MODULAR FORMS: II
CHING-LI CHAI, CHANG-SHOU LIN, AND CHIN-LUNG WANG ABSTRACT. A “pre-modular form” Zn(σ; τ)of weight12n(n+1)is intro-duced for each n∈ N, where(σ, τ) ∈C×H. Let Eτ = C/(Z+Zτ).
Then the pre-modular form has the property that every non-trivial zero of Zn(σ; τ), namely σ6∈Eτ[2], corresponds exactly to the solution to the
non-linear mean field equation (MFE)
4u+eu=8πn δ0
on the flat torus Eτ. This paper is a sequel to [3], where the
hyperellip-tic curve ¯Xn(τ) ⊂ SymnEτ associated to the MFE is constructed. Our
construction of Zn(σ; τ)relies on a detailed study of the correspondence P1←X¯
n(τ) →Eτ where the former map is the hyperelliptic projection
and the latter map is induced from the addition law.
CONTENTS
0. Introduction 1
1. Geometry on Xnfor ρ=8nπ 9
2. Pre-modular forms Zn(σ; τ) 15
3. Explicit determination of Zn 26
4. A remark on the classical approach to Zn 32
5. Future perspectives 39
Appendix A. A counting formula for Lam´e equations 43
References 47
0. INTRODUCTION
Consider the flat torus E= Eτ =C/Λτ, τ= a+bi, b>0 andΛ=Λτ =
Zω1+Zω2with ω1=1 and ω2=τ. Let G be the Green function on E:
(0.1) −4G=δ0− 1 |E| on E, R EG=0, With Appendix A written by You-Cheng Chou.
1
where δ0 is the Dirac measure at the lattice point 0 ∈ E. In this paper, we
continue our study initiated in Part I [3] to study the equations (0.2) n∇G(ai) =
n
∑
j=1,6=i
∇G(ai−aj), i=1, . . . , n,
with unknown a= (a1,· · · , an), ai ∈ E×= E\{0}and ai 6= ajif i 6= j. The
main theme is the connection between (0.2) and the following mean field equation:
(0.3) 4u+eu= ρ δ0 on E, ρ∈R+.
Equation (0.3) is originated from the prescribed curvature problem in conformal geometry. It is also related to the higher dimensional complex Monge–Amp`ere equation: (0.4) det ∂ 2w ∂zi∂ ¯zj d i,j=1 =e −w on(E\{0})d.
For any solutions ui(z) to (0.3) corresponding to ρ = ρi, i = 1, . . . , d, the
function w(z1,· · · , zd) = − d
∑
i=1 ui(zi) + (log 4)dsatisfies (0.4) with a logarithmic singularity along the normal crossing divi-sor D= Ed\(E\{0})d. In particular, bubbling solutions to (0.3) give
exam-ples of bubbling solutions to (0.4).
The above discussions indicate the fundamental importance to study the concentration phenomenon of bubbling solutions to equation (0.3) in details. Indeed this is the heart inside the connection between (0.2) and (0.3) which we would like to explore. Suppose that{uk}is a sequence of
bubbling solutions to (0.3) corresponding to ρ = ρk. Then it was proved
in [4] by PDE method that ρk → 8πn with n ∈ N, and the blow-up set
{a1,· · · , an}of {uk}must satisfy (0.2). The PDE method actually applies
to a more general class of non-linear equations, but it is in general subtle to treat the reverse direction to construct solutions from{a1,· · · , an}. For
(0.3) with ρ= 8πn, it turn out that a sequence of bubbling solutions could be straightforwardly constructed from a non-trivial solution {a1,· · · , an}
to (0.2) through the developing map f and the Liouville formula: (0.5) uλ(z) =log 8e
2λ|f0(z)|2
(1+e2λ|f(z)|2)2, λ∈R,
where f is a meromorphic function onC which has zeros precisely on z≡ai
(mod Λ), i=1, . . . , n, and satisfies the type II constraints: (0.6) f(z+ωi) =e2iθjf(z), θj ∈R, j=1, 2.
