Zeros of certain modular functions and an application
Tetsuya Asai, Masanobu Kaneko and Hirohito Ninomiya
1. Main result
Let j(τ ) be the classical elliptic modular invariant, which is a holomorphic function
in the upper half-plane H, is invariant under the action of the modular group SL2 (Z),
and has a simple pole at infinity. Let ϕn (j) be the monic polynomial in j = j(τ )
obtained from j(τ ) − 744 by the action of the n-th Hecke operator (the precise definition
of which will be recalled later),
ϕn (j(τ )) = n (j(τ ) − 744) |0 T (n) (n = 1, 2, 3, . . .).
(1)
In this paper, we prove the following:
Theorem 1. For each n, all the zeros of the polynomial ϕn (j) are simple and lie in the
interval (0, 1728).
As has been known since the work of F.K.C. Rankin and H.P.F. Swinnerton-Dyer
[8], the values of j(τ ) at the zeros in H of the Eisenstein series Ek (τ ) of any weight k
on SL2 (Z) always lie in the interval [0, 1728], or equivalently, all the zeros of Ek (τ ) in
the standard fundamental domain lie on the unit circle. This result was generalized by
R.A. Rankin [7] to certain Poincaré series. Furthermore, the zeros of Atkin’s orthogonal
polynomials, as well as of certain “hypergeometric modular form”, both of which are
studied in a joint paper by D. Zagier and Kaneko [5] and have an intimate connection
to the j-invariants of supersingular elliptic curves, have the same property. (Here we
mention that the Eisenstein series is also related to the supersingular j-invariants [9].)
Our Theorem 1 supplies another example with this seemingly peculiar, and not yet fully
understood property of zeros.
As an application of this theorem, we can give an interesting proof of the following
fact. Let J(q) denote the Laurent series in q = e2πiτ of the Fourier expansion of j(τ ),
J(q) =
1
+ 744 + 196884q + 21493760q 2 + 864299970q 3 + · · · ,
q
and rn the coefficient of q n of the reciprocal of J(q),
X
1
rn q n = q − 744q 2 + 356652q 3 − 140361152q 4 + 49336682190q 5 − · · · .
=
J(q) n≥1
Corollary 2. The signs of the coefficients of 1/J(q) are strictly alternating. In other
words, (−1)n−1 rn is always a positive integer.
1991 Mathematics Subject Classification 11F03, 11F11, 11F30
1
We derive this from Theorem 1 using the following expansion formula of the reciprocal of J(q) − j, where j represents a variable:
∞
X
1
qn
=
ϕ0n (j) ,
J(q) − j n=1
n
(2)
where the symbol 0 denotes differentiation with respect to j. The formula (2) is
obtained as a corollary of a special case of an expansion formula for certain Green’s
kernel functions, which, as will be recalled in §3, has appeared in several places, notably
in R. Borcherds’s work on the Moonshine Conjecture in an equivalent form as a product
formula.
From Theorem 1 it is obvious that the sign of ϕ0n (0) is plus for odd n and minus
for even n, which readily proves Corollary 2 by virtue of formula (2). We note that,
as Borcherds and Zagier pointed out to us, the corollary can also be proven directly by
investigating asymptotic behaviour of rn through residue caluculation.
In the next section, we prove Theorem 1. In §3, we discuss the expansion formula
mentioned above, and also discuss briefly some results similar to Corollary 2.
2. The location of zeros of ϕn (j)
By definition, the Hecke operator T (n) (n = 1, 2, 3, . . .) acts on a modular form
f (τ ) of weight k (on SL2 (Z)) as
(f |k T (n)) (τ ) = n
X
k−1
ad=n,d>0
0≤b≤d−1
Ã
−k
d f
!
aτ + b
,
d
(3)
or, in terms of Fourier series,

(f |k T (n)) (q) =
X
m∈Z

X
mn

dk−1 a( 2 ) q m ,
d
0<d|(m,n)
(4)
P
where f (q) = m∈Z a(m)q m is a Fourier expansion of f (τ ). (See for instance Serre [10].)
In this section we prove Theorem 1, which asserts that the zeros of ϕn (j) (defined by
(1)) are all real, simple, and moreover lie in the interval (0, 1728). To illustrate, we give
the first few ϕn (j) and their zeros:
ϕ1 (j) = j − 744
; 744.000,
ϕ2 (j) = j 2 − 1488j + 159768
; 116.491, 1371.509,
3
2
ϕ3 (j) = j − 2232j + 1069956j − 36866976 ; 37.312, 632.482, 1562.205.
