PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 135, Number 2, February 2007, Pages 313–328
S 0002-9939(06)08353-5
Article electronically published on September 11, 2006
EQUILIBRIUM POINT OF GREEN’S FUNCTION
FOR THE ANNULUS AND EISENSTEIN SERIES
AHMED SEBBAR AND THÉRÈSE FALLIERO
(Communicated by Richard A. Wentworth)
Abstract. We study the motion of the equilibrium point of Green’s function
and give an explicit parametrization of the unique zero of the Bergman kernel
of the annulus. This problem is reduced to solving the equation ℘(z, τ ) =
2
− π3 E2 (τ ), where E2 (τ ) is the usual Eisenstein series.
1. Introduction
In [2], [3], Eichler and Zagier gave an explicit formula for the solution z0 (τ ) for
τ > 0 of the equation ℘(z, τ ) = 0, ℘ being the Weierstrass function of periods
1 and τ . They also observe that the method developed can be applied to find
the solution to any equation of the form ℘(z, τ ) = φ(τ ), where φ is a modular
2
form of weight 2. In this work, we solve the equation ℘(z, τ ) = − π3 E2 (τ ), where
E2 (τ ) is an Eisenstein series which is not a modular form. One motivation for
studying such an equation is that a close connection is established between equilibrium points (critical points) of Green’s functions of the annulus and Eisenstein
series. The links between the apparently distant themes rely on the role played
by the Green’s function for multiply connected domains, their equilibrium points
with their motions, and by classical kernel functions such as the Bergman kernel or
the Schiffer kernel on Riemann surfaces. This connection brings together concepts
from analysis, geometry and number theory.
More precisely, our main result (Theorem 3.2) is the following: if CR = {R <
|z| < 1} is a given annulus, as the pole of the Green’s function of CR goes to the
boundary, the limiting position of the equilibrium point is given by an equation of
2
the form ℘(z, τ ) = − π3 E2 (τ ). We will explain how the equilibrium point depends
on the Bergman kernel of the annulus CR . In particular, the zeros of the Bergman
kernel will exhibit, for a given R > 1, two real numbers R1 , R2 ∈ (1, R) with
R1 R2 = R and such that the annulus {R1 < |z| < R2 } contains all the equilibrium
points of the Green’s function of CR . The important fact here is that the radii of
the two annulus have equal geometrical means. We recover in an effective manner
the results of Maria [10] and complete a result of Rudin [12].
Received by the editors January 25, 2005 and, in revised form, June 1, 2005.
2000 Mathematics Subject Classification. Primary 11F03, 11F11, 30C40, 34B30.
Key words and phrases. Equilibrium points, Eisenstein series, Bergman kernel.
We are grateful to Henri Cohen and Don Zagier for teaching us some facts about the zeros of
the Eisenstein series E2 .
c
2006
American Mathematical Society
Reverts to public domain 28 years from publication
313
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314
AHMED SEBBAR AND THÉRÈSE FALLIERO
2
A second motivation for studying the equation ℘(z, τ ) = − π3 E2 (τ ) relies on an
important theorem of Schiffer and Hawley [14] and also Hejhal [7]: For a bounded
plane domain D with analytic boundary ∂D and connectivity 1 ≤ p ≤ ∞, there
exists a relationship between the curvature of the Bergman metric, the Schiffer
kernels and their zeros. Finally, a physical interpretation may be given to the
notion of equilibrium point; if the boundary of CR is kept at potential zero (the
potential at an arbitrary point W ∈ CR is due to a unit charge at a fixed Z ∈ CR
and the charge on the boundary induced by it is precisely G(Z, W )), the equilibrium
point is the point where the intensity of the corresponding field of force is zero. It
seems of interest to study the link between this physical picture, Eisenstein series,
the Bergman kernel and the Bergman metric in higher connectivity.
2. Statement of the problem
Let Dp be a bounded open set of the complex plane C bordered by p analytic
curves Γ1 , Γ2 , · · · , Γp−1 and an outer boundary Γ0 . By the classical Hopf’s lemma,
the normal derivative of the the Dirichlet Green’s function is positive on the boundary Γ0 ∪ Γ1 ∪ · · · ∪ Γp−1 of Dp , and one may ask if there is a compact set K in Dp ,
independent of z0 , containing all the equilibrium points of the (complex) Green’s
function G(z0 , z) of Dp . Our objective in this work is to give a full answer in the
\ E, where E is a given union
case p = 1. The same question should be asked in C
of intervals E = [E1 , E2 ] ∪ [E3 , E4 ] ∪ · · · ∪ [E2n−1 , E2n ]. In fact a theorem of Koebe
\ E can be mapped conformally into a domain Dn and such that each
asserts that C
component Γi is a circle, 0 ≤ i ≤ n − 1. On the other hand, the complex Green’s
\ E with a pole
function (analytic completion of the classical Green’s function) of C
at x0 is given by the abelian integral [17] (p. 227):
z
1
1
(2.1)
h(ζ)q(ζ)− 2
dζ,
G(x0 , z) =
ζ
−
x0
E2n
where q(ζ) = (ζ −E1 )(ζ −E2 ) · · · (ζ −E2n ) and h(ζ) is a monic polynomial of degree
n − 1, h(ζ) = ζ n−1 + hn−2 ζ n−2 + · · · + h0 with coefficients satisfying the following
system of the Jacobi inversion problem:
E2j+1
1
dζ
(2.2)
hk
ζ k |q(ζ)|− 2
= 0 , 1 ≤ j ≤ n − 1,
ζ
−
x0
E2j
0≤k≤n−1
and
(2.3)
1
hk xk0 = −q(x0 ) 2 .