The function f could be explicitly written down in terms of {a1,· · · , an},
The aim of this paper is to develop a theory towards understanding the structure of solutions to (0.2) and how it changes under deformations of Eτ in the moduli space. Our starting point is the hyperelliptic geometry
developed in Part I [3] which encodes the essential algebraic constraints among the n equations in (0.2). Based on it, the complete information turns out depends on certain pre-modular forms, which is the main theme of this paper. To be more precise, the main task of this paper is to prove that if (0.2) has a non-trivial solution for E = Eτ, then this τ must be a zero of
a pre-modular form Zn(σ; τ)of weight 12n(n+1), with σ ∈ E\Eτ[2]and
τ ∈ H = {τ ∈ C | Im τ > 0}. That is, Zn(σ; τ) is a modular form of
weight 12n(n+1)with respect toΓ(N)whenever σ ∈Eτ[N](the N-torsion
points). Our function Zn(σ; τ)is holomorphic in τ∈H and real analytic in σ∈ Eτ, therefore it also provides a tool to study the corresponding modular
form at any N torsion points via deformations in σ. Recently, the idea of deformations was successfully applied in [2] in the case n = 1 (ρ= 8π) to provide a complete solution to (0.3). It is our hope that results of this paper and Part I [3] will lay the foundation to extend the study to all n∈N.
Now we give the detailed descriptions. We start by explaining the notion “non-trivial solutions” to (0.2). In [3] we proved the following result:
Theorem A. Let a = (a1,· · · , an) ∈ En be a solution to (0.2), then either
{a1,· · · , an} ∩ {−a1,· · · ,−an} = ∅ or {a1,· · · , an} = {−a1,· · · ,−an}. Moreover, (0.2) is equivalent to (0.7) n
∑
j=1 ∇G(aj) =0,and the following holomorphic system (0.8)
n
∑
j=1,6=i
(ζ(ai−aj) +ζ(aj) −ζ(ai)) =0, i=1,· · · , n.
We remark that the Weierstrass ζ function has singularities only at the lattice points and thus (0.8) is meaningful only if ai 6=0 for all i and ai 6=aj
for i6= j (as points in E). From the system of equations (0.8) we introduce a hyperelliptic curve ¯Ynas follows. Let
Yn= { (a1,· · · , an) | ai ∈ E×for all i, ai 6=ajfor all i6=j,
and(a1,· · · , an)satisfies (0.8)}/Sn,
(unordered n-tuples), and ¯Ynbe the closure of Ynin SymnE = En/Sn, the
n-th symmetric product of E. Then ¯
Yn=Yn∪ {(0,· · · , 0)}
Define the map B : Yn→C by
a7→ Ba = (2n−1) n
∑
i=1
and let Xn = { {a1,· · · , an} ∈ Yn | {a1,· · · , an} ∩ {−a1,· · · ,−an} = ∅}.
Then a classical result says that
(i) Yn\Xnconsists of 2n+1 points (counting with multiplicities),
(ii) B|Xn : Xn →C is a two to one map (onto its image),
(iii) ¯Xn=Y¯nis a hyperelliptic curve which can be parametrized by
Yn∼= {(B, C) |C2= `n(B)}
where `n(B) is a polynomial in B of degree 2n+1. The branch
points of the hyperelliptic curve is precisely ¯Xn\Xn. In particular,
the algebraic curve ¯Xnis smooth if`n(B)has no multiple roots.
A proof of the above statements with rigorous details can be found in [3], which is based on a detailed study of the (integral) Lam´e equation on Eτ
(0.9) w00 = (n(n+1)℘ +B)w
from both the analytic and algebraic point of views. Indeed, let (0.10) wa(z):=ez∑ ζ(ai) n
∏
i=1 σ(z−ai) σ(z)be the classical Hermite–Halphen ansatz. Then wa and w−a are
indepen-dent solutions to (0.9) with B= Baif and only if a∈ Xn. Historically ¯Xnis
also known as the Lam´e curve.
By Theorem A, it is natural to separate the study of (0.2) into two stages. We first study the algebraic geometry associated to the hyperelliptic curve
¯
Xn and then study the remaining single equation on Green functions (0.7)
for a∈ X¯n. Notice that, by the anti-symmetry of∇G, (0.7) holds automati-cally if a is a branch point of ¯Xn. Hence the system (0.2) contains all branch
points of ¯Xnas its solutions. We say that a solution a to (0.2) is non-trivial if
a is not a branch point of ¯Xn.