For the proof, we may assume n ≥ 2. Let D denote the standard fundamental
domain in the upper half-plane H under the action of the modular group:
D = {τ ∈ H : |τ | ≥ 1, −1/2 < Re(τ ) ≤ 1/2, and |τ | > 1 if − 1/2 < Re(τ ) < 0}.
2
Then let C be a part of the boundary of D defined by
C = {τ ∈ H : |τ | = 1, 0 ≤ Re(τ ) ≤ 1/2}.
In the following, we consider the function
Fn (τ ) = ϕn (j(τ ))
on the arc C. Recall that the map τ 7→ j(τ ) gives a 1-1 correspondence between C and
the interval [0, 1728]; in particular, the function Fn (τ ) takes real values on C because
the polynomial ϕn (j) has retional integer coefficients. Since the degree of ϕn (j) is n, it
is sufficient to show that the function Fn (τ ) has at least n distinct zeros on the arc C.
The essential point in the proof is the following estimate, which we refer to as the
Key Lemma.
Key Lemma. Let τ0 = x0 + iy0 ∈ C. Then we have
|Fn (τ0 )e−2πny0 − 2 cos(2πnx0 )| < 2.
This lemma implies that the function Fn (τ ) changes sign at least once in each part of
ν−1
ν
the arc C with
< Re(τ ) <
for ν = 1, 2, · · · , n, and hence Fn (τ ) has at least
2n
2n
n distinct zeros on C, as desired.
Proof of Key Lemma. By the definition (3) of the Hecke operators, we have
à Ã
X
Fn (τ ) =
ad=n,d>0
0≤b≤d−1
!
!
aτ + b
j
− 744 .
d
(5)
Let M be the maximum of |j(τ ) − 744 − e−2πiτ | in the closure of D:
M = max |j(τ ) − 744 − e−2πiτ |.
τ ∈D̄
The following estimate based on the positivity of the√ Fourier coefficients cn of j(τ )
provides the inequality M < 1335 (note that Im(τ ) ≥ 23 when τ ∈ D̄):
|j(τ ) − 744 − e−2πiτ | ≤
X
cn |q|n
n≥1
=
X
cn e−2πIm(τ )n
n≥1
≤
X
cn e
−2π
√
3
n
2
n≥1
=
¯
¯ Ã√
!
√ ¯
¯
3¯
−3
¯
− 744 − e2π 2 ¯ = 1334.813 · · · .
¯j
¯
¯
2
For any τ ∈ H, let τ ∗ denote the unique point in D which is equivalent to τ under
the action of the modular group. We claim that the following estimates concerning the
values of each summand in (5) hold for any τ0 = x0 + iy0 ∈ C and any n ≥ 2.
3
(i) |j(nτ0 ) − 744 − e−2πinτ0 | ≤ M.
¯ µ ¶
¯
¯
¯
τ0
2πinτ̄0 ¯
¯
(ii) ¯j
− 744 − e
¯ ≤ M.
n
aτ0 + b
τ0
τ0 + n − 1
(iii) Assume
is distinct from nτ0 , , and
. Then
d
n
n
¯ Ã
¯
!
¯
¯
aτ0 + b
¯
¯
− 744¯ ≤ eπny0 + M.
¯j
¯
¯
d
The inequality (i) is easily demonstrated because nτ0 − (nτ0 )∗ ∈ Z, and thus
∗
|j(nτ0 ) − 744 − e−2πinτ0 | = |j((nτ0 )∗ ) − 744 − e−2πi(nτ0 ) | ≤ M,
the last inequality following directly from the µdefinition
of M , since (nτ0 )∗ ∈ D. The
¶∗
n
τ0
n
inequality (ii) is similarly derived from − −
∈ Z and nτ¯0 = :
τ0
n
τ0
¯ µ ¶
¯
¯
¯
τ0
2πinτ̄0 ¯
¯j
− 744 − e
¯
¯
n
¯ µ
¯
¶
¯
n
−2πi(− τn ) ¯¯
¯
0
= ¯j −
− 744 − e
¯
τ
¯ µµ 0¶∗ ¶
¯
∗
τ
¯
τ0
−2πi( n0 ) ¯¯
¯
= ¯j
− 744 − e
¯ ≤ M.
n
As for (iii), we proceed as follows. Put z =
Im(z ∗ ) =
αz + β
aτ0 + b
and z ∗ =
. We then have
d
γz + δ
ny0
.
