)
p(p+ 1− λf(p)
(2.10)
where we use αf(p)
3 + βf (p)
3 = λf(p)
3 − 3λf(p) and αf (p)2 + βf (p)2 = λf(p)2 − 2. Writing
(p+1−λf (p)
p)−1 as (p+1)−1
1− λf (p)
, using the geometric series formula and collecting
terms, we find
M3(p) =
p(p+ 1)
p(3p+ 1)λf(p)
p(p+ 1)2
2 + 3p+ 1)λf(p)
p(p+ 1)3
pr/2(p− 1)λf(p)r
(p+ 1)r+1
(2.11)
We use (2.8) to replace φ̂(log p/ logR) in (2.9) with φ̂(0) + O(1/ log2R) and the above expansion
for M3(p); the proof is then completed by simple algebra and recalling the definitions of Ar,F(p)
and A′r,F(p), (1.12). �
2.3. Formulas for the r ≥ 3 Terms. For many families we either know or conjecture a distribution
for the (weighted) Fourier coefficients. If this were the case, then we could replace the Ar,F(p) with
the rth moment. In many applications (for example, using the Petersson formula for families of
cuspidal newforms of fixed weight and square-free level tending to infinity) we know the moments
up to a negligible correction.
In all the cases we study, the known or conjectured distribution is even, and the moments have a
tractable generating function. Thus we may show
Lemma 2.1. Assume for r ≥ 3 that
Ar,F(p) =
Mℓ +O
log2 R
if r = 2ℓ
log2 R
otherwise,
(2.12)
and that there is a nice function gM such that
gM(x) = M2x
2 +M3x
3 + · · · =
ℓ. (2.13)
Then the contribution from the r ≥ 3 terms in Theorem 1.1 is
− 2φ̂(0)
(p+ 1)2
· (p− 1) log p
log3R
. (2.14)
12 STEVEN J. MILLER
Proof. The big-Oh term in Ar,F(p) yields an error of size 1/ log
3R. The contribution from the
r ≥ 3 terms in Theorem 1.1 may therefore be written as
− 2φ̂(0)
(p− 1) log p
(p+ 1)2
log3R
. (2.15)
The result now follows by using the generating function gM to evaluate the ℓ-sum. �
Remark 2.2. In the above lemma, note that gM(x) has even and odd powers of x, even though the
known or conjectured distribution is even. This is because the expansion in Theorem 1.1 involves
pr/2, and the only contribution is when r = 2ℓ.
Lemma 2.3. If the distribution of the weighted Fourier coefficients satisfies Sato-Tate (normalized
to be a semi-circle) with errors in the moments of size O(1/ log2R), then the contribution from the
r ≥ 3 terms in Theorem 1.1 is
ST; eA φ̂(0)
log3R
, (2.16)
where
ST; eA =
(2p+ 1)(p− 1) log p
p(p+ 1)3
≈ .4160714430. (2.17)
If the Fourier coefficients vanish except for primes congruent to a mod b (where φ(b) = 2) and the
distribution of the weighted Fourier coefficients for p ≡ a mod b satisfies the analogue of Sato-
Tate for elliptic curves with complex multiplication, then the contribution from the r ≥ 3 terms in
Theorem 1.1 is
− 2γCM,a,b φ̂(0)
log3R
, (2.18)
where
γCM,a,b =
p≡a mod b
2(3p+ 1) log p
(p+ 1)3
. (2.19)
In particular,
γCM1,3 ≈ .38184489, γCM1,4 ≈ 0.46633061. (2.20)
Proof. If the distribution of the weighted Fourier coefficients satisfies Sato-Tate (normalized to be
a semi-circle here), then Mℓ = Cℓ =
, the ℓth Catalan number. We have (see sequence
A000108 in [Sl])
gST(x) =
1− 4x
− 1− x = 2x2 + 5x3 + 14x4 + · · · =
(p+ 1)2
2p+ 1
p(p+ 1)2
. (2.21)
The value for γ
ST; eA was obtained by summing the contributions from the first million primes.
For curves with complex multiplication, Mℓ = Dℓ = 2 · 12
; while the actual sequence is just(
= (ℓ+1)Cℓ, we prefer to write it this way as the first 2 emphasizes that the contribution is zero
LOWER ORDER TERMS IN 1-LEVEL DENSITIES 13
for half the primes, and it is 1
that is the natural sequences to study. The generating function is
gCM(x) =
1− 4x√
1− 4x
− 2x = 6x2 + 20x3 + 70x4 + · · · =
(p+ 1)2
2(3p+ 1)
(p− 1)(p+ 1)2
; (2.22)
these numbers are the convolution of the Catalan numbers and the central binomial (see sequence
A000984 in [Sl]). The numerical values were obtained by calculating the contribution from the first
million primes. �
Remark 2.4. It is interesting how close the three sums are. Part of this is due to the fact that these
sums converge rapidly. As the small primes contribute more to these sums, it is not surprising that
γCM1,4 > γCM1,3 (the first primes for γCM1,4 are 5 and 11, versus 7 and 13 for γCM1,3).
Remark 2.5. When we investigate one-parameter families of elliptic curves over Q(T ), it is im-
plausible to assume that for each p the rth moment agrees with the rth moment of the limiting
distribution up to negligible terms. This is because there are at most p data points involved in the
weighted averages Ar,F(p); however, it is enlightening to compare the contribution from the r ≥ 3
terms in these families to the theoretical predictions when we have instantaneous convergence to
the limiting distribution.
We conclude by sketching the argument for identifying the presence of the Sato-Tate distribution
for weight k cuspidal newforms of square-free level N → ∞. In the expansion of λf(p)r, to first
order all that often matters is the constant term; by the Petersson formula this is the case for cuspidal
newforms of weight k and square-free level N → ∞, though this is not the case for families of
elliptic curves with complex multiplication. If r is odd then the constant term is zero, and thus to
first order (in the Petersson formula) these terms do not contribute. For r = 2ℓ even, the constant
term is 1
(2ℓ)!
ℓ!(ℓ+1)!
= Cℓ, the ℓ
th Catalan number. We shall write
λf (p)
br,r−2kλf(p
r−2k), (2.23)
and note that if r = 2ℓ then the constant term is b2ℓ,0 = Cℓ. We have
Ar,F(p) =
WR(F)
f∈S(p)
wR(f)λf(p)
WR(F)
f∈S(p)
wR(f)
br,r−2kλf(p
r−2k) =
br,r−2kAr,F ;k(p), (2.24)
where
Ar,F ;k(p) =
WR(F)
f∈S(p)
wR(f)λf(p
r−2k). (2.25)
We expect the main term to be A2ℓ,F ;0, which yields the contribution described in (2.16).
14 STEVEN J. MILLER
3. FAMILIES OF CUSPIDAL NEWFORMS
Let F be a family of cuspidal newforms of weight k and prime level N ; perhaps we split by
sign (the answer is the same, regardless of whether or not we split). We consider the lower order
correction terms in the limit as N → ∞.
3.1. Weights. Let
ζN(s) =
Z(s, f) =
ζN(s)L(s, f ⊗ f)
; (3.1)
L(s, sym2f) =
ζ(2s)Z(s, f)
ζN(2s)
, Z(1, f) =
ζN(2)
L(1, sym2f). (3.2)
To simplify the presentation, we use the harmonic weights10
wR(f) = ζN(2)/Z(1, f) = ζ(2)/L(1, sym
2f), (3.4)
and note that
WR(F) =
wR(f) =
(k − 1)N
+O(N−1); (3.5)
we may take R to be the analytic conductor, so R = 15N/64π2. We have introduced the harmonic
weights to facilitate applying the Petersson formula to calculate the average moments Ar,F(p) from
studying Ar,F ;k(p). The Petersson formula (see Corollary 2.10, Equation (2.58) of [ILS]) yields,
for m,n > 1 relatively prime to the level N ,
WR(F)
wR(f)λf(m)λf(n) = δmn + O
(mn)1/4
log 2mnN
k5/6N
, (3.6)
where δmn = 1 if m = n and 0 otherwise.