We start by reviewing of the simplest case n=1. Then ¯X1∼= E and there
is no new hyperelliptic geometry. Also a is a solution to (0.2) if and only if
∇G(a) = 0, namely a ∈ E is a critical point of G. The branch points of E consist of all half periods 12ωj, j = 1, 2, 3. Hence a non-trivial solution p is
simply a non half-period critical point of G. How many such points might G have? This had been answered in [11]:
Theorem B. For any τ ∈H, the Green function G(z; τ)on the flat torus Eτ has
at most five critical points.
Since G(z)is an even function, the extra critical points appear in pair±p. The following result was announced in [11] and recently proved in [12]:
Theorem C. Suppose that the pair of non half-period critical points{±p}of G exists, the±p are the minimal points of G. In fact any solution to (0.3) must be a minimizer of the non-linear functional
J8π(u) = 1 2 Z E |∇u|2−8π logZ Ee −8πG+u on u∈ H1(E) ∩ {u|R Eu=0}.
To proceed, we need the detailed analytic description of Green functions in [11]. Recall the theta function ϑ :=ϑ1, where
ϑ1(z; τ) = −i
∞
∑
n=−∞
(−1)nq(n+12)2e(2n+1)πiz, q=e2πiτ.
Then the Green function is given by (0.11) G(z; τ) = − 1
2π log|ϑ(z; τ)| + y2
2b+C(τ),
where z= x+iy=tω1+sω2, ηi =2ζ(12ωi), i =1, 2, are quasi-periods and
C(τ)is a constant so thatRE
τG = 0. Using(log ϑ)z = ζ(z) −η1z and the
Legendre relation η1ω2−η2ω1=2πi, we have
(0.12) Gz(z; τ) = − 1
4π ζ(z; τ) −tη1(τ) −sη2(τ).
For any z = tω1+sω2, t, s ∈ R, we define the Hecke function Z(z; τ) =
Zt,s(τ)by
Zt,s(τ) =Z(tω1+sω2; τ) =ζ(tω1+sω2; τ) −tη1(τ) −sη2(τ).
Therefore z is a critical point of G(z; τ)if and only if Zt,s(τ) = 0. Since
Zt+1,s =Zt,s =Zt,s+1, we may assume that 0≤t, s<1.
We found Zt,s(τ)in [11]. But it first appeared in [9] where Hecke showed
that if z = tω1+sω2is an N torsion point then Zt,s(τ)is a modular form
of weight one with respect toΓ(N) = {A∈SL(2,Z) | A≡ I2 (mod N) }.
Z(z; τ)is holomorphic in τ, but not in z. Zt,s(τ) ≡ 0 for z a half-period,
thus we are interested in equation Zt,s(τ) =0 with(t, s) 6∈ 12Z2.
Now we turn to (0.2) and (0.3) for general n ≥ 1. A central issue is to study the following
Conjecture 0.1. Under the deformations in τ, equation (0.2) has no bifurcations
at any non-trivial solution.
For n=1, Conjecture 0.1 can be deduced from Theorem B (c.f. [12]). For n≥2, we do not expect an extension of Theorem B to work.
Our approach to Conjecture 0.1 for n ≥ 2 is to construct a pre-modular form Zn(σ; τ)with σ∈ Eτ which generalizes the Hecke function (Z1= Z),
and study this single analytic function Zninstead. The pre-modular form
Zn(σ; τ)is naturally associated to the family of hyperelliptic curves ¯Xn(τ), τ∈H. The goal is to show that any non-trivial solution a = (a1,· · · , an)to
(0.2) comes from the zero of Zn(σ; τ)with σ=∑ni=1ai, and vice versa.
Consider the Green function equation (0.7) in Theorem A. Let zn(a) =
ζ(∑ni=1ai) −∑ni=1ζ(ai). If∑ni=1ai 6=0 then
−4π
∑
∇G(ai) =∑
(ζ(tiω1+siω2) −tiη1−siη2)In §1 we will show that in fact ∑ni=1ai 6∈ Eτ[2]for any a ∈ Xn(τ). Hence (0.7) is equivalent to (0.13) zn(a) =Z n
∑
i=1 ai . This motivates us to study the mapσ: ¯Xn→E, a7→ σ(a):=
n
∑
i=1
ai
induced from the addition map En → E. The algebraic curve ¯Xn(τ)might
be singular for some τ, but it must be irreducible (c.f. Theorem 1.5 (3)). In particular, σ is a finite morphism and deg σ is defined. The function field K(X¯n)is also defined and is finite over K(E) ∼= C[℘,℘0]. It is well known
that deg σ= [K(X¯n): K(E)] =number of points for a general fiber. Theorem 0.2. The map σ: ¯Xn→ E has degree12n(n+1).