|γaτ0 + γb + δd|2
ny0
if (a, b, d) satisfies the
2
∗
condition in (iii), because we have |j(z)−744| ≤ e2πIm(z ) +M by the definition
√ of M and
the triangle inequality. Put L = |γaτ0 + γb + δd|; we now show that L ≥ 2. We may
assume γ ≥ 0.√If γ = 0 , then δ = ±1 and L = |d| ≥ 2. If γ ≥ 2 or a ≥ 2, then we easily
see that L ≥ 3. Suppose γ = a = 1. Then we have d = n and L = |τ0 + b + nδ|. In this
√
case, noting that b + nδ is a non-zero integer because 1 ≤ b ≤ n − 1, we have L ≥ 2
unless b + nδ = −1, which is possible only when b = n − 1, the case being excluded by
the assumption. This proves (iii). From (i), (ii) , (iii) and the trivial estimate
In order to prove (iii), it is sufficient to show that Im(z ∗ ) ≤
¯ µ
¯
¶
¯
¯
τ0 + n − 1
¯j
¯ ≤ e2πny0 + M
−
744
¯
¯
n
for the excluded case, we obtain
|Fn (τ0 ) − (e−2πinτ0 + e2πinτ̄0 )| ≤ σ1 (n)M + (σ1 (n) − 3)eπny0 + e2πny0 ,
where σ1 (n) is the sum of positive divisors of n. Multiplying both sides by e−2πny0 , we
have
|Fn (τ0 )e−2πny0 − 2 cos(2πnx0 )| ≤ σ1 (n)M e−2πny0 + (σ1 (n) − 3)e−πny0 + 1.
4
2
Using the bound M <
√ 1335 and the trivial estimate σ1 (n) ≤ n √(and so σ1 (n) − 3 ≤
√
πn 3
3
n2 ), as well as y0 ≥
and the fact that n2 e−πn 3 and n2 e− 2 are monotonically
2
decreasing for n ≥ 2, we finally obtain
|Fn (τ0 )e−2πny0 − 2 cos(2πnx0 )| ≤ n2 (M e−πn
√
3
+ e−
√
−2π 3
≤ 4(1335e
+e
= 1.1176 · · · < 2,
√
πn 3
2
√
−π 3
)+1
(6)
)+1
which completes our proof of Theorem 1.
Remark. Theorem 1 is valid when we replace j(τ ) − 744 in the definition (1) of ϕn (j)
by any j(τ ) − a with 0 < a < 1728; the proof is completely analogous. Moreover, our
method of proof implies that when we take any real number a, the zeros of the resulting
polynomials n(j(τ ) − a)|0 T (n) have the property
stated in Theorem 1 for all sufficiently
√
√
− πn2 3
2
−πn 3
large n. This is because n (M e
+e
) in (6) tends to zero as n becomes large.
3. Expansion formula for certain Green’s kernel functions
For each integer k in the set S := {0, 4, 6, 8, 10, 14} and each positive integer n, let
fn(k) (q) and gn(k) (q) be the Fourier series of the unique meromorphic modular forms on
SL2 (Z) of weight k and 2 − k, respectively, characterized by the following properties:
(i) They are holomorphic in τ (q = e2πiτ ) in the upper half-plane.
(ii) fn(k) (q) − q −n ∈ qZ[[q]], whereas gn(k) (q) − q −n ∈ Z[[q]].
The uniqueness of fn(k) (q) (resp., gn(k) (q)) follows from the fact that no holomorphic cusp
(resp., modular) forms of weight k (resp., 2 − k) exist when k is in the set S. As for
existence, we can construct the forms fn(k) (q) and gn(k) (q) in the manner described below;
the verification of the properties (i) and (ii) will then be straightforward.
For n = 1, put
(k)
f1 (q) = (J(q) − 744 +
E14−k (q)
2k
(k)
) · Ek (q), g1 (q) =
,
Bk
∆(q)
where Ek (q) is the Eisenstein series of weight k (E0 (q) = 1),
Ek (q) = 1 −
2k X
σk−1 (n)q n
Bk n≥1
(Bk = k-th Bernoulli number, σk−1 (n) =
X
dk−1 )
d|n
2k
is an integer when k ∈ S), and ∆(q) is the discriminant
Bk
function of weight 12 defined by
(note that the number
E4 (q)3 − E6 (q)2
∆(q) =
.