0≤k≤n−1
\ E with a pole at ∞ is now given by
If x0 = ∞, the Green’s function of C
z
1
G(z) =
(2.4)
h(ζ)q(ζ)− 2 dζ
E2n
and the system
(2.5)
0≤k≤n−1
E2j+1
hk
ζ k |q(ζ)|− 2 dζ = 0 ,
1
1 ≤ j ≤ n − 1;
E2j
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hn−1 = 1.
EQUILIBRIUM POINT OF GREEN’S FUNCTION
315
Definition 2.1. The equilibrium (or critical) points of the Green’s function with
pole at a finite point x0 are the zeros c1 = c1 (x0 ), c2 = c2 (x0 ), · · · , cn−1 = cn−1 (x0 )
of the polynomial h(ζ) = hn−1 ζ n−1 +hn−2 ζ n−2 +· · ·+h0 , with the above conditions
(2.2), (2.3).
For x0 = ∞, the equilibrium points ci , 1 ≤ i ≤ n − 1, are all real and simple,
located in the gaps (E2j , E2j+1 ), 1 ≤ j ≤ n − 1. Furthermore, their images are
given by [17]:
1
1 E2j+1
G(cj ) =
|h(ζ)|q(ζ)− 2 dζ.
2 E2j
From now on, we consider only the case of two intervals and without any loss of
generality we take E = [−1, α] ∪ [β, 1], −1 < α < 0 < β < 1. In this case, the
\ E with pole x0 is
Green’s function of C
z
k(ζ) dζ
G(x0 , z) =
1
1 q(ζ) 2 ζ − x0
with q(ζ) = (ζ 2 −1)(ζ −α)(ζ −β), and k(ζ) = k0 +k1 ζ is a polynomial with complex
coefficients determined by the two conditions:
β
α
k(ζ)
dζ
= 0,
ζ
−
x0
q(ζ)
1
k(x0 ) = −q(x0 ) 2 .
1
2
The next result is an immediate consequence of the definition of the equilibrium
point c(x0 ).
Proposition 2.2. Let ρ = max (|α|, |β|) and define the sequence of moments an =
β ζn
1 dζ. Then for |x0 | > ρ the equilibrium point is given by
α
q(ζ) 2
c(x0 ) = (2.6)
n≥1
n≥1
in particular
an x−n
0
an−1 x−n
0
β
c(∞) =
ζ
α q(ζ) 12
β 1
α q(ζ) 12
;
dζ
.
dζ
This is easily obtained from the representation of c(x0 ) as a quotient of two
transforms of Cauchy type:
β ζ
dζ
α q(ζ) 12 ζ−x0
c(x0 ) = β
α
1
1
q(ζ) 2
dζ
ζ−x0
.
The function x0 → c(x0 ) is an analytic function of x0 on the simply connected
\ [α, β], and it can be given a Laurent expansion in the neighborhood of
domain C
infinity in C,
−(n+1)
c(x0 ) =
An x−n
A n = a0
Hn (a0 , a1 , · · · , an+1 ), |x0 | > ρ,
0 ,
n≥0
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316
AHMED SEBBAR AND THÉRÈSE FALLIERO
where H0 (a0 , a1 ) = a1 and for n ≥ 1:
a0 0
a1 a0
a1
Hn (a0 , a1 , · · · , an+1 ) = an
.
an an+1 a1
a2
0
..
.
..
..
.
..
.
..
.
.
a1
The determinant Hn (a0 , a1 , · · · , an+1 ) is a homogeneous polynomial of degree n+1,
H1 (a0 , a1 , a2 ) = a0 a2 − a21 , H2 (a0 , a1 , a2 , a3 ) = a20 a3 − 2a0 a1 a2 + a31 , · · · . They are
actually related to Faber polynomials. For a formal Laurent series
b1
+ · · · , b = 0,
g(t) = bt + b0 +
t
the Faber polynomials Φn (w) are defined by
log
∞
Φn (w) −n
g(t) − w
=−
t .
bt
n
n=1
If we take the derivative in w and choose b = 1, bk = ak+1
a0 , k ≥ 0, we find from
(2.6) that
Φ (0)
An = − n+2 .
n+2
Now, we would like to take up the problem of the variation of the function c(x0 ) from
\ [−1, α] ∪ [β, 1]
the annulus model point of view. We conformally map the region C
−πK/K . The point at infinity ∞ is sent
on the annulus Ar,1 = {r < |v| < 1}, r = e
onto a point s of the positive real half line. This conformal map is given by ([1], p.