In §1, we will apply the Theorem of the Cube in the theory of abelian va-rieties (see e.g. [15]) to prove it. From Theorem 0.2, there is a polynomial Wn(z) ∈C[g2, g3,℘(σ),℘0(σ)][z]of degree12n(n+1)which defines the
cov-ering map σ between algebraic curves.
The major question is to find a natural primitive element of this cover-ing map. Namely a rational function on ¯Xn which has Wnas its minimal
polynomial. This is achieved by the following fundamental theorem:
Theorem 0.3. The rational function zn ∈ K(X¯n)is a primitive generator for the
field extension K(X¯n)over K(E)which is integral over the affine curve E×.
Moreover, the induced map zn : ¯Xn → P1also has degree 12n(n+1), with its
fibration structure over∞∈P1analytically isomorphic to σ : ¯X
n →E over 0.
This means that Wn(zn) = 0, and conversely for general τ and σ=σ0∈
Eτ, the roots of Wn(z) = 0 are precisely those 12n(n+1)values z = zn(a)
with σ(a) =σ0.
The proof is given in §2, Theorem 2.1 and 2.9. A major tool used in the proof is the notion of tensor product of two Lam´e equations w00 = I1w and
w0 = I2w, where I=n(n+1)℘(z), I1 = I+Baand I2= I+Bb. Indeed, for
given τ with ¯Xnnot necessarily being smooth, and for a given general point
σ0 ∈ E, we need to show that the 12n(n+1)points on the fiber of ¯Xn → E
above σ0 has distinct zn values. From the definition of zn, it is enough to
show that for σ(a) = σ(b) = σ0, the condition ∑ ζ(ai) = ∑ ζ(bi) implies
Ba =Bband then a= b if σ0 6∈E[2].
If w001 = I1w1 and w002 = I2w2, then the product q = w1w2 satisfies the
fourth order ODE (tensor product) given by
We remark that if Ba = Bb, then I1= I2and q actually satisfies a third order
ODE as the second symmetric product of a Lam´e equation. This is a useful tool in Part I [3] in the study of the Lam´e curve.
If however a 6= b, by (0.10) and addition law we find that q = waw−b+
w−awb is an even elliptic function solution to (0.14), namely a polynomial in
x = ℘(z). This leads to strong constraints on (0.14) in variable x and even-tually leads to a contradiction for generic choices of σ0.
Now we set
(0.15) Zn(σ; τ) =Wn(Z(σ; τ)).
Then Zn(σ; τ)is pre-modular of weight 12n(n+1), by which we mean that
it is modular with respect toΓ(N)whenever σ∈ Eτ[N]being an N-torsion
point. From the construction and (0.13) it is readily seen that Zn(σ; τ)is the
generalization of the Hecke function Z=Z1we are looking for. In fact, for
general n≥1, we establish the following correspondences in §2: (1) Solution u to the mean field equation (0.3) for ρ=8πn. (2) Periods integral Z L f0 f ∈ √
−1R for any one cycle L (c.f. (1.5)). (3) Green equation
n
∑
i=1
∇G(ai) =0 on the hyperelliptic curve Xn.
(4) Coincidence equation zn(a) =Z(σ(a))for a∈Xn.
(5) Zero of pre-modular form Zn(σ; τ):=Wn(Z)with σ6∈Eτ[2].
(For more precise statements on (5) see Theorem 2.13.)
For σ being an N-torsion point, the modular form Z2(σ; τ)and Z3(σ; τ)
were first constructed by Dahmen [6] in his study on integral Lam´e equa-tions (0.9) with algebraic soluequa-tions (i.e. with finite monodromy group). In-deed, for σ∈ Eτ[N],
Z2(σ; τ) =Z3(σ) −3℘(σ; τ)Z(σ) − ℘0(σ; τ),
Z3(σ; τ) =Z6−15℘Z4−20℘0Z3+ (274 g2−45℘2)Z2−12℘0℘Z− 54℘02.