3.2. Results. From Theorem 1.1, there are five terms to analyze: SA′(F), S0(F), S1(F), S2(F)
and SA(F). One advantage of our approach (replacing sums of αf(p)r + βf(p)r with moments of
λf(p)
r) is that the Fourier coefficients of a generic cuspidal newform should follow Sato-Tate; the
Petersson formula easily gives Sato-Tate on average as we vary the forms while letting the level
tend to infinity, which is all we need here. Thus Ar,F(p) is basically the r
th moment of the Sato-Tate
distribution (which, because of our normalizations, is a semi-circle here). The odd moments of the
semi-circle are zero, and the (2ℓ)th moment is Cℓ. If we let
P (ℓ) =
(p− 1) log p
(p + 1)2
, (3.7)
10The harmonic weights are essentially constant. By [I1, HL] they can fluctuate within the family as
N−1−ǫ ≪k ωR(f) ≪k N−1+ǫ; (3.3)
if we allow ineffective constants we can replace N ǫ with logN for N large.
LOWER ORDER TERMS IN 1-LEVEL DENSITIES 15
then we find
SA,0(F) = −
2φ̂(0)
CℓP (ℓ), (3.8)
and we are writing the correction term as a weighted sum of the expected main term of the moments
of the Fourier coefficients; see Lemma 2.3 for another way of writing this correction. These expan-
sions facilitate comparison with other families where the coefficients do not follow the Sato-Tate
distribution (such as one-parameter families of elliptic curves with complex multiplication).
Below we sketch an analysis of the lower order correction terms of size 1/ logR to families of
cuspidal newforms of weight k and prime levelN → ∞. We analyze the five terms in the expansion
of S(F) in Theorem 1.1.
The following lemma is useful for evaluating many of the sums that arise. We approximated γPNT
below by using the first million primes (see Remark 3.3 for an alternate, more accurate expression
for γPNT). The proof is a consequence of the prime number theorem; see Section 8.1 of [Yo1] for
details.
Lemma 3.1. Let θ(t) =
p≤t log p and E(t) = θ(t)− t. If φ̂ is a compactly support even Schwartz
test function, then
2 log p
p logR
log p
2φ̂(0)
log3R
, (3.9)
where
γPNT = 1 +
dt ≈ −1.33258. (3.10)
Remark 3.2. The constant γPNT also occurs in the definition of the constants c4,1 and c4,2 in [Yo1],
which arise from calculating lower order terms in two-parameter families of elliptic curves. The
constants c4,1 and c4,2 are in error, as the value of γPNT used in [Yo1] double counted the +1.
Remark 3.3. Steven Finch has informed us that γPNT = −γ −
(log p)/(p2 − p); see
http://www.research.att.com/∼njas/sequences/A083343 for a high precision
evaluation and [Lan, RoSc] for proofs.
Theorem 3.4. Let φ̂ be supported in (−σ, σ) for some σ < 4/3 and consider the harmonic weights
wR(f) = ζ(2)/L(1, sym
2f). (3.11)
S(F) = φ(0)
2(−γST;0 + γST;2 − γST; eA + γPNT)φ̂(0)
log3R
(3.12)
where
γST;0 =
2 log p
p(p+1)
≈ 0.7691106216
γST;2 =
(4p2+3p+1) log p
p(p+1)3
≈ 1.1851820642
ST; eA =
ℓ=2CℓP (ℓ) ≈ 0.4160714430
γPNT = 1 +
dt ≈ −1.33258
(3.13)
16 STEVEN J. MILLER
− γST;0 + γST;2 − γST; eA = 0. (3.14)
The notation above is to emphasize that these coefficients arise from the Sato-Tate distribution.
The subscript 0 (resp. 2) indicates that this contribution arises from the A0,F(p) (resp. A2,F(p))
terms, the subscript Ã indicates the contribution from S eA(F) (the Ar,F(p) terms with r ≥ 3),
and we use PNT for the final constant to indicate a contribution from applying the Prime Number
Theorem to evaluate sums of our test function.
Proof. The proof follows by calculating the contribution of the five pieces in Theorem 1.1. We
assume φ̂ is an even Schwartz function such that supp(φ̂) ⊂ (−σ, σ), with σ < 4/3, F is the
family of weight k and prime level N cuspidal newforms (with N → ∞), and we use the harmonic
weights of §3.1. Straightforward algebra shows11
(1) SA′(F) ≪ N−1/2.
(2) SA(F) = −
ST; eA
bφ(0)
R.11 log2 R
N .73
N3σ/4 logR
. In particular, for test
functions supported in (−4/3, 4/3) we have SA(F) = −
ST; eA
bφ(0)
+O (R−ǫ), where γ
ST; eA
≈ .4160714430 (see Lemma 2.3).
(3) S0(F) = φ(0)+ 2(2γPNT−γST;0)
bφ(0)
log3 R
, where γST;0 =
2 log p
p(p+1)
≈ 0.7691106216,
γPNT = 1 +
dt ≈ −1.33258.
(4) S1(F) ≪ logNN
≪ N 34σ−1 logN .
(5) Assume σ < 4. Then
S2(F) = −
− 2γPNT φ̂(0)
γST;2 φ̂(0)
log3R
γST;2 =
(4p2 + 3p+ 1) log p
p(p+ 1)3
≈ 1.1851820642 (3.15)
and γPNT is defined in (3.10).
The SA′(F) piece does not contribute, and the other four pieces contribute multiples of γST;0,
γST;2, γST;3 and γPNT. �
Remark 3.5. Numerical calculations will never suffice to show that −γST;1 + γST;2 − γST; eA is
exactly zero; however, we have
− γST;0 + γST;2 − γST; eA =
p(p+ 1)
4p2 + 3p+ 1
p(p+ 1)3
(2p+ 1)(p− 1)
p(p+ 1)3
log p
0 · log p = 0. (3.16)
This may also be seen by calculating the lower order terms using a different variant of the explicit
formula. Instead of expanding in terms of αf (p)
m + βf(p)
m we expand in terms of λf (p
m). The
11Except for the SA(F) piece, where a little care is required; see Appendix A for details.
LOWER ORDER TERMS IN 1-LEVEL DENSITIES 17
terms which depend on the Fourier coefficients are given by
WR(F)
wR(f)
λf(p)
m log p
pm/2 logR
log p
log p
p logR
log p
WR(F)
wR(f)
m) log p
pm/2 logR
log p
(m+ 2)
log p
(3.17)
this follows from trivially modifying Proposition 2.1 of [Yo1]. ForN a prime, the Petersson formula
shows that only the second piece contributes for σ < 4/3, and we regain our result that the lower
order term of size 1/ logR from the Fourier coefficients is just 2γPNTφ̂(0)/ logR. We prefer our
expanded version as it shows how the moments of the Fourier coefficients at the primes influence
the correction terms, and will be useful for comparisons with families that either do not satisfy
Sato-Tate, or do not immediately satisfy Sato-Tate with negligible error for each prime.
4. PRELIMINARIES FOR FAMILIES OF ELLIPTIC CURVES
4.1. Notation. We review some notation and results for elliptic curves; see [Kn, Si1, Si2] for more
details. Consider a one-parameter family of elliptic curves over Q(T ):
E : y2 = x3 + A(T )x+B(T ), A(T ), B(T ) ∈ Z[T ]. (4.1)
For each t ∈ Z we obtain an elliptic curve Et by specializing T to t. We denote the Fourier
coefficients by at(p) = λt(p)
p; by Hasse’s bound we have |at(p)| ≤ 2
p or |λt(p)| ≤ 2. The
discriminant and j-invariant of the elliptic curve Et are
∆(t) = −16(4A(t)3 + 27B(t)2), j(t) = −1728 · 4A(t)3/∆(t). (4.2)
Consider an elliptic curve y2 = x3 + Ax + B (with A,B ∈ Z) and a prime p ≥ 5. As p ≥ 5,
the equation is minimal if either p4 does not divide A or p6 does not divide B. If the equation is
minimal at p then
at(p) = −
x mod p
x3 + A(t)x+B(t)
= p+ 1−Nt(p), (4.3)
where Nt(p) is the number of points (including infinity) on the reduced curve Ẽ mod p. Note that
at+mp(p) = at(p). This periodicity is our analogue of the Petersson formula; while it is significantly
weaker, it will allow us to obtain results for sufficiently small support.