282; [4])
cn2 M sn2 u + sn2 M cn2 u
\ [−1, α] ∪ [β, 1],
, z∈C
z=
sn2 u − sn2 M
with classical notations for
x
dx
, x = snw
w=
2
(1 − x )(1 − k2 x2 )
0
and
k2
K
M
τ
2(β − α)
2
∈ (0, 1), k = 1 − k2 ,
(1 + β)(1 − α)
1
1
dx
dx
=
,
, K =
2
2
2
(1 − x )(1 − k x )
0
0
(1 − x2 )(1 − k 2 x2 )
√ 1−α
2
dt
K log s
=
,
=
π
(1 − t2 )(1 − k2 t2 )
0
K log v
,
= iK /K, , r = e−iπ/τ , u =
π
=
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EQUILIBRIUM POINT OF GREEN’S FUNCTION
317
and the Jacobi theta functions which will be used later:
∞
2
(−1)n eiπτ (n+1/2) sin(2n + 1)πz,
θ1 (z, τ ) = 2
θ2 (z, τ ) = 2
n=0
∞
2
eiπτ (n+1/2) cos(2n + 1)πz,
n=0
θ3 (z, τ ) = 1 + 2
θ4 (z, τ ) = 1 + 2
∞
2
eiπτ n cos 2nπz,
n=1
∞
2
(−1)n eiπτ n cos 2nπz,
n=1
1 θ1 (u)
,
cn2 u = 1 − sn2 u.
snu = √
k θ4 (u)
\ ([−1, α] ∪ [β, 1]) has the form
Therefore the Green’s function of C
θ ( u−M , τ ) 1 2K
(2.7)
G(x) = G(x, ∞) = − log .
θ1 ( u+M
,
τ
)
2K
On the other hand, the Green’s function of the annulus Ar,1 = {r < |v| < 1}, 0 <
r < 1, can also be given by (see [4] and the references therein)
(2.8)
∞
w1/2 z − log w/2 log r (1 − z/w) m=1 (1 − r 2m z/w)(1 − r 2m w/z)
G(z, w) = log
.
2m z w̄)(1 − r 2m /z w̄)
w̄−1/2 z log w̄/2 log r (1 − z w̄) ∞
m=1 (1 − r
In the next section, we derive closed expressions for Green’s functions and prove
our main result.
3. A formula for the zero of the Bergman kernel
We recall some essential facts concerning the Bergman kernel that we need for
our investigation on the motion of the equilibrium point. For a finite not necessarily
simply connected open Riemann surface Ω, the Bergman kernel of Ω, denoted by
K(z, t̄)dz dt̄, is the kernel characterized by the reproducing property for holomorphic differentials α(t)dt, where
1
K(z, t̄)dt̄ ∧ α(t)dt.
α(z) =
2i
Ω
It has been shown by Schiffer [13] that the Bergman kernel is given by the second
derivative of the Green’s function:
2 ∂ 2 G(z, t)
(3.1)
.
K(z, t̄) = −
π ∂z∂ t̄
The adjoint kernel or the Schiffer kernel is defined by
2 ∂ 2 G(z, t)
,
π ∂z∂t
and we have the important relation [6] and [15]
(3.2)
(3.3)
L(z, t) = −
K(z, t̄)dz = −L(z, t)dz,
z ∈ Ω, t ∈ ∂Ω.
In their answer to the Lu Qi-keng conjecture for finite Riemann surfaces, Suita
and Yamada [15] proved that for a finite Riemann surface Ω which is not simply
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318
AHMED SEBBAR AND THÉRÈSE FALLIERO
connected, the Bergman kernel K(z, t̄) has exactly n+2p−1 zeros as a function of z,
for t ∈ ∂Ω and for n and p being respectively the number of boundary contours and
the genus of Ω. In particular, if Ω is a plane domain with two boundary components,
the Bergman kernel is a function and has only one zero. Our objective is to give
a complete description of this zero by using some results on Jacobi forms. On the
other hand, Zarankiewicz in [18] and Fay in [5] (p. 133) gave an expression of the
kernel-function for the annulus Ar = {r < |z| < 1} in a closed form
ζ(iπ; 2 log 1r , 2iπ)
1
1
K(z, t̄) =
℘(log(z t̄); 2 log , 2iπ) +
πz t̄
r
iπ
or
(3.4)
1
K(z, t̄) =
πz t̄
1
η
℘ (log(z t̄); 2ω, 2ω ) + − ω 2ω
,
where ℘ and ζ are the Weierstrass functions determined by the periods 2ω = 2iπ
and 2ω = 2 log r. From (2.8) and (3.2) we obtain at once:
Proposition 3.1. For z, t ∈ Ar , z = w,
(3.5)
K(z, t̄) =
1
1
L
z,
.
|t|2
t
In particular the Schiffer kernel has no zero in Ar .