The construction of Z2is based on the addition formula, and the
construc-tion of Z3is based on the n= 2 case and a technical intermediate step: for z3=ζ(σ) −∑3i=1ζ(ai)with σ=∑3i=1ai, a classical cubic formula
(0.16) z33=3(℘(σ) +
∑
℘(ai))z3+ (℘0(σ) −∑
℘0(ai))holds (c.f. Lemma 4.3). By eliminating terms involving ai’s, a degree six
polynomial equation W3(z3) =0 is then achieved. The existence of a
mod-ular form Zn of weight 12n(n+1) is also conjectured in [6], though it is
unclear how his method would proceed for n≥4.
For all σ and n, the existence of the (pre)-modular forms Zn(σ; τ)now
follows from Theorem 0.3 and (0.15). However the explicit construction is still a challenging problem.
In a different direction, the Lam´e curve had also been studied extensively in the finite band integration theory. In the complex case, this theory concerns about the eigenvalue problem on a second order ODE Lw := w00−Iw =
Bw with eigenvalue B. The potential I = I(z)is called a finite-gap (band) potential if the ODE has only logarithmic free solutions except for a finite number of B∈C. The integral Lam´e equations (with I(z) =n(n+1)℘(z)) provide good (indeed earliest) examples of them. Using this theory, Maier [14] had recently written down an explicit map π : ¯Xn → E in terms of
coordinates(B, C)on ¯Xn(in our notations). It turns out we can prove Theorem 0.4(c.f. Theorem 3.5). The map π agrees with σ : ¯Xn→ E.
In principle this allows us to compute the polynomial Wn(z) for any
n∈N by eliminating variables(B, C), though in practice the needed calcu-lations are very demanding and time consuming. §3 is devoted to this ex-plicit construction. In particular the weight 10 pre-modular form Z4(σ; τ)is
explicitly written down by a couple hours Mathematica calculations (c.f. Ex-ample 3.10): Z4(σ; τ) =Z10−45℘Z8−120℘0Z7+ (3994 g2−630℘2)Z6−504℘℘0Z5 − 15 4(280℘ 3−49g 2℘ −115g3)Z4+15(11g2−24℘2)℘0Z3 −9 4(140℘ 4−245g 2℘2+190g3℘ +21g22)Z2 − (40℘3−163g2℘ +125g3)℘0Z+ 34(25g2−3℘2)(℘0)2. (0.17)
One might be curious on the possibility to extend the more classical con-struction of Zn(σ; τ)for n ≤3 to n≥4. In §4.2 we explain why no classical
formula like (0.16) with degree n can exist for n ≥ 4, and thus it is not possible to derive the explicit expression of Zn(σ; τ)via a naive induction
for n ≥ 4. For this purpose, and for the sake of completeness, we present in §4.1 an alternative treatment of Dahmen’s construction for n = 2, 3 in a format that is needed for our discussions in §4.2. At this point, it might be insightful to point out that K(X¯n)is in general not Galois over K(E)(under
the morphism σ : ¯Xn→ E, c.f. Example 4.4).
The existence and effective construction of Zn(σ; τ) opens the door to
extend our results on equations (0.2) and (0.3) for n = 1 (established in [11, 12, 2]) to general n ∈ N. As a related example, the explicit expression
of Z4(σ; τ)can be used to solve Dahmen’s conjecture on a counting formula
for integral Lam´e equations with finite monodromy for n = 4 (c.f. §5.2 and Appendix A). Further applications of our pre-modular forms will be discussed in subsequent works. Some of them, especially those related to equations (0.2) and (0.3), are briefly described in §5.
1. GEOMETRY ON XnFORρ=8nπ
We would like to extend some of the results proved in [11] for ρ=8π to the case ρ=8nπ for all n∈N. Especially we will generalize Hecke’s
func-tion Z to Znwhose non-trivial zeros again encode the structure of solutions
to the corresponding mean field equation
(1.1) 4u+eu =8πn δ0 in E=Eτ.
We first recall the structural results from Part I [3].
Proposition 1.1(Periods integrals and type II evenness [3]).
(1) If solutions exist for ρ=8nπ, then there is a unique even solution within each type II scaling family.
(2) The solution u is determined by the zeros a1,· · · , anof its developing map
f (through expression (0.5) with λ=0). In fact for g= f0/ f , g(z) = n
∑
i=1 ℘0(ai) ℘(z) − ℘(ai) = n∑
i=1 Ω(z, ai), f(z) = f(0)exp Z z g(ξ)dξ = f(0) n∏
i=1 exp Z z Ω(ξ, ai)dξ, (1.2)subject to the non-degenerate conditions ai 6∈E[2], ai 6= ±ajfor i6=j.