Let E be an elliptic curve with minimal Weierstrass equation at p, and assume p divides the
discriminant (so the reduced curve modulo p is singular). Then aE(p) ∈ {−1, 0, 1}, depending on
the type of reduction. By changing coordinates we may write the reduced curve as (y − αx)(y −
βx) = x3. If α = β then we say E has a cusp and additive (or unstable) reduction at p, and aE(p) =
0. If α 6= β then E has a node and multiplicative (or semi-stable) reduction at p; if α, β ∈ Q we
say E has split reduction and aE(p) = 1, otherwise it has non-split reduction and aE(p) = −1. We
shall see later that many of our arguments are simpler when there is no multiplicative reduction,
which is true for families with complex multiplication.
Our arguments below are complicated by the fact that for many p there are t such that y2 =
x3 + A(T )x+ B(T ) is not minimal at p when we specialize T to t. For the families we study, the
specialized curve at T = t is minimal at p provided pk (k depends on the family) does not divide a
18 STEVEN J. MILLER
polynomial D(t) (which also depends on the family, and is the product of irreducible polynomial
factors of ∆(t)). For example, we shall later study the family with complex multiplication
y2 = x3 +B(6T + 1)κ, (4.4)
whereB|6∞ (i.e., p|B implies p is 2 or 3) and κ ∈ {1, 2}). Up to powers of 2 and 3, the discriminant
is ∆(T ) = (6T + 1)2κ, and note that (6t + 1, 6) = 1 for all t. Thus for a given t the equation is
minimal for all primes provided that 6t + 1 is sixth-power free if κ = 1 and cube-free if κ = 2. In
this case we would take D(t) = 6t+ 1 and k = 6/κ. To simplify the arguments, we shall sieve our
families, and rather than taking all t ∈ [N, 2N ] instead additionally require that D(t) is kth power
free. Equivalently, we may take all t ∈ [N, 2N ] and set the weights to be zero if D(t) is not kth
power free. Thus throughout the paper we adopt the following conventions:
• the family is y2 = x3 + A(T )x + B(T ) with A(T ), B(T ) ∈ Z[T ], and we specialize T to
t ∈ [N, 2N ] with N → ∞;
• we associate polynomials D1(T ), . . . , Dd(T ) and integers k1, . . . , kd ≥ 3, and the weights
are wR(t) = 1 if t ∈ [N, 2N ] and Di(t) is kith power free, and 0 otherwise;
• logR is the average log-conductor of the family, and logR = (1 + o(1)) logN (see [DM2,
Mil2]).
4.2. Sieving. For ease of notation, we assume that we have a family where D(T ) is an irreducible
polynomial, and thus there is only one power, say k; the more general case proceeds analogously.
We assume that k ≥ 3 so that certain sums are small (if k ≤ 2 we need to assume either the ABC
of Square-Free Sieve Conjecture). Let δkNd exceed the largest value of |D(t)| for t ∈ [N, 2N ]. We
say a t ∈ [N, 2N ] is good if D(t) is kth power free; otherwise we say t is bad. To determine the
lower order correction terms we must evaluate S(F), which is defined in (1.10). We may write
S(F) =
WR(F)
wR(t)S(t). (4.5)
As wR(t) = 0 if t is bad, for bad t we have the freedom of defining S(t) in any manner we may
choose. Thus, even though the expansion for at(p) in (4.3) requires the elliptic curve Et to be
minimal at p, we may use this definition for all t. We use inclusion - exclusion to write our sums
in a more tractable form; the decomposition is standard (see, for example, [Mil2]). Letting ℓ be an
integer (its size will depend on d and k), we have
S(F) = 1
WR(F)
D(t) k−power free
wR(t)S(t)
WR(F)
logℓ N∑
D(t)≡0 mod dk
S(t) +
WR(F)
δNd/k∑
d=1+logℓ N
D(t)≡0 mod dk
S(t),
(4.6)
where µ is the Möbius function. For many families we can show that
D(t)≡0 mod dk
S(t)2 = O
. (4.7)
LOWER ORDER TERMS IN 1-LEVEL DENSITIES 19
If this condition12 holds, then applying the Cauchy-Schwarz inequality to (4.6) yields
S(F) = 1
WR(F)
logℓ N∑
D(t)≡0 mod dk
S(t) +O
WR(F)
δNd/k∑
d=1+logℓ N
WR(F)
logℓ N∑
D(t)≡0 mod dk
S(t) +O
WR(F)
· (logN)−(
k−1)·ℓ
. (4.8)
For all our families WR(F) will be of size N (see [Mil2] for a proof). Thus for ℓ sufficiently
large the error term is significantly smaller than 1/ log3R, and hence negligible (remember logR =
(1 + o(1)) logN). Note it is important that k ≥ 3, as otherwise we would have obtained logN to
a non-negative power (as we would have summed 1/d). For smaller k we may argue by using the
ABC or Square-Free Sieve Conjectures.
The advantage of the above decomposition is that the sums are over t in arithmetic progressions,
and we may exploit the relation at+mp(p) = at(p) to determine the family averages by evaluating
sums of Legendre symbols. This is our analogue, poor as it may be, to the Petersson formula.
There is one technicality that arises here which did not in [Mil2]. There the goal was only
to calculate the main term in the n-level densities; thus “small” primes (p less than a power of
logN) could safely be ignored. If we fix a d and consider all t with D(t) ≡ 0 mod dk, we ob-
tain a union of arithmetic progressions, with each progression having step size dk. We would
like to say that we basically have (N/dk)/p complete sums for each progression, with summands
at0(p), at0+dkp(p), at0+2dkp(p), and so on. The problem is that if p|d then we do not have a complete
sum, but rather we have the same term each time! We discuss how to handle this obstruction in the
next sub-section.
4.3. Moments of the Fourier Coefficients and the Explicit Formula. Our definitions imply that
Ar,F(p) is obtained by averaging λt(p)
r over all t ∈ [N, 2N ] such that p |r ∆(t); the remaining t
yield A′r,F(p). We have sums such as
WR(F)
logℓ N∑
D(t)≡0 mod dk
S(t). (4.9)
In all of our families D(T ) will be the product of the irreducible polynomial factors of ∆(T ). For
ease of exposition, we assume D(T ) is given by just one factor.
We expand S(F) and S(t) by using Theorem 1.1. The sum of S(t) over t with D(t) ≡ 0 mod dk
breaks up into two types of sums, those where ∆(t) ≡ 0 mod p and those where ∆(t) 6≡ 0 mod p.
For a fixed d, the goal is to use the periodicity of the t-sums to replace Ar,F(p) with complete sums.
Thus we need to understand complete sums. If t ∈ [N, 2N ], d ≤ logℓN and p is fixed, then the
set of t such that D(t) ≡ 0 mod dk is a union of arithmetic progressions; the number of arithmetic
progressions equals the number of distinct solutions to D(t) ≡ 0 mod dk, which we denote by
k). We have (N/dk)/p complete sums, and at most p summands left over.
12Actually, this condition is a little difficult to use in practice. It is easier to first pull out the sum over all primes p
and then square; see [Mil2] for details.