In order to formulate the result on the parametrization of the zero of the kernel
function of the annulus, let us introduce the classical Eisenstein series and other
arithmetical functions:
∞
∞
nq n
=
1
−
24
σ1 (n)q n ,
n
1
−
q
n=1
n=1
(3.6)
E2 (τ ) = 1 − 24
(3.7)
E4 (τ ) = 1 + 240
∞
∞
n3 q n
=
1
+
240
σ3 (n)q n ,
n
1
−
q
n=1
n=1
(3.8)
E6 (τ ) = 1 − 504
∞
∞
n5 q n
=
1
−
504
σ5 (n)q n
n
1
−
q
n=1
n=1
with
σk (n) =
dk ,
q = e2iπτ .
d|n
It is well known that E4 (τ ) and E6 (τ ) are modular forms of weights 4 and 6 for
the group Γ = SL2 (Z). E2 (τ ) is not modular, but
(3.9) E2∗ (τ ) = 1−24
∞
3
nq n
3
= 1−
−24q −72q 2 −96q 3 −168q 4 −· · ·
−
n
1
−
q
πτ
πτ
n=1
is the non-holomorphic Eisenstein series of weight 2 for Γ. We recall the classical
discriminant function ∆ defined by the infinite product
∆(τ ) = q
∞
(1 − q n )24 = q − 24q 2 + 252q 3 − 1472q 4 + · · · .
n=1
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EQUILIBRIUM POINT OF GREEN’S FUNCTION
319
Then ∆(τ ) is a cusp form for Γ of weight 12 and verifies the relation 1728∆ =
E43 − E62 . We introduce, following Ramanujan, the functions
Φr,s (q) =
(3.10)
m=∞
n=∞
mr ns q mn .
m=1 n=1
We refer to [11] for the following properties of these functions:
Φr,s (q)
Φr,s (q)
=
Φs,r (q),
r
d
=
q
Φ0,s−r (q).
dq
Sometimes, we simply write Φr,s (τ ) for Φr,s (q) and we have the algebraic relation
Φr,s (τ ) =
Kl,m,n E2 (τ )l E4 (τ )m E6 (τ )n .
2l+4m+6n=r+s+1
l−1≤inf(r,s)
In the sequel, we consider new half-periods ω1 , ω3 with ω1 = −ω = log
and ω3 = ω = iπ.
1
r
= log R
Theorem 3.2. For each R > 1 and for each Z ∈ CR = {1 < |Z| < R}, the
Bergman kernel K(Z, W ), as a function of W , has a unique zero W = W (Z) in
CR given by
℘(log(ZW ); 2 log R, 2iπ) =
Moreover, u(τ ) =
Z + τ Z:
1
2ω1
1
φ(τ ),
(2ω1 )2
φ(τ ) = −
π2
E2 (τ ),
3
τ=
iπ
.
log R
log(ZW ) has the following integral representation modulo
√ i∞
3 1728 3
Φ5,6 (t) − ∆(t)
τ
π
u(τ ) = ± ± ±
(t − τ )dt.
3
2 4
5
Φ2,3 (t) 2
τ
In particular arg Z − arg W (Z) = π for every Z ∈ CR .
Proof. Let us rewrite equation (3.4) in a convenient form for our calculations, which
will also explain the apparition of the Eisenstein series E2 . From the Legendre
relation ηω − η ω = i π2 , (3.4) can be simply written as
1
η
(3.11)
K(z, t̄) =
℘(log(z t̄); 2ω, 2ω ) + .
πz t̄
ω
If R = 1/r, the holomorphic involution i : z → Z = 1/z transforms the annulus Ar
into the annulus CR = {1 < |z| < R}, and the kernel functions of the two domains
are related by the identity
KAr (z, w̄) = KCR (Z, W )i (z)i (w).
In particular we obtain that
KCR (Z, W ) =
1
πZW
℘(log(ZW ); 2ω, 2ω ) +
η
ω
.
Now, ℘(u; 2ω, 2ω ) = ℘(u; 2ω1 , 2ω3 ) and in general, if ω, ω are given with ωω > 0
and if
α β
ω
ω1
=
ω3
γ δ
ω
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320
AHMED SEBBAR AND THÉRÈSE FALLIERO
with
α
γ
β
δ
∈ SL2 (Z), the periods undergo the same transformation as ω, ω and
η
η1
αη + βη η
β(ηω − η ω)
βπi
−
= −
=
=
.
ω
ω1
ω αω + βω
ωω1
2ωω1
In our case, the period of the Weierstrass ζ−function transforms as
iπ
1
η
η
η1
−
− .
=
=
2ω1
2ω 4ωω
2ω 4ω
η
η1
1
3
We will write τ = − ωω = ω
ω1 , hence ℘(u; 2ω, 2ω ) + ω − 2ω = ℘(u; 2ω1 , 2ω3 ) + ω1 ,
and substituting into (3.4), we get the more convenient formula
1
η1
(3.12)
KCR (Z, W ) =
.
℘(log(ZW ); 2ω1 , 2ω3 ) +
ω1
πZW
The Eisenstein series E2 appears when we make explicit the local behavior of
℘(u; 2ω1 , 2ω3 ) from its Fourier expansion. Indeed the function u → ℘(u; 2ω1 , 2ω3 )
is an even function with pole of order two at the origin. The development at u = 0
does not contain a constant term and can be obtained for instance from the Fourier
series
℘(u; 2ω1 , 2ω3 )
η1
+
=−
ω1
π
2ω1
2 ∞
nq n
πu
1
−8
cos 2n(
) ,
πu
1 − qn
2ω1
sin2 ( 2ω
)
n=1
1
q = e2iπτ ,
and according to (3.6),
⎛
⎞
2
nq n
π
η1
1
⎝1 − 24
⎠ = 1 ( π )2 E2 (τ ).