The condition ordz=0g(z) = 2n leads to n−1 equations for a1, . . . , an:
Under the notations(w, xj, yj) = (℘(z),℘(pj),℘0(pj)),
g(z) = n
∑
j=1 1 w yj 1−xj/w = n∑
j=1 yj w+ n∑
j=1 yjxj w2 + · · · + n∑
j=1 yjxrj wr+1 + · · · .Since g(z)has a zero at z =0 of order 2n and 1/w has a zero at z =0 of order two, we get xi 6=xj for i6= j and
n
∑
j=1 yjxrj = n∑
j=1 ℘0(aj)℘(aj)r =0, 0≤r ≤n−2.This gives the polynomial system describing the developing maps. Notice that the condition yi 6=0 follows automatically.
Theorem 1.2(Green/polynomial system [3]). For ρ = 8nπ, n ∈ N, the n
equations for a1,· · · , anwith ai 6= ±aj for i6=j are precisely
(1.3) ℘0(a1)℘r(a1) + · · · + ℘0(an)℘r(an) =0,
where r=0, . . . , n−2, and
Indeed the Green function equation (1.4) is equivalent to the type II con-dition (0.6). From Proposition 1.1 (2), this means that the periods integral
R
LiΩ∈
√
−1R for L1, L2being the two fundamental periods of E. Let f be
given by (1.2). By evaluating the periods integrals (c.f. Part I [3, (5.4) and (5.7)]), we have
f(z+ω1) =e−4πi∑isif(z),
f(z+ω2) =e4πi∑itif(z),
(1.5)
where ai = tiω1+siω2for i =1, . . . , n. Then by the Liouville theorem, we
obtain solutions uλto (1.1) by (0.5).
Note that the algebraic conditions in (1.3) are equivalent to (1.6)
∑
j6=i
℘0(aj) + ℘0(ai)
℘(aj) − ℘(ai) =0, i=1, . . . , n.
(See Part I [3, Theorem 0.6].) By the addition formula for℘, (1.6) is identical with (0.8) provided that ai 6= ±aj for i 6= j, or equivalently℘(ai) 6= ℘(aj)
for i6= j. Thus, a scaling family of solutions to (1.1) corresponds exactly to a non-trivial solution to (0.2).
Proposition 1.3(Hyperelliptic geometry, the Liouville curve [3]). For xi :=
℘(ai)with xi 6=xjfor i6=j and yi := ℘0(ai), the first n−1 algebraic equations
(1.7)
∑
yixri =0, r =0, . . . , n−2,defines a non-singular open algebraic curve Xn ⊂ SymnE, called the Liouville
curve. It has a hyperelliptic structure under the 2 to 1 map a 7→ ∑℘(ai), or
algebraically: Xn→P1, (xi, yi)ni=1 7→ n
∑
i=1 xi.This is closely related to (integral) Lam´e equations: Write f =exp Z g dz=exp Z n
∑
i=1 (2ζ(ai) −ζ(ai−z) −ζ(ai+z))dz =e2∑ni=1ζ(ai)z n∏
i=1 σ(z−ai) σ(z+ai) = wa w−a , where (1.8) w(z) =wa(z):= ez∑ ζ(ai) n∏
i=1 σ(z−ai) σ(z)is the Hermite–Halphen ansatz for solutions to integral Lam´e equations (see [16, 3]).
Proposition 1.4(Explicit map a7→ Ba[3]). a ∈ Xnif and only if waand w−a
are independent solutions of the Lam´e equation (1.9) d
2w
dz2 − n(n+1)℘(z) +Baw=0,
where Ba = (2n−1)∑ni=1℘(ai).
The full information on the compactified hyperelliptic curve ¯Xn → P1,
especially on the branch points added, is described by
Theorem 1.5(Hyperelliptic geometry on ¯Xn, the Lam´e curve [3]).
(1) The natural compactification ¯Xn ⊂ SymnE coincides with the, possibly
singular, projective model of the hyperelliptic curve defined by C2= `n(B, g2, g3)
=4Bs2n+4g3sn−2sn−g2sn−1sn−g3s2n−1,
(1.10)
in(B, C), where sk = sk(B, g2, g3) = rkBk+ · · · ∈ Q[B, g2, g3], is a
recursively defined polynomial of homogeneous degree k with deg g2 =2,
deg g3 =3, and B= (2n−1)s1.