20 STEVEN J. MILLER
Recall
Ar,F(p) =
WR(F)
f∈S(p)
wR(f)λf(p)
r, A′r,F(p) =
WR(F)
f 6∈S(p)
wR(f)λf(p)
r, (4.10)
and set
Ar,F(p) =
t mod p
p |r∆(t)
at(p)
r = pr/2
t mod p
p |r∆(t)
λt(p)
r, A′r,F(p) =
t mod p
p|∆(t)
at(p)
r. (4.11)
Lemma 4.1. Let D be a product of irreducible polynomials such that (i) for all t no two factors are
divisible by the same prime; (ii) the same k ≥ 3 (see the conventions on page 18) is associated to
each polynomial factor. For any ℓ ≥ 7 we have
Ar,F(p) =
Ar,F(p)
p · pr/2
logℓ/2N
A′r,F(p) =
A′r,F(p)
p · pr/2
logℓ/2N
. (4.12)
Proof. For our family, the d ≥ logℓN terms give a negligible contribution. We rewrite Ar,F(p) as
Ar,F(p) =
WR(F)
t∈[N,2N],p |rD(t)
D(t) k−power free
λt(p)
WR(F)
logℓ N∑
t∈[N,2N],p |rD(t)
D(t)≡0 mod dk
λt(p)
log−ℓ/2N
WR(F)
logℓ N∑
k)N/dk
t mod p
p |rD(t)
λt(p)
WR(F)
logℓ N∑
WR(F)
logℓ N∑
µ(d)δp|d
k)N/dk
t mod p
p |rD(t)
λt(p)
 , (4.13)
where δp|d = 1 if p|d and 0 otherwise. For sufficiently small support the big-Oh term above is
negligible. As k ≥ 3, we have
WR(F) = N
1− νD(d
logℓ/2N
logℓ N∑
µ(d)νD(d
logℓ/2N
. (4.14)
LOWER ORDER TERMS IN 1-LEVEL DENSITIES 21
For the terms with µ(d)δp|d in (4.13), we may write d as d̃p, with (d̃, p) = 1 (the µ(d) factor forces
d to be square-free, so p||d). For sufficiently small support, (4.13) becomes
Ar,F(p)
p · pr/2
1− νD(p
log−ℓ/2N
; (4.15)
this is because
WR(F)
logℓ N∑
µ(d)νD(d
µ(p)νD(p
logℓ N∑
p |rd̃
µ(d̃)νD(d̃
= −νD(p
1− νD(p
logℓ/2N
(4.16)
(the last line follows because of the multiplicativity of νD (see for example [Nag]) and the fact that
we are missing the factor corresponding to p). The proof for A′r,F(p) follows analogously. �
We may rewrite the expansion in Theorem 1.1. We do not state the most general version possible,
but rather a variant that will encompass all of our examples.
Theorem 4.2 (Expansion for S(F) for many elliptic curve families). Let y2 = x3+A(T )x+B(T )
be a family of elliptic curves over Q(T ). Let ∆(T ) be the discriminant (and the only primes dividing
the greatest common divisor of the coefficients of ∆(T ) are 2 or 3), and let D(T ) be the product of
the irreducible polynomial factors of ∆(T ). Assume for all t that no prime simultaneously divides
two different factors of D(t), that each specialized curve has additive reduction at 2 and 3, and that
there is a k ≥ 3 such that for p ≥ 5 each specialized curve is minimal provided that D(T ) is kth
power free (if the equation is a minimal Weierstrass equation for all p ≥ 5 we take k = ∞); thus
we have the same k for each irreducible polynomial factor of D(T ). Let νD(d) denote the number
of solutions to D(t) ≡ 0 mod d. Set wR(t) = 1 if t ∈ [N, 2N ] and D(t) is kth power free, and 0
otherwise. Let
Ar,F(p) =
t mod p
p |r∆(t)
at(p)
r = pr/2
t mod p
p |r∆(t)
λt(p)
r, A′r,F(p) =
t mod p
p|∆(t)
at(p)
ÃF(p) =
t mod p
p|r∆(t)
at(p)
p3/2(p+ 1− at(p))
t mod p
p |r∆(t)
λt(p)
p+ 1− λt(p)
HD,k(p) = 1 +
1− νD(p
. (4.17)
22 STEVEN J. MILLER
We have
S(F) = −2φ̂(0)
A′m,F(p)HD,k(p) log p
pm+1 logR
−2φ̂(0)
2A0,F(p)HD,k(p) log p
p2(p+ 1) logR
2A0,F(p)HD,k(p) log p
p2 logR
log p
A1,F(p)HD,k(p)
log p
log p
+ 2φ̂(0)
A1,F(p)HD,k(p)(3p+ 1)
p2(p+ 1)2
log p
A2,F(p)HD,k(p) log p
p3 logR
log p
+ 2φ̂(0)
A2,F(p)HD,k(p)(4p2 + 3p+ 1) log p
p3(p+ 1)3 logR
−2φ̂(0)
ÃF(p)HD,k(p)p3/2(p− 1) log p
p(p+ 1)3 logR
log3R
= SA′(F) + S0(F) + S1(F) + S2(F) + S eA(F) +O
log3R
. (4.18)
If the family only has additive reduction (as is the case for our examples with complex multiplica-
tion), then the A′m,F(p) piece contributes 0.
Proof. The proof follows by using Lemma 4.1 to simplify Theorem 1.1, and (2.8) to replace the
φ̂(m log p/ logR) terms with φ̂(0) + O(log−2R) in the A′m,F(p) terms. See Remark 1.2 for com-
ments on the need to numerically evaluate the ÃF(p) piece. �
For later use, we record a useful variant of Lemma 3.1.
Lemma 4.3. Let ϕ be the Euler totient function, and
θa,b(t) =
p≡a mod b
log p, Ea,b(t) = θa,b(t)−
. (4.19)
If φ̂ is a compactly support even Schwartz test function, then
2 log p
p logR
log p
2φ̂(0)
2E1,3(t)
log3R
, (4.20)
where
γPNT;1,3 = 1 +
2E1,3(t)
dt ≈ −2.375
γPNT;1,4 = 1 +
2E1,4(t)
dt ≈ −2.224; (4.21)
γPNT;1,3 and γPNT;1,4 were approximated by integrating up to the four millionth prime, 67,867,979.
LOWER ORDER TERMS IN 1-LEVEL DENSITIES 23
Remark 4.4. Steven Finch has informed us that, similar to Remark 3.3, using results from [Lan,
Mor] yields formulas for γPNT;1,3 and γPNT;1,4 which converge more rapidly:
γPNT;1,3 = −2γ − 4 log 2π + log 3 + 6 log Γ
p≡1,2 mod 3
log p
p2 − pδ1,3(p)
≈ −2.375494
γPNT;1,4 = −2γ − 3 log 2π + 4 log Γ
p≡1,3 mod 4
log p
p2 − pδ1,4(p)
≈ −2.224837; (4.22)
here γ is Euler’s constant and δ1,n(p) = 1 if p ≡ 1 mod n and 0 otherwise.
5. EXAMPLES: ONE-PARAMETER FAMILIES OF ELLIPTIC CURVES OVER Q(T )
We calculate the lower order correction terms for several one-parameter families of elliptic curves
over Q(T ), and compare the results to what we would obtain if there was instant convergence (for
each prime p) to the limiting distribution of the Fourier coefficients. We study families with and
without complex multiplication, as well as families with forced torsion points or rank. We perform
the calculations in complete detail for the first family, and merely highlight the changes for the other
families.
5.1. CM Example: The family y2 = x3 +B(6T + 1)κ over Q(T ).
5.1.1. Preliminaries. Consider the following one-parameter family of elliptic curves over Q(T )
with complex multiplication:
y2 = x3 +B(6T + 1)κ, B ∈ {1, 2, 3, 6}, κ ∈ {1, 2}, k = 6/κ. (5.1)
We obtain slightly different behavior for the lower order correction terms depending on whether
or not B is a perfect square for all primes congruent to 1 modulo 3. For example, if B = b2 and
κ = 2, then we have forced a torsion point of order 3 on the elliptic curve over Q(T ), namely
(0, b(6T + 1)). The advantage of using 6T + 1 instead of T is that (6T + 1, 6) = 1, and thus we
do not need to worry about the troublesome primes 2 and 3 (each at(p) = 0 for p ∈ {2, 3}). Up to
powers of 2 and 3 the discriminant is (6T + 1)κ; thus we take D(T ) = 6T + 1. For each prime p
the specialized curve Et is minimal at p provided that p
2k |r 6t + 1. If p2k|6t + 1 then wR(t) = 0,
so we may define the summands any way we wish; it is convenient to use (4.3) to define at(p),
even though the curve is not minimal at p. In particular, this implies that at(p) = 0 for any t where
p3|6t+ 1.
One very nice property of our family is that it only has additive reduction; thus if p|D(t) but
p2k |rD(t) then at(p) = 0. As our weights restrict our family to D(t) being k = 6/κ power free, we
always use (4.3) to define at(p).
It is easy to evaluate A1,F(p) and A2,F(p). While these sums are the average first and second
moments over primes not dividing the discriminant, as at(p) = 0 for p|∆(t) we may extend these
sums to be over all primes.