1728
5
For general n, we put
(k)
(k)
fn(k) (q) = n1−k · f1 (q)|k T (n), gn(k) (q) = nk−1 · g1 (q)|2−k T (n),
(k)
where T (n) is the Hecke operator. Further, we put f0 (q) = Ek (q). Note in particular
that fn(0) (q) = ϕn (J(q)).
Now we have
Theorem 3. Let k ∈ S. Then
∞
∞
X
X
Ek (p)E14−k (q)∆(q)−1
(k)
=
fn(k) (p)q n = −
gm
(q)pm ,
J(q) − J(p)
n=0
m=1
where p and q are independent formal variables.
This theorem is fairly well-known. When k ≥ 4, the function on the left-hand side
is essentially the Green’s function in the sense of Eichler [3], fn(k) (q) is a Poincaré series
and is the k − 1-st derivative of gn(k) (q) up to a constant. The case of k = 0 constitutes
a restatement of a product formula for the j-function (equivalently, the denominator
formula for the Monster Lie algebra) stated in the introduction of Borcherds [2] and
also appearing in Norton [6] and Alexander-Cummins-Mckay-Simons [1]. Furthermore,
the polynomials ϕn (j)(= fn(0) (q)) are viewed as Faber polynomials, the subject of vast
study since the original work of G. Faber [4], mainly from analytical points of view.
We also mention that Zagier (in preparation) obtained a similar formula which involves
meromorphic modular forms of half-integral weight. Here, for the reader’s convenience,
we give a simple and elementary unified proof for all k in question.
Proof. First we note that any meromorphic modular form on SL2 (Z) of weight 2 which
is holomorphic in H is a derivative (with respect to τ ) of a polynomial in j(τ ). In
particular, the constant term of the Fourier series of such a form always vanishes (recall:
1 d
d
= q ). Now let us put
2πi dτ
dq
J(q) = q −1 +
∞
X
(k)
cn q n , f1 (q) = q −1 +
n=0
∞
X
(k)
−1
n
a(k)
+
n q , g1 (q) = q
n=1
∞
X
n
b(k)
n q .
n=0
From (4), we have
fn(k) (q) = q −n + n1−k a(k)
n q + ···
∀n ≥ 1.
(k)
As a consequence of the preceding remark, looking at the constant terms of fn(k) (q)g1 (q)
(k)
and g1 (q)Ek (q), we obtain
1−k (k)
b(k)
an
n = −n
and
(k)
b0 =
2k
.
Bk
∀n ≥ 1,
(7)
(8)
(k)
(q) are uniquely determined by their principal
Since the forms J(p)fn(k) (p) and J(q)gm
parts (terms with non-positive exponents), we obtain, by comparing the coefficients
6
(k)
and using relations (7) and (8) as well as the fact that the constant term of gm
(q) is
(k)
b0 σk−1 (m), the recurrence relations
(k)
J(p)fn(k) (p) = fn+1 (p) +
n
X
(k)
(k)
cn−` f` (p) − b(k)
n f0 (p)
∀n ≥ 0,
(9)
`=0
and
(k)
(k)
(q) = gm+1 (q) +
J(q)gm
m
X
(k)
cm−` g` (q) +
`=1
2k
(k)
σk−1 (m)g1 (q)
Bk
∀m ≥ 1.
(10)
Multiplying both sides of (9) (resp., (10)) by q n (resp., pm ) and summing, we have
Ã
!
Ã
´
1³
1
1
(k)
(k)
(k)
J(p)F (p, q) =
F (p, q) − f0 (p) + J(q) −
F (p, q) + f0 (p) −g1 (q) +
q
q
q
!
(resp.,
Ã
!
´
1³
1
(k)
(k)
G(p, q) − g1 (q)p + J(p) −
G(p, q) + g1 (q) (1 − Ek (p)) ),
J(q)G(p, q) =
p
p
P
(k)
n
where F (p, q) = ∞
n=0 fn (p)q (resp., G(p, q) =
transformed into the formula in Theorem 3.
P∞
m=1
(k)
gm
(q)pm ). This can easily be
Corollary 4. Let k ∈ S. Then
∞
X
E14−k (q)∆(q)−1
n
=
ϕ(k)
n (j)q , and
J(q) − j
n=0
∞
X
Ek (q)
=
ψn(k) (j)q n ,
J(q) − j
n=1
where ϕn(k) (j) and ψn(k) (j) are monic polynomials of respective degrees n and n − 1 which
are determined by
fn(k) (q) = ϕ(k)
n (j(q))Ek (q)
and
gn(k) (q) = ψn(k) (j(q))E14−k (q)∆(q)−1 .