(3.13)
=
ω1
3 2ω1
1 − qn
3 2ω1
n≥1
We now solve our problem in the annulus CR rather in Ar , that is, for a fixed
Z ∈ CR we solve
η1
=0
(3.14)
℘(log(ZW ); 2ω1 , 2ω3 ) +
ω1
in W by using the methods in [2] and [3]. We introduce the new variable u, 2uω1 =
log(ZW ), and by the homogeneity properties of the ℘−function the equation (3.14)
simply becomes
(3.15)
℘(u; τ ) = φ(τ ),
2
where ℘(u; τ ) = ℘(u; 1, τ ), φ(τ ) = − π3 E2 (τ ). From the differential equation for
the ℘−function, Eichler and Zagier obtained an expression of the second derivative
u (τ ) of the solution of (3.15):
− 3
1 3
4φ − g2 φ − g3 2 E(τ )
(3.16)
u (τ ) = uφ (τ ) = ±
2
72π
with
E(τ ) = −2 4φ3 − g2 φ − g3 φ∗∗ + (12φ2 − g2 )φ∗ 2 + (36g3 φ + 2g22 )φ∗
(3.17)
+ 12g2 φ4 + 3g22 φ2 + 6g2 g3 φ − g23 + 27g32
and
(3.18)
∗
φ = 12π
2
1 d
1 d
1
1
∗∗
2
− E2 φ , φ = 12π
− E 2 φ∗ .
2iπ dτ
6
2iπ dτ
3
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EQUILIBRIUM POINT OF GREEN’S FUNCTION
321
1 d
k
It is known that the differential operator 2iπ
dτ − 12 E2 sends the space of modular
forms of weight k for some subgroup of SL2 (Z) into the space of modular forms of
weight k + 2, hence for φ(τ ) = ek (τ ), φ∗ is a modular form of weight 4 and φ∗∗ (τ )
is a modular form of weight 6 for the subgroup Γ(2) of SL2 (Z).
By using the fundamental fact [11], [16] that every modular form on Γ is uniquely
expressible as a polynomial in E4 and E6 and the extension C[E2 , E4 , E6 ] of
C[E4 , E6 ] is closed under differentiation,
⎧
1 d
1
⎪
E4 = (E2 E4 − E6 ),
⎪
⎪
⎪
2iπ
dτ
3
⎪
⎨
1
1 d
(3.19)
E6 = (E2 E6 − E42 ),
⎪
2iπ
dτ
2
⎪
⎪
⎪
⎪
⎩ 1 d E = 1 (E 2 − E ).
2
4
2iπ dτ
12 2
Equation (3.16) becomes, after some calculations,
√
3 −E26 + 15E24 E4 − 40E23 E6 + 45E22 E42 − 24E2 E4 E6 − 27E43 + 32E62
π
u (τ ) = ±
.
3
144
(3E2 E4 − 2E6 − E 3 ) 2
2
With the function Φr,s as given in (3.10), we use the following identities of Ramanujan [11]:
d
1728Φ2,3 = −E23 + 3E2 E4 − 2E6 , Φ5,6 = q
Φ4,5
dq
to obtain
d 20736Φ5,6 = (q ) 15E2 E42 − 20E22 E6 + 10E23 E6 − 4E4 E6 − E25 .
dq
Consequently, the previous derivatives formulas (3.19) give
248832Φ5,6 = 9E43 + 16E62 + 5(−E26 + 15E24 E4 − 40E23 E6 + 45E22 E42 − 24E2 E4 E6 ).
Using the relation 1728∆ = E43 − E62 , we deduce
√
1728 3 Φ5,6 (τ ) − ∆(τ )
π
u (τ ) = ±
3
5
Φ2,3 (τ ) 2
√
1728 3 π 120eiπτ − 1440e3iπτ + 21600e5iπτ + · · · .
= ±
5
Let us denote the constant
expression for u , we get
√
1728 3
π
5
by C0 . If we integrate twice in the above
i∞
u(τ ) = C1 + C2 τ ± C0
Φ5,6 (t) − ∆(t)
3
τ
Φ2,3 (t) 2
(t − τ )dt,
where the path of integration may be any curve which does not pass through a zero
of Φ2,3 (t). For τ > 1
C0 u(τ ) = C1 + C2 τ ± 2 120eiπτ − 160e3iπτ + 4320e5iπτ + · · · .
π
The constants C1 , C2 are determined by analyzing the behavior of u(τ ) for τ near
∞. The equation (3.15) has the particular solution
φ(τ )
dx
u0 (τ ) = −
.