24 STEVEN J. MILLER
We use Theorem 4.2 to write the 1-level density in a tractable manner. Straightforward calcula-
tion (see Appendix B.1 for details) shows that
A0,F(p) =
p− 1 if p ≥ 5
0 otherwise
A1,F(p) = 0
A2,F(p) =
2p2 − 2p if p ≡ 1 mod 3
0 otherwise.
(5.2)
Not surprisingly, neither the zeroth, first or second moments depend on B or on κ; this universality
leads to the common behavior of the main terms in the n-level densities. We shall see dependence
on the parameters B and κ in the higher moments Ar,F(p), and this will lead to different lower
order terms for the different families.
As we are using Theorem 4.2 instead of Theorem 1.1, each prime sum is weighted by
HD,k(p) = 1 +
= HmainD,k (p) +H
sieve
D,k (p), (5.3)
with HmainD,k (p) = 1. H
sieve
D,k (p) arises from sieving our family to D(t) being (6/κ)-power free. We
shall calculate the contribution of these two pieces separately. We expect the contribution from
HsieveD,k (p) to be significantly smaller, as each p-sum is decreased by approximately 1/p
5.1.2. Contribution from HmainD,k (p).
We first calculate the contributions from the four pieces of HmainD,k (p). We then combine the
results, and compare to what we would have had if the Fourier coefficients followed the Sato-Tate
distribution or for each prime immediately perfectly followed the complex multiplication analogue
of Sato-Tate.
Lemma 5.1. Let supp(φ̂) ⊂ (−σ, σ). We have
S0(F) = φ(0) +
2φ̂(0) · (2γPNT − γ(≥5)CM;0 − γ
log3R
+O(Nσ−1), (5.4)
where
CM;0 =
4 log p
p(p+ 1)
≈ 0.709919
2,3 =
2 log 2
2 log 3
≈ 1.4255554, (5.5)
and γPNT is defined in Lemma 3.1.
Note γ(≥5)CM;0 is almost 2γST;0 (see (3.13)); the difference is that here p ≥ 5.
Proof. Substituting for A0,F(p) and using (2.8) yields
S0(F) = −
2φ̂(0)
4 log p
p(p+ 1)
2 log p
p logR
log p
log3R
. (5.6)
LOWER ORDER TERMS IN 1-LEVEL DENSITIES 25
The first prime sum converges; using the first million primes we find γ
CM;0 ≈ 0.709919. The
remaining piece is
2 log p
p logR
log p
− 2φ̂(0)
2 log 2
2 log 3
log3R
. (5.7)
The claim now follows from the definition of γ
2,3 and using Lemma 3.1 to evaluate the remaining
sum. �
Lemma 5.2. Let supp(φ̂) ⊂ (−σ, σ) and
(1,3)
CM;2 =
p≡1 mod 3
2(5p2 + 2p+ 1) log p
p(p+ 1)3
≈ 0.6412881898. (5.8)
S2(F) = −
2φ̂(0) · (−γPNT;1,3 + γ(1,3)CM;2)
log3R
+O(Nσ−1), (5.9)
where γPNT;1,3 = −2.375494 (see Lemma 4.3 for its definition).
Proof. Substituting our formula for A2,F(p) and collecting the pieces yields
S2(F) = −2
p≡1 mod 3
2 log p
log p
2φ̂(0)
p≡1 mod 3
2(5p2 + 2p+ 1) log p
p(p+ 1)3
. (5.10)
The first sum is evaluated by Lemma 4.3. The second sum converges, and was approximated by
taking the first four million primes. �
Lemma 5.3. For the families FB,κ: y2 = x3 + B(6T + 1)κ with B ∈ {1, 2, 3, 6} and κ ∈ {1, 2},
we have SÃ(F) = −2γ
(1,3)
CM;Ã,B,κ
φ̂(0)/ logR + O(log−3R), where
(1,3)
CM;Ã;1,1
≈ .3437
(1,3)
CM;Ã;1,2
≈ .4203
(1,3)
CM;Ã;2,2
≈ .5670
(1,3)
CM;Ã;3,2
≈ .1413
(1,3)
CM;Ã;6,2
≈ .2620; (5.11)
the error is at most .0367.
Proof. As the sum converges, we have written a program in C (using PARI as a library) to approxi-
mate the answer. We used all primes p ≤ 48611 (the first 5000 primes), which gives us an error of
at most about 8√
p+1−2√p ≈ .0367. The error should be significantly less, as this is assuming no
oscillation. We also expect to gain a factor of 1/2 as half the primes have zero contribution. �
Remark 5.4. When κ = 1 a simple change of variables shows that all four values of B lead to the
same behavior. The case of κ = 2 is more interesting. If κ = 2 and B = 1, then we have the torsion
point (0, 6T + 1) on the elliptic surface. If B ∈ {2, 3, 6} and
= 1 then (0, 6t+ 1 mod p) is on
the curve Et mod p, while if
= −1 then (0, 6t+ 1 mod p) is not on the reduced curve.
26 STEVEN J. MILLER
5.1.3. Contribution from HsieveD,k (p).
Lemma 5.5. Notation as in Lemma 5.3, the contributions from the HsieveD,k (p) sieved terms to the
lower order corrections are
(1,3)
CM, sieve;012 + γ
(1,3)
CM, sieve;B,κ)φ̂(0)
log3R
, (5.12)
(1,3)
CM, sieve;012 ≈ −.004288
(1,3)
CM, sieve;1,1 ≈ .000446
(1,3)
CM, sieve;1,2 ≈ .000699
(1,3)
CM, sieved;2,2 ≈ .000761
(1,3)
CM, sieve;3,2 ≈ .000125
(1,3)
CM, sieve;6,2 ≈ .000199, (5.13)
where the errors in the constants are at most 10−15 (we are displaying fewer digits than we could!).
Proof. The presence of the additional factor of 1/p3 ensures that we have very rapid convergence.
The contribution from the r ≥ 3 terms was calculated at the same time as the contribution in Lemma
5.3, and is denoted by γ
(1,3)
CM,sieve;B,κ. The other terms (r ∈ {0, 1, 2}) were computed in analogous
manners as before, and grouped together into γ(1,3)CM, sieve;012. �
5.1.4. Results. We have shown
Theorem 5.6. For σ < 2/3, the HmainD,k (p) terms contribute φ(0)/2 to the main term. The lower
order correction from the HmainD,k (p) and H
sieve
D,k (p) terms is
2φ̂(0) · (2γPNT − γ(≥5)CM;0 − γ
2,3 − γPNT;1,3 + γ
(1,3)
CM;2 − γ
(1,3)
CM;Ã,B,κ
− γ(1,3)CM, sieve;012 − γ
(1,3)
CM, sieve;B,κ)
log3R
. (5.14)
Using the numerical values of our constants for the five choices of (B, κ) gives, up to errors of size
O(log−3R), lower order terms of approximately
B = 1, κ = 1 : −2.124 · 2φ̂(0)/ logR,
B = 1, κ = 2 : −2.201 · 2φ̂(0)/ logR,
B = 2, κ = 2 : −2.347 · 2φ̂(0)/ logR
B = 3, κ = 2 : −1.921 · 2φ̂(0)/ logR
B = 6, κ = 2 : −2.042 · 2φ̂(0)/ logR. (5.15)
These should be contrasted to the family of cuspidal newforms, whose correction term was
γPNT ·
2φ̂(0)
≈ −1.33258 · 2φ̂(0)
. (5.16)
LOWER ORDER TERMS IN 1-LEVEL DENSITIES 27
Remark 5.7. The most interesting piece in the lower order terms is from the weighted moment
sums with r ≥ 3 (see Lemma 5.3); note the contribution from the sieving is significantly smaller
(see Lemma 5.5). As each curve in the family has complex multiplication, we expect the limiting
distribution of the Fourier coefficients to differ from Sato-Tate; however, the coefficients satisfy
a related distribution (it is uniform if we consider the related curve over the quadratic field; see
[Mur]). This distribution is even, and the even moments are: 2, 6, 20, 70, 252 and so on. In
general, the 2ℓth moment is Dℓ = 2 · 12
(the factor of 2 is because the coefficients vanish for
p ≡ 2 mod 3, so those congruent to 2 modulo 3 contribute double); note the 2ℓth moment of the
Sato-Tate distribution is Cℓ =
. The generating function is
gCM(x) =
1− 4x√
1− 4x
− 2x = 6x2 + 20x3 + 126x4 + · · · =
ℓ (5.17)
(see sequence A000984 in [Sl]). The contribution from the r ≥ 3 terms is
− 2φ̂(0)
p≡1 mod 3
(p− 1) log p
(p+ 1)2
. (5.18)
Using the generating function, we see that the ℓ-sum is just 2(3p + 1)/(p − 1)(p + 1)2, so the
contribution is
2φ̂(0)
p≡1 mod 3
2(3p+ 1) log p
(p+ 1)3
(1,3)
CM;Ã
φ̂(0)
, (5.19)
where taking the first million primes yields
(1,3)
CM;Ã
≈ .38184489. (5.20)
It is interesting to compare the expected contribution from the Complex Multiplication distribution
(for the moments r ≥ 3) and that from the Sato-Tate distribution (for the moments r ≥ 3). The
contribution from the Sato-Tate, in this case, was shown in Lemma 2.3 to be
SA,0(F) = −
ST; eA φ̂(0)
, γST ≈ 0.4160714430. (5.21)
Note how close this is to .38184489, the contribution from the Complex Multiplication distribution.