(2) deg`n=2n+1 and ¯Xnhas arithmetic genus g= n.
(3) The curve ¯Xn is smooth except for a finite number of τ, namely the
dis-criminant loci of`n(B, g2, g3)so that`n(B)has multiple roots. ¯Xnis an
irreducible curve which is smooth at infinity.
(4) The 2n+2 branch points a ∈ X¯n\Xnare characterized by−a = a. In
fact{−ai} ∩ {ai} 6=∅⇒ −a= a. Also 0∈ {ai} ⇒a= (0, 0,· · · , 0).
(5) The limiting system of (1.7) at a=0nis given by (1.11)
n
∑
i=1
t2ri +1 =0, r=1, . . . , n−1 under the constraints
(1.12) ti 6=0, and ti 6= −tj.
Moreover, the system (1.11) and (1.12) has a unique solution inPn−1 up to permutations.
The point a = 0n ∈ X¯n is referred as the point at infinity. For the other 2n+1 finite branch points with a = −a, wa = w−awhich is still a solution
to the Lam´e equation. In the literature, these 2n+1 functions are known as the Lam´e functions. Indeed, there are four species of them, depending on the number of half periods contained in{ai}. We call them being of type O,
I, II, and III respectively. For n = 2k being even, a must be of type O or II. For n=2k+1 being odd, a must be of type I or III. There are factorizations of the polynomial`n(B)according to the types:
Proposition 1.6. [8, 16] In terms of ei = ℘(12ωi), we may write
where cn>0 is a constant, li(B)’s are monic polynomials in B such that
(1) For n = 2k, l0(B)consists of type O roots with deg l0(B) = 12n+1 =
k+1. For i = 1, 2, 3, li(B) consists of type II roots a which does not
contain 12ωi. Moreover, deg li(B) = 12n= k.
(2) For n =2k+1, l0(B)consists of type III roots with deg l0(B) = 12(n−
1) =k. For i= 1, 2, 3, li(B)consists of type I roots a which contains12ωi.
Moreover, deg li(B) = 12(n+1) =k+1.
We remark that both Proposition 1.6 and Theorem 1.5 (5) will be used in the proof of Theorem 0.2 (= Theorem 1.10 later in this section). Here are some examples to illustrate Proposition 1.6:
Example 1.7. Decomposition`n(B) =c2nl0(B)l1(B)l2(B)l3(B)for 1≤n≤5.
(1) n=1, k=0, ¯X1∼= E, C2= `1(B) =4B3−g2B−g3=4 3
∏
i=1 (B−ei). (2) n=2, k=1, (notice that e1+e2+e3=0) C2= `2(B) = 814B5−277g2B3+13g3B2+13g22B−g2g3 = 2 2 34(B 2−3g 2) 3∏
i=1 (B+3ei). (3) n=3, k=1, deg li(B) =2 for i=1, 2, 3, C2= `3(B) = 1 223454B(16B 6−504g 2B4+2376g3B3 +4185g22B2−36450g2g3B+91125g23−3375g32) = 2 2 3454B 3∏
i=1 (B2−6eiB+15(3e2i −g2)). (4) n=4, k=2, deg l0(B) =3, C2= `4(B) = 1 385474(B 3−52g 2B+560g3) 3∏
i=1 (B2+10eiB−7(5e2i +g2)). (5) n=5, k=2, deg li(B) =3 for i=1, 2, 3, C2= `5(B) = 1 3125474112(B 2−27g 2) × 3∏
i=1(B3−15eiB2+ (315e2i −132g2)B+ei(2835e2i −540g2)).
Now we study the last equation (1.4) in Theorem 1.2, namely the Green function equation on ¯Xn: (1.13) n
∑
i=1 ∇G(ai) =0.Definition 1.8(Fundamental rational function). Consider the function on En: zn(a1, . . . , an):=ζ(a1+ · · · +an) − n
∑
i=1 ζ(ai).znis a rational function on Ensince it is meromorphic and periodic in each variable
ai, and Enis an abelian variety.
It satisfies the reduction property along half periods: If a = a0∪ {12ωi}then zn(a) =zn−1(a0).