2 (x − e1 )(x − e2 )(x − e3 )
∞
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322
AHMED SEBBAR AND THÉRÈSE FALLIERO
The integral is on any path which does not contain any of the real zeros e3 < e2 < e1
of 4x3 − g2 x − g3 . We would like to investigate the behavior of this integral when
3
τ goes to i∞. Making the substitution z = ex−e
, we obtain
2 −e3
where λ =
φ(τ )
dx
2 (x − e1 )(x − e2 )(x − e3 )
∞
)−e3
φ(τ
e2 −e3
dz
1
= −√
e1 − e3 ∞
2 z(1 − z)(1 − λz)
)−e3
φ(τ
e2 −e3
dz
1
τ
,
= −√
2
e1 − e3 0
2 z(1 − z)(1 − λz)
−
e2 −e3
e1 −e3
is the elliptic modular function. The importance of the above
)−e3
change of variables is that |λ φ(τ
e2 −e3 | < 1 for |τ | large. We will analyze the integral
1
as an incomplete hypergeometric function. We expand (1 − λz)− 2 in power series
and integrate term by term using the identity
x
dz
1
= (n + )
z n+1 2
z(1 − z)
(n + 1)
0
We get with y =
0
x
dz
− xn x(1 − x).
zn z(1 − z)
φ(τ )−e3
e2 −e3 :
u0 (τ ) =
a
τ
− √
2 2 e1 − e3
0
y
dz
− y(1 − y)
an y n ,
z(1 − z)
n≥0
where
a
an
1 · 3 · · · (2m − 1) 2
λm ,
= 1+
2 · 4 · · · 2m
m≥1
1 · 3 · · · (2m − 1) (2n + 3) · · · (2m − 1) =
λm .
2 · 4 · · · 2m
(2n + 2) · · · 2m
m≥n+1
\ (−∞, 0] ∪ [1, +∞) the branch of
We consider in C
z
(0, 1) and the branch of 0 √ du
given by
y(1 − y) which is positive in
u(1−u)
z
0
du
π
= + i log 2 z(1 − z) − i(2z − 1) .
2
u(1 − u)
√
This corresponds to the choice of the branch of sin−1 z = −i log( 1 − z 2 +iz) which
is a conformal map between C \ (−∞, −1] ∪ [1, +∞) and {z ∈ C, |z| < π2 }, so we
may write
u0 (τ ) =
π
a
τ
− √
+ i log(2 y(1 − y) − i(2y − 1)) + y(1 − y)
an y n .
2 2 e1 − e3 2
n≥0
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EQUILIBRIUM POINT OF GREEN’S FUNCTION
323
The general value of u(τ ) is m + nτ ± u0 (τ ). The behavior of u0 (τ ) as τ tends to
i∞ is a consequence of the following known identities:
e2iπnτ
,
e1 = φ(τ ) + π 2 + 8π 2
(1 + e2iπnτ )2
n≥1
e2
= φ(τ ) + 8π 2
e3
= φ(τ ) − 8π 2
and the approximation λ(τ ) =
tends to i∞,
e2 −e3
e1 −e3
u0 (τ ) =
eiπ(2n−1)τ
,
(1 + eiπ(2n−1)τ )2
n≥1
eiπ(2n−1)τ
,
(1 − eiπ(2n−1)τ )2
n≥1
∼ 16eiπτ . An easy computation yields, as τ
1
τ
− + O(eiπτ ).
2 4
1
It follows that C1 = −1
4 , C2 = 2 .
iπ
The equality arg Z −arg W (Z) = π is a consequence of the fact that for τ = log
R,
i∞ Φ5,6 (t)−∆(t)
the integral τ
(t − τ )dt is real. We have thus obtained a precise
3
Φ2,3 (t) 2
analysis of the motion of the unique zero of the Bergman kernel of the annulus.
By symmetry, we suppose that Z = eα , 0 < α < log R, and we denote by WR the
corresponding zero. We observe that
√ i∞
3 1728 3
Φ5,6 (t) − ∆(t)
1
(t − τ )dt ∈
/ {0, } + Z,
+
π
3
4
5
2
Φ2,3 (t) 2
τ
otherwise
℘(u0 ; τ ) = φ(τ ) = e3 = φ(τ ) − 8π 2
or
℘(u0 ; τ ) = φ(τ ) = e2 = φ(τ ) + 8π 2
eiπ(2n−1)τ
(1 − eiπ(2n−1)τ )2
n≥1
eiπ(2n−1)τ
.
(1 + eiπ(2n−1)τ )2
n≥1
This is impossible for τ = iπ/ log R.
As usual, the fractional part x of x ∈ R is such√that 0 ≤ x < 1, x − x ∈
i∞ Φ5,6 (t)−∆(t)
(t − τ )dt.
Z. We denote by β0 the fractional part of 34 + 17285 3 π τ
3
Φ2,3 (t) 2
An easy computation shows that if β0 < 1/2, then the extreme positions of the
equilibrium points of the Green’s function, W1 , WR , or the zeros of the Bergman
kernel for Z = 1 and Z = R are W1 = −R2β0 , WR = −R1−2β0 . Whereas if
β0 > 1/2, W1 = −R2−2β0 , WR = −R2β0 −1 , with W1 .WR = R.