5.2. CM Example: The family y2 = x3−B(36T +6)(36T +5)x over Q(T ). The analysis of this
family proceeds almost identically to the analysis for the families y2 = x3+B(6T +1)κ over Q(T ),
with trivial modifications because D(T ) has two factors; note no prime can simultaneously divide
both factors, and each factor is of degree 1. The main difference is that now at(p) = 0 whenever
p ≡ 3 mod 4 (as is seen by sending x → −x). We therefore content ourselves with summarizing
the main new feature.
There are two interesting cases. If B = 1 then the family has rank 1 over Q(T ) (see Lemma
B.5); note in this case that we have the point (36T + 6, 36T + 6). If B = 2 then the family
has rank 0 over Q(T ). This follows by trivially modifying the proof in Lemma B.5, resulting in
A1,F(p) = −2p
if p ≡ 1 mod 4 and 0 otherwise (which averages to 0 by Dirichlet’s Theorem
for primes in arithmetic progressions).
As with the previous family, the most interesting pieces are the lower order correction terms from
S eA(F), namely the pieces from HmainD,k (p) and HsieveD,k (p) (as we must sieve). We record the results
28 STEVEN J. MILLER
from numerical calculations using the first 10,000 primes. We write the main term as γ
(1,4)
CM; eA,B
(1, 4) denotes that there is only a contribution from p ≡ 1 mod 4) and the sieve term as γ(1,4)CM,sieve;B.
We find that
(1,4)
CM; eA,1
≈ −0.1109 γ(1,4)CM,sieve;1 ≈ −.0003
(1,4)
CM; eA,2
≈ 0.6279 γ(1,4)CM,sieve;2 ≈ .0013.
(5.22)
What is fascinating here is that, when B = 1, the value of γ
(1,4)
CM; eA,B
is significantly lower than
what we would predict for a family with complex multiplication. A natural explanation for this is
that the distribution corresponding to Sato-Tate for curves with complex multiplication cannot be
the full story (even in the limit) for a family with rank. Rosen and Silverman [RoSi] prove
Theorem 5.8 (Rosen-Silverman). Assume Tate’s conjecture holds for a one-parameter family E of
elliptic curves y2 = x3+A(T )x+B(T ) over Q(T ) (Tate’s conjecture is known to hold for rational
surfaces). Let AE(p) =
t mod p at(p). Then
−AE(p) log p = rank E(Q(T )). (5.23)
Thus if the elliptic curves have positive rank, there is a slight bias among the at(p) to be negative.
For a fixed prime p the bias is roughly of size −r for each at(p), where r is the rank over Q(T ) and
each at(p) is of size
p. While in the limit as p → ∞ the ratio of the bias to at(p) tends to zero, it
is the small primes that contribute most to the lower order terms. As γ
(1,4)
CM; eA,B
arises from weighted
sums of at(p)
3, we expect this term to be smaller for curves with rank; this is born out beautifully
by our data (see (5.22)).
5.3. Non-CM Example: The family y2 = x3 − 3x + 12T over Q(T ). We consider the family
y2 = x3−3x+12T over Q(T ); note this family does not have complex multiplication. For all t the
above is a global minimal Weierstrass equation, and at(2) = at(3) = 0. Straightforward calculation
(see Appendix B.3 for details) shows that
A0,F(p) =
p− 2 if p ≥ 5
0 otherwise
A1,F(p) =
if p ≥ 5
0 otherwise
A2,F(p) =
p2 − 2p− 2− p
if p ≥ 5
0 otherwise.
(5.24)
Unlike our families with complex multiplication (which only had additive reduction), here we
have multiplicative reduction13, and must calculate A′m,F(p). We have
A′m,F(p) =
0 if p = 2, 3
2 if m is even(
if m is odd;
(5.25)
13As we have multiplicative reduction, for each t as p → ∞ the at(p) satisfy Sato-Tate; see [CHT, Tay].
LOWER ORDER TERMS IN 1-LEVEL DENSITIES 29
this follows (see Appendix B.3) from the fact that for a given p there are only two t modulo p such
that p|∆(t), and one has at(p) =
and the other has at(p) =
We sketch the evaluations of the terms from (4.18) of Theorem 4.2; for this family, note that
HD,k(p) = 1. We constantly use the results from Appendix B.3.
Lemma 5.9. We have SA′(F) = −2γ(3)A′ φ̂(0)/ logR +O(log
−3R), where
A′ = 2
log p
p3 − p
p≡1 mod 12
log p
p2 − 1
p≡5 mod 12
log p
p2 − 1
 ≈ −0.082971426. (5.26)
Proof. As A′m,F(p) =
, the result follows by separately evaluating m even and odd,
and using the geometric series formula. �
Lemma 5.10. We have
S0(F) = φ(0)−
2φ̂(0) · (γ(3)0 + γ
2,3 − 2γPNT)
log3R
, (5.27)
where
(4p− 2) log p
p2(p+ 1)
≈ 0.331539448, (5.28)
γPNT is defined in Lemma 3.1 and γ
2,3 is defined in Lemma 5.1.
Proof. For p ≥ 5 we have A0,F(p) = p− 2. The γ(3)0 term comes from collecting the pieces whose
prime sum converges for any bounded φ̂ (and replacing φ̂(2 log p/ logR) with φ̂(0) at a cost of
O(log−2R)), while the remaining pieces come from using Lemma 3.1 to evaluate the prime sum
which converges due to the compact support of φ̂. �
Lemma 5.11. We have S1(F) = −2γ(3)1 φ̂(0)/ logR +O(log−3R), where
· (p− 1) log p
p2(p+ 1)2
= −0.013643784. (5.29)
Proof. As the prime sums decay like 1/p2, we may replace φ̂(log p/ logR) with φ̂(0) at a cost of
O(log−2R). The claim follows from A1,F(p) =
and simple algebra. �
Lemma 5.12. We have
S2(F) = −
2φ̂(0) · (γ(3)2 − 12γ
2,3 + γPNT)
log3R
, (5.30)
where
p4 − (13 + 7)
p3 − (25 + 6
)p2 − (16 + 2
)p− 4) log p
p3(p+ 1)3
≈ .085627. (5.31)
30 STEVEN J. MILLER
Proof. For p ≥ 5 we have A0,F(p) = p2−2p−2−
p. The γ
2 term comes from collecting the
pieces whose prime sum converges for any bounded φ̂ (and replacing φ̂(2 log p/ logR) with φ̂(0)
at a cost of O(log−2R)), while the remaining pieces come from using Lemma 3.1 to evaluate the
prime sum which converges due to the compact support of φ̂. �
Lemma 5.13. We have S eA(F) = −2γ
φ̂(0)/ logR +O(log−3R), where
≈ .3369. (5.32)
Proof. As the series converges, this follows by direct evaluation. �
We have shown
Theorem 5.14. The S0(F) and S2(F) terms contribute φ(0)/2 to the main term. The lower order
correction terms are
2φ̂(0) ·
A′ + γ
0 + γ
1 + γ
2 + γ
2,3 − γPNT
log3R
; (5.33)
using the calculated and computed values of these constants gives
− 2.703 ·
2φ̂(0)
log3R
. (5.34)
Our result should be contrasted to the family of cuspidal newforms, where the correction term
was of size
γPNT ·
2φ̂(0)
≈ −1.33258 · 2φ̂(0)
. (5.35)
Remark 5.15. It is not surprising that our family of elliptic curves has a different lower order
correction than the family of cuspidal newforms. This is due, in large part, to the fact that we do not
have immediate convergence to the Sato-Tate distribution for the coefficients. This is exasperated
by the fact that most of the contribution to the lower order corrections comes from the small primes.