(k)
In fact, the polynomials ϕ(k)
n (j) and ψn (j) are determined inductively using the relations
ϕ0n (j)
(0)
. The
(9) and (10). In particular, ϕ(0)
(j)
is
identical
to
ϕ
(j)
in
§1,
and
ψ
(j)
=
n
n
n
n
latter is because we have
d
−q
dq
Ã
ϕn (j(q))
n
!
ϕ0n (j(q)) E14 (q)
=
,
n
∆(q)
and the left-hand side of this clearly satisfies the conditions that characterize gn(0) (q).
We therefore obtain the formula (2) in §1.
7
If we substitute j = 1728 in the formula (2), we can derive a result for the co1
∆
efficients of
= 2 demonstrating that they are all positive. Similarly,
J(q) − 1728
E6
substituting j = 0 or j = 1728 into the formula
∞
X
E14 (q)∆(q)−1
=
ϕn (j)q n
J(q) − j
n=0
which represents the case k = 0 of the first formula in the Corollary 4, we conclude
E6
by Theorem 1 that the sign of the coefficient in the Fourier expansion of
is strictly
E4
E2
alternating, while that of 4 is always positive; the latter case is, however, obvious from
E6
the expansions of E4 and E6 .
(k)
We note that the statement of Theorem 1 also holds for all ϕ(k)
n (j) and ψn (j) with
k ∈ S, which is proved along the same line, though each evaluation becomes rather complicated. We can therefore obtain similar results for the signs of the Fourier expansions
1 ∆ E6 ∆
of various meromorphic modular forms, such as
,
,
, . . . (alternating type)
E4 E4 E4
∆ E4 ∆ E4 ∆
and
,
,
, . . . (always positive).
E6 E6
E62
Acknowledgement
It is our pleasure to thank Prof. John Mckay for his reading the first version of our
paper and making various comments, including the reference to the work on Faber
polynomials. We would like to thank Professors Richard Borcherds and Don Zagier,
who pointed out the direct proof of Corollary 2. Our thanks also go to Prof. Noriko
Yui, from whom we learned that the series 1/J(q) appeared as a “Mirror map” of a
certain family of K3 surfaces.
References
[1] Alexander, D., Cummins, C, Mckay, J. and Simons, C. : Completely Replicable
Functions, in “Groups, Combinatorics and Geometry (LMS Lecture Note Series
165)”, edited by M.W. Liebeck and J. Saxl, Cambridge University Press (1991), 87
– 95.
[2] Borcherds, R. : Monstrous moonshine and monstrous Lie superalgebras, Invent.
Math. 109 (1992), 405 – 444.
[3] Eichler, M. : The basis problem for modular forms and the traces of the Hecke
operators, Modular functions of one variable I, LNM 320 (1973), 75 – 151.
[4] Faber, G. : Über polynomische Entwickelungen, Math. Ann. 57 (1903), 389 – 408.
8
[5] Kaneko, M. and Zagier, D. : Supersingular j-invariants, hypergeometric series, and
Atkin’s orthogonal polynomials, preprint.
[6] Norton, S. : More on Moonshine, in “Computational Group Theory”, edited by
M.D. Atkinson, Academic Press (1984), 185 – 193.
[7] Rankin, R.A. : The zeros of certain Poincaré seires, Compositio Math. 46 (1982),
255–272.
[8] Rankin, F.K.C. and Swinnnerton-Dyer, H.P.F. : On the zeros of Eisnstein series,
Bull. London Math. Soc. 2 (1970), 169 – 170.
[9] Serre, J.-P. : Congruences et formes modulaires (d’après H.P.F. Swinnerton-Dyer),
Sém. Bourbaki 416 (1971/72), or Œuvres III 74 – 88.
[10] Serre, J.-P. : Cours d’Arithmétique, Presses Universitaires de France, 1970; English
translation: Springer, GTM 7, 1973.
Tetsuya Asai
Department of Mathematics, Shizuoka University, Shizuoka 422, Japan
E-mail: [email protected]
Masanobu Kaneko Graduate School of Mathematics, Kyushu University 33, Fukuoka 812, Japan
E-mail: [email protected]
Hirohito Ninomiya Department of Mathematics, Kyoto University, Kyoto 606, Japan
E-mail: [email protected]
9