Remark 3.3. The same method with another change of variables can give another
expression to the result of Eichler and Zagier for the solution of ℘(z0 , τ ) = 0. In
fact, we write a solution in the following form:
∞
dx
z0 (τ ) = −
,
2
(x
−
e
)(x
− e2 )(x − e3 )
0
1
but here we make the substitution
x − e3 =
e1 − e3
,
z
z=
e1 − e3
x − e3
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324
AHMED SEBBAR AND THÉRÈSE FALLIERO
2
and use the fact that |λ e3e−e
| is very small for |τ | large. The same method gives
3
e3 −e1
with y = e3
y
dz
a
+ y(1 − y)
z0 (τ ) = √
an y n .
2 e1 − e3 0
z(1 − z)
n≥0
The term
√ 1
2 e1 −e3
y
0
√
dz
z(1−z)
tends to ± 12 ±
√
log(2 6+5)
2iπ
when τ tends to i∞ as in
[2].
4. Some identities
The constant −π /3 in (3.15) seems very special, but actually it is very natural.
It is classical ([9], p. 415) that the Weierstrass sigma function associated to the
periods 2ω1 and 2ω3 is related to the theta function θ1 (z, τ ) by
2
σ(u, 2ω1 , 2ω3 ) = 2ω1 e2η1 ω1 v
2
θ1 (v)
,
θ1 (0)
u = 2ω1 v.
Consequently the function Φ(v, τ ) = σ(u, 2ω1 , 2ω3 )e−2η1 ω1 v admits the following
infinite product decomposition ([9], p. 424):
2
Φ(v, τ ) =
∞
(1 − q n z 2 )(1 − q n z −2 )
−iω1
(z − z −1 )
,
π
(1 − q n )2
n=1
z = eiπv .
From (3.13), we have precisely
1
d2
π2
E2 (τ ) = −
log Φ(v, τ ) = −℘(v, τ ) −
inπτ +v 2 .
2
dv
3
n∈Z sinh
2
These functions are studied from a very different point of view in [8]. Our result
d2
gives the zero of the second derivative dv
2 log Φ(v, τ ).
For k ≥ 4, the zeros of Eisenstein series Ek have interested several authors,
but to our knowledge no result is available for the zeros of the series E2 (or the
critical points of the discriminant
function ∆(t)). As a function of the real variable
∞
0 ≤ q < 1, the series 1 − 24 n=1 σ1 (n)q n is a decreasing function from 1 to
−∞, hence it has a unique zero q0 in (0, 1) corresponding to a pure imaginary
number z0 = iγ0 , γ0 ∈ R of the Eisenstein series E2 (τ ). According to D. Zagier,
there are infinitely many zeros of E2 (τ ) in the half-strip {0 ≤ τ < 1} (private
communication, via H. Cohen). The following two zeros were computed for us by
H. Cohen:
z0 = 0.52352170001799926680053440480610976968 . . . i,
1
z1 = + 0.13091903039676244690411482601971302060 . . . i
2
q0
with z0 = log
2iπ . By comparing the formulas in Theorem 3.2 and in [2] (corollary)
when τ is a zero of the series E2 , we obtain some identities. In particular
Corollary 4.1. Let z0 = iγ0 be a zero of the Eisenstein series E2 , γ0 ∈ R. Then
γ0 verifies the two integral equations with some integers n and m:
√
∞
(2n + 1)γ0 π − log(5 + 2 6)
∆(it)
√
=
3 (t − γ0 )dt
288π 2 6
E
γ0
6 (it) 2
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EQUILIBRIUM POINT OF GREEN’S FUNCTION
and
325
√ ∞
Φ5,6 (it) − ∆(it)
1728 3
1
+m=
π
(t − γ0 )dt.
3
4
5
Φ2,3 (it) 2
γ0
We observe that from its infinite product expansion, ∆(t) does not vanish on
the upper half plane and moreover the series E6 (τ ) has on the imaginary axis a
unique zero at i. The integral in the preceding corollary must be taken first on a
segment [iγ0 , i − i] followed by the half circle centered at i , of radius and finally
on [i + i, i∞).
We refer to [3] for the complete definition of Jacobi forms. These are holomorphic
functions Φ : HxC → C such that
2
α β
k 2iπmcz
+b
z
cτ +d ,
=
(cτ
+
d)
(1) Φ aτ
,
e
∈
SL
(Z)
,
2
cτ +d cτ +d
γ δ
2
(λ, µ) ∈ Z2 ,
(2) Φ (τ, z + λτ + µ) = e−2iπm(λ τ +2λz) Φ (τ, z) ,
(3) Φ (τ, z) has a Fourier expansion of the form
Φ (τ, z) =
∞
c(n, r)e2iπ(nτ +rz) ,
n=0 r∈Z,r 2 ≤4nm
k and m are natural numbers called the weight and index of Φ. The ring of Jacobi
forms of index 1 is generated by the Jacobi-Eisenstein series E4,1 of weight 4 and
E6,1 of weight 6 [3] (p. 23):
E4,1 = 1 + ζ 2 + 56ζ + 126 + 56ζ −1 + ζ −2 q + · · · ,
E6,1 = 1 + ζ 2 − 88ζ − 330 − 88ζ −1 + ζ −2 q + · · ·
with q = e2iπτ , ζ = e2iπz and
1
(E6 E4,1 − E4 E6,1 ) ,
144
1 2
E4 E4,1 − E6 E6,1 .