APPENDIX A. EVALUATION OF SA(F) FOR THE FAMILY OF CUSPIDAL NEWFORMS
Lemma A.1. Notation as in §3, we have
SA(F) = −
ST; eA φ̂(0)
R.11 log2R
N .73
N3σ/4 logR
(A.36)
In particular, for test functions supported in (−4/3, 4/3) we have
SA(F) = −
ST; eA φ̂(0)
, (A.37)
where γ
ST; eA ≈ .4160714430 (see Lemma 2.3).
LOWER ORDER TERMS IN 1-LEVEL DENSITIES 31
Proof. Recall
SA(F) = −2φ̂(0)
Ar,F(p)p
r/2(p− 1) log p
(p+ 1)r+1 logR
. (A.38)
Using |Ar,F(p)| ≤ 2r, we may easily bound the contribution from r large, say r ≥ 1 + 2 logR.
These terms contribute
r=1+2 logR
2rpr/2(p− 1) log p
(p+ 1)r+1 logR
log p
r=1+2 logR
log p
)2 logR
2007 ·
)2 logR
p≥2008
log p
p(2 logR)/3
R.77 logR
; (A.39)
note it is essential that 2
2/3 < 1. Thus it suffices to study r ≤ 2 logR.
SA(F) = −2φ̂(0)
2 logR∑
br,r−2k
Ar,F ;k(p)p
r/2(p− 1) log p
(p+ 1)r+1 logR
R.77 logR
= −2φ̂(0)
(p− 1) log p
logR∑
(p+ 1)2
R.77 logR
− 2φ̂(0)
2 logR∑
k 6=r/2
br,r−2k
Ar,F ;k(p)p
r/2(p− 1) log p
(p+ 1)r+1
. (A.40)
In Lemma 2.3 we handled the first p and ℓ-sum when we summed over all ℓ ≥ 2; however, the
contribution from ℓ ≥ logR is bounded by (8/9)logR ≪ R−.11. Thus
SA(F) = −
2γST;3 φ̂(0)
R.11 logR
2φ̂(0)
2 logR∑
(r−2)/2∑
br,r−2k
Ar,F ;k(p)p
r/2(p− 1) log p
(p+ 1)r+1
. (A.41)
To finish the analysis we must study the br,r−2kAr,F ;k(p) terms. Trivial estimation suffices for all
r when p ≥ 13; in fact, bounding these terms for small primes is what necessitated our restricting
to r ≤ 2 logR. From (3.6) (the Petersson formula with harmonic weights) we find
Ar,F ;k(p) ≪
p(r−2k)/4 log
p(r−2k)/4N
k5/6N
rpr/4 log(pN)
. (A.42)
32 STEVEN J. MILLER
∑(r−2)/2
k=0 br,r−2k| ≤ 2r, we have
SA(F) = −
ST; eA φ̂(0)
R.11 logR
2 logR∑
r2rp3r/4 log(pN)
(p+ 1)r logR
. (A.43)
As our Schwartz test functions restrict p to be at most Rσ, the second error term is bounded by
N logR
log(pN)
2 logR∑
2p3/4
≪ logR
p≤2007
2 logR∑
2p3/4
p≥2008
2 logR∑
2p3/4
2 · 33/4
)2 logR
logR +
p≥2008
2p3/4
p + 1
.27 log2R
p=2011
p−1/4 ≪ log
N .73
N3σ/4 logR
, (A.44)
which is negligible provided that σ < 4/3. �
APPENDIX B. EVALUATION OF Ar,F FOR FAMILIES OF ELLIPTIC CURVES
The following standard result allows us to evaluate the second moment of many one-parameter
families of elliptic curves over Q (see [ALM, BEW] for a proof).
Lemma B.1 (Quadratic Legendre Sums). Assume a and b are not both zero mod p and p > 2. Then
at2 + bt + c
(p− 1)
if p |r b2 − 4ac
otherwise.
(B.1)
B.1. The family y2 = x3 +B(6T + 1)κ over Q(T ).
In the arguments below, we constantly use the fact that if p|∆(t) then at(p) = 0. This allows us
to ignore the p |r∆(t) conditions. We assume B ∈ {1, 2, 3, 6} and κ ∈ {1, 2}.
Lemma B.2. We have
A0,F(p) =
p− 1 if p ≥ 5
0 otherwise.
(B.2)
Proof. We have A0,F(p) = 0 if p = 2 or 3 because, in these cases, there are no t such that p |r∆(t).
If p ≥ 5 then p |r ∆(t) is equivalent to p |r B(6t + 1) mod p. As 6 is invertible mod p, as t ranges
over Z/pZ there is exactly one value such that B(6t+ 1) ≡ 0 mod p, and the claim follows. �
Lemma B.3. We have A1,F(p) = 0.
LOWER ORDER TERMS IN 1-LEVEL DENSITIES 33
Proof. The claim is immediate for p = 2, 3 or p ≡ 2 mod 3; it is also clear when κ = 1. Thus we
assume below that p ≡ 1 mod 3 and κ = 2:
−A1,F(p) =
t mod p
at(p)
t mod p
x mod p
x3 +B(6t+ 1)2
t mod p
x mod p
x3 +Bt2
. (B.3)
The x = 0 term gives
(p−1), and the remaining p−1 values of x each give −
by LemmaB.1.
Therefore A1,F(p) = 0. �
Lemma B.4. We have A2,F(p) = 2p2 − 2p if p ≡ 1 mod 3, and 0 otherwise.
Proof. The claim is immediate for p = 2, 3 or p ≡ 2 mod 3. We do the proof for the harder case of
κ = 2; the result is the same when κ = 1 and follows similarly. For p ≡ 1 mod 3:
A2,F(p) =
t mod p
a2t (p) =
t mod p
x mod p
y mod p
x3 +B(6t+ 1)2
y3 +B(6t+ 1)2
t mod p
x mod p
y mod p
x3 +Bt2
y3 +Bt2
y mod p
x3 +Bt2
y3 +Bt2
x mod p
y mod p
tx3 +B
ty3 +B
x mod p
y mod p
t mod p
tx3 +B
ty3 +B
. (B.4)
We use inclusion / exclusion to reduce to xy 6= 0. If x = 0, the t and y-sums give p
If y = 0, the t and x-sums give p
. We subtract the doubly counted contribution from
x = y = 0, which gives p
. Thus
A2,F(p) =
t mod p
tx3 +B
ty3 +B
+ 2p− p− p2. (B.5)
By Lemma B.1, the t-sum is (p − 1)
if p|B2(x3 − y3)2 and −
otherwise; as B|6∞
we have p |r B. As p = 6m + 1, let g be a generator of the multiplicative group Z/pZ. Solving
g3a ≡ g3b yields b = a, a + 2m, or a + 4m, so x3 ≡ y3 three times (for x, y 6≡ 0 mod p). In each
instance y equals x times a square (1, g2m, g4m). Thus
A2,F(p) =
y3≡x3
+ p− p2
= (p− 1)3p+ p− p2 = 2p2 − 2p. (B.6)
34 STEVEN J. MILLER
B.2. The family y2 = x3−(36T+6)(36T+5)x over Q(T ). In the arguments below, we constantly
use the fact that if p|∆(t) then at(p) = 0. This allows us to ignore the p |r∆(t) conditions.
Lemma B.5. We have A0,F(p) = p− 2 if p ≥ 3 and 0 otherwise.
Proof. We have A0,F(p) = 0 if p = 2 because there are no t such that p|r∆(t). If p ≥ 3 then p|r∆(t)
is equivalent to p |r (36t + 6)(36t + 5) mod p. As 36 is invertible mod p, as t ranges over Z/pZ
there are exactly two values such that (36t+ 6)(36 + 5) ≡ 0 mod p, and the claim follows. �
Lemma B.6. We have A1,F(p) = −2p if p ≡ 1 mod 4 and 0 otherwise.
Proof. The claim is immediate if p = 2 or p ≡ 3 mod 4. If p ≡ 1 mod 4 then we may replace
36t+ 6 with t in the complete sums, and we find that
A1,F(p) = −
t mod p
x mod p
x3 − t(t− 1)x
x mod p
t mod p
t2 − t− x2
. (B.7)
As p ≡ 1 mod 4, −1 is a square, say −1 ≡ α2 mod p. Thus
above. Further by Lemma
B.1 the t-sum is p− 1 if p divides the discriminant 1 + 4x2, and is −1 otherwise. There are always
exactly two distinct solutions to 1 + 4x2 ≡ 0 mod p for p ≡ 1 mod 4, and both roots are squares
modulo p.
To see this, letting w denote the inverse of w modulo p we find the two solutions are ±2α. As(
= 1, we have
. Let p = 4n+1. Then
= (−1)(p2−1)/8 = (−1)n,
and by Euler’s criterion we have
≡ α(p−1)/2 ≡
)(p−1)/4 ≡ (−1)n mod p. (B.8)
= 1, and the two roots to 1 + 4x2 ≡ 0 mod p are both squares. Therefore
A1,F(p) = −2p +
x mod p
= −2p. (B.9)
Remark B.7. By the results of Rosen and Silverman [RoSi], our family has rank 1 over Q(T ); this
is not surprising as we have forced the point (36T + 6, 36T + 6) to lie on the curve over Q(T ).
Lemma B.8. Let E denote the elliptic curve y2 = x3 − x, with aE(p) the corresponding Fourier
coefficient. We have
A2,F(p) =
2p(p− 3)− aE(p)2 if p ≡ 1 mod 4
0 otherwise.
(B.10)
Proof. The proof follows by similar calculations as above. �
LOWER ORDER TERMS IN 1-LEVEL DENSITIES 35
B.3. The family y2 = x3 − 3x+ 12T over Q(T ). For the family y2 = x3 − 3x+ 12T , we have
c4(T ) = 2
4 · 32
c6(T ) = 2
7 · 34T
∆(T ) = 26 · 33(6T − 1)(6T + 1); (B.11)
further direct calculation shows that at(2) = at(3) = 0 for all t. Thus our equation is a global
minimal Weierstrass equation, and we need only worry about primes p ≥ 5. Note that c4(t) and
∆(t) are never divisible by a prime p ≥ 5; thus this family can only have multiplicative reduction
for primes exceeding 3.
If p|6t−1, replacing xwith x+1 (to move the singular point to (0, 0)) gives y2−3x2 ≡ x3 mod p.
The reduction is split if
3 ∈ Fp and non-split otherwise. Thus if p|6t − 1 then at(p) =
similar argument (sending x to x − 1) shows that if p|6t + 1 then at(p) =
. A straightforward
calculation shows
1 if p ≡ 1, 11 mod 12
−1 if p ≡ 5, 7 mod 12,
1 if p ≡ 1, 7 mod 12
−1 if p ≡ 5, 11 mod 12.
(B.12)
Lemma B.9. We have A0,F(p) = p− 2 if p ≥ 3 and 0 otherwise.
Proof. We have A0,F(p) = 0 if p = 2 or 3 by direct computation. As 12 is invertible mod p, as t
ranges over Z/pZ there are exactly two values such that (6t− 1)(6t+1) ≡ 0 mod p, and the claim
follows. �
Lemma B.10. A1,F(2) = A1,F(3) = 0, and for p ≥ 5 we have
A1,F(p) =
2 if p ≡ 1 mod 12
0 if p ≡ 7, 11 mod 12
−2 if p ≡ 5 mod 12.
(B.13)
Proof. The claim is immediate for p ≤ 3. We have
A1,F(p) = −
t mod p
∆(t) 6≡0 mod p
at(p)
t mod p
x3 − 3x+ 12t
t mod p
∆(t)≡0 mod p
x3 − 3x+ 12
= 0 +
; (B.14)
the last line follows from our formulas for at(p) for p|∆(t). �
Lemma B.11. A2,F(2) = A2,F(3) = 0, and for p ≥ 5 we have A2,F(p) = p2 − 3p− 4− 2
Proof. The claim is immediate for p ≤ 3. For p ≥ 5 we have at(p)2 = 1 if p|∆(t). Thus
A2,F(p) =
t mod p
∆(t) 6≡0 mod p
at(p)
t mod p
x mod p
y mod p
x3 − 3x+ 12t
y3 − 3y + 12t
− 2. (B.15)
36 STEVEN J. MILLER
Sending t→ 12−1t mod p, we have a quadratic in t with discriminant
(x3 − 3x)− (y3 − 3y)
= (x− y)2 · (y2 + xy + x2 − 3)2 = δ(x, y). (B.16)
We use Lemma B.1 to evaluate the t-sum; it is p − 1 if p|δ(x, y), and −1 otherwise. Letting
η(x, y) = 1 if p|δ(x, y) and 0 otherwise, we have
A2,F(p) =
x mod p
y mod p
η(x, y)p− p2 − 2. (B.17)
For a fixed x, p|δ(x, y) if y = x or if y2 + xy + x2 − 3 ≡ 0 mod p (we must be careful about
double counting). There are two distinct solutions to the quadratic (in y) if its discriminant 12−3x2
is a non-zero square in Z/pZ, one solution (namely −2−1x, which is not equivalent to x) if it is
congruent to zero (which happens only when x ≡ ±2 mod p), and no solutions otherwise. If the
discriminant 12−3x2 is a square, the two solutions are distinct from x provided that x 6≡ ±1 mod p
(if x ≡ ±1 mod p then one of the solutions is x and the other is distinct). Thus, for a fixed x, the
number of y such that p|δ(x, y) is 2 +
12−3x2
if x 6≡ ±1,±2 and 2 if x ≡ ±1,±2. Therefore
A2,F(p) =
x mod p
x 6≡±1,±2 mod p
12− 3x2
x≡±1,±2 mod p
2 · p− p2 − 2
= 2(p− 4)p+ p
x mod p
x 6≡±1,±2 mod p
12− 3x2
+ 4 · 2p− p2 − 2
= p2 − 2 + p
t mod p
12− 3x2
− 2p = p2 − 2p− 2− p
, (B.18)
where we used Lemma B.1 to evaluate the x-sum (as p ≥ 5, p does not divide its discriminant). �
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E-mail address: Steven.J.Miller@williams.edu
DEPARTMENT OF MATHEMATICS AND STATISTICS, WILLIAMS COLLEGE, WILLIAMSTOWN, MA 01267
	1. Introduction
	2. Explicit Formulas
	2.1. Standard Explicit Formula
	2.2. The Alternate Explicit Formula
	2.3. Formulas for the r 3 Terms
	3. Families of cuspidal newforms
	3.1. Weights
	3.2. Results
	4. Preliminaries for Families of Elliptic Curves
	4.1. Notation
	4.2. Sieving
	4.3. Moments of the Fourier Coefficients and the Explicit Formula
	5. Examples: One-parameter families of elliptic curves over Q(T)
	5.1. CM Example: The family y2 = x3 + B (6T+1) over Q(T)
	5.2. CM Example: The family y2 = x3 -B(36T+6)(36T+5)x over Q(T)
	5.3. Non-CM Example: The family y2 = x3 -3x + 12T over Q(T)
	Appendix A. Evaluation of SA(F) for the family of cuspidal newforms
	Appendix B. Evaluation of Ar,F for families of elliptic curves
	B.1. The family y2 = x3 + B (6T+1) over Q(T)
	B.2. The family y2 = x3 -(36T+6)(36T+5)x over Q(T)
	B.3. The family y2 = x3 -3x+12T over Q(T)
	References