=
144
Φ10,1 =
Φ12,1
By Theorem 3.6 of [3],
Φ12,1 (τ,z)
Φ10,1 (τ,z)
= − π32 ℘(τ, z). By (3.19), if z = z(τ ) is the solution
2
of ℘(z, τ ) = − π3 E2 as given in Theorem 3.2, we have 3E4 E6,1 (z, τ ) = 2E6 E4,1 (z, τ ).
5. Location and motions of the equilibrium point
We now give the result on the location of the equilibrium point for the Green’s
function.
Theorem 5.1. For each R > 1 and for each z ∈ CR = {1 < |z| < R}, the Green’s
function G(z, w) has a unique equilibrium point w = F (z) located on the same
diameter as z and that satisfies the inequalities
√
√
−|z| ≤ −|w| ≤ − R,
R ≤ |z| ≤ R,
√
√
1 ≤ |z| ≤ R.
− R ≤ −|w| ≤ −|z|,
Furthermore, the function F (z) approaches at the boundary the zero w(z) of the
Bergman kernel. If z = eα is real, this zero w = eu is given by the equation
℘(u; 2ω1 , 2ω3 ) = ℘(α; 2ω1 , 2ω3 ) +
℘ (α; 2ω1 , 2ω3 )
.
2ζ(α; 2ω1 , 2ω3 ) − 2λα
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326
AHMED SEBBAR AND THÉRÈSE FALLIERO
Proof. The two inequalities between z and w = F (z) are proved in [12] (p. 64).
Without loss of generality, we suppose from the start that z = eα is real so that
0 < α < log R. The Green’s function of CR with pole at z has the following form,
derived from (2.8):
G(z, w) = V (log z),
V (u) =
σ(u + α) −2λαu
e
,
σ(u − α)
where σ is the Weierstrass sigma function associated to the periods ω1 = log R ,
2
ω3 = iπ and λ = ωη11 = 13 2ωπ 1 E2 (τ ). The equilibrium point w = eu for G(z, w)
is given by the equation
ζ(u + α) − ζ(u − α) = 2αλ = 2α
1
3
π
2ω1
2
E2 (τ ).
As is shown in [10], there is a solution corresponding to the unique equilibrium point
in the annulus CR . By elementary transformations, this equation is equivalent to
℘(u; 2ω1 , 2ω3 ) = ℘(α; 2ω1 , 2ω3 ) +
(5.1)
℘ (α; 2ω1 , 2ω3 )
,
2ζ(α; 2ω1 , 2ω3 ) − 2λα
hence it is of the type considered before, but of a more complicated nature. We
would like to point out that (5.1) must be in accordance with the Schiffer relation
2
G(z,t)
(3.1) K(z, t̄) = − π2 ∂ ∂z∂
t̄ . For a real z = x0 ∈ CR , the equilibrium point w = x
is also real, and the limiting position of x when x0 tends to the boundary of CR is
dx
0)
= 0. We simply denote F (x, x0 ) = ∂G(x,x
so that
given by dx
∂x
0
∂F (x,x0 )
∂ 2 G(x,x0 )
∂x
∂ x
dx
0
0
= − ∂F∂x
= − ∂ 2∂x∂x
,
(x,x0 )
G(x,x0 )
dx0
2
and
dx
dx0
= 0 if and only if
∂F (x, x0 )
∂ 2 G(x, x0 )
=
= 0.
∂x0
∂x∂x0
dx
According to the Schiffer relation and (3.3), we obtain dx
= 0 if and only if
0
K(x0 , x) = 0. This explains the relation between the limiting positions of the equilibrium points and the zeros of the Bergman kernel. It is also possible to interpret
(5.1) as a perturbed form of (3.15). We use the classical formula giving the functions
πα
as follows:
℘ and ζ in terms of theta function with z = 2ω
1
η1
π d
1
lnθ
(z, τ ),
α+
1
ω1
2ω1 dz
2 2
π
d
η1
1
= −
−
lnθ
(z, τ ).
1
ω1
4ω1
dz 2
ζ(α; ω1 , ω3 )) =
℘(α; 2ω1 , 2ω3 )
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EQUILIBRIUM POINT OF GREEN’S FUNCTION
327
The equation (5.1) becomes
℘(u; 2ω1 , 2ω3 )
(5.2)
= −
η1
+ F(α; τ ),
ω1
where a perturbative term
(5.3)
F(α; τ ) = −
π
4ω1
2
d2
lnθ
dz 2
1
1
(z, τ ) −
1
2
π
4ω1
2
d3
dz 3 lnθ
d
dz lnθ
1
1
1
1
(z, τ )
.
(z, τ )
Remark 5.2. The same analysis can be done for the equilibrium point of the Green’s
function of a domain bounded by two non-concentric circles C1 , C2 , the first being
inside the second. For a given z in this domain, there is a unique circle C through
z orthogonal to the boundary circles. The equilibrium point w is on this circle.
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LABAG, Laboratoire Bordelais d’Analyse et Géométrie, Institut de Mathématiques,
Université Bordeaux I, 33405 Talence, France
E-mail address:
[email protected]
Faculté des Sciences, Université d’Avignon, 84000 Avignon, France
E-mail address: [email protected]
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