Consider the identity (where ai =tiω1+siω2):
−4π
∑
∇G(ai) =∑
Z(ai)=
∑
(ζ(tiω1+siω2) −tiη1−siη2)=ζ(
∑
ai) − (∑
ti)η1− (∑
si)η2−zn(a)=Z(
∑
ai) −zn(a).Hence the Green equation (1.13) is equivalent to (1.14) zn(a) = Z(
∑
ai).This motivates us to study the branched covering map (1.15) σ: ¯Xn→E, a7→ σ(a):=
n
∑
i=1
ai
induced from the addition map En→ E.
Now we may move to the discussions concerning new modular func-tions. We start by determining deg σ.
Remark 1.9. For the reader’s convenience we recall some definitions and facts. The function field K(C)is defined for any irreducible algebraic curve C. Two irreducible curves which are isomorphic outside a finite number of points (birational) have the same function field. For a finite morphism of irreducible curves f : X → Y, K(X)is a finite extension of K(Y)and the degree of f is defined by deg f = [K(X) : K(Y)]. Geometrically deg f is also the number of points for a general fiber f−1(p), p∈Y. Indeed, for any L∈Pic Y (line bundle), we have deg f∗L= deg f deg L. Taking L=OY(p)
with p a general smooth point of Y then gives the result.
A standard reference is [10, II.6, Proposition 6.9], where nonsingular curves are treated. The irreducible case is reduced to the nonsingular case through normalizations ˜X → X and ˜Y → Y, since it is clear that the in-duced finite morphism ˜f : ˜X→ Y has the same degree as f . Furthermore,˜ the definition also extends to the case f : X → Y where X = Sk
i=1Xi has
a finite number of irreducible components. We require that f|Xi is a
fi-nite morphism for each i and then deg f := ∑ki=1deg f|Xi. Since all curves
considered here are proper (projective), it is enough to require f|Xi to be non-constant to ensure that it is a finite morphism.
Theorem 1.10. The map σ: ¯Xn →E has degree12n(n+1).
Proof. The idea is to apply Theorem of the Cube [15, p.58, Corollary 2] for mor-phisms from an arbitrary variety V (not necessarily smooth) into abelian varieties (here the torus E): For any three morphisms f , g, h : V →E and a line bundle L∈Pic E, we have
(f+g+h)∗L∼= (f+g)∗L⊗ (g+h)∗L⊗ (h+ f)∗L
⊗ f∗L−1⊗g∗L−1⊗h∗L−1.
(1.16)
We will apply it to the algebraic curve V = Vn ⊂ Enwhich consists of the
ordered n-tuples a’s so that Vn/Sn =X¯n.
Take L be any line bundle with deg L6=0. Using the fact that deg F∗L =
deg F deg L for any finite morphism F : V →E, (1.16) implies that deg(f+g+h) =
deg(f+g) +deg(g+h) +deg(h+ f) −deg f−deg g−deg h. (1.17)
We prove inductively that for j = 1, . . . , n the branched covering map fj : Vn →E defined by
fj(a):=a1+ · · · +aj
has degree 12j(j+1)n!. The case j = n then gives the result since fn
de-scends to σ under the Snaction. (Notice that the map fj can not descend
to a map on ¯Xn for all j < n.) Since the curve Vn is not necessarily
irre-ducible, we need to make sure that fj is non-constant on each irreducible
component to guarantee that fjis a finite morphism. This will nevertheless
be clear during the following proof.
Assuming first that it has been proved for j =1, 2. To go from j to j+1, we let f(a) = fj−1(a), g(a) = aj, and h(a) = aj+1. Then by (1.17), fj+1has
degree n! times
1
2j(j+1) +3+21j(j+1) −12(j−1)j−1−1= 21(j+1)(j+2)
as expected.
It remains to investigate the case j=1 and j=2.
For j = 1, by Theorem 1.5 (4), the inverse image of 0 ∈ E under f1 :
Vn → E consists of a single point~0. By Theorem 1.5 (5), the limiting
sys-tem of equations (1.11) and (1.12) has a unique non-degenerate solution in
Pn−1 up to permutations. From this, we conclude that there are precisely
n! branches of Vn→ E near~0. For a point b∈ E close to 0, each branch will
contribute a point a with a1=b. Thus the degree of f1is n!.
For j = 2, we consider the inverse image of 0 ∈ E under f2 : Vn → E.
Namely Vn 3a7→ a1+a2 =0.
The point a = 0 again contributes degree n! by a similar branch argu-ment: Indeed, over each branch near~0 we may represent a = (ai(t))as an
analytic curve in t. Then t7→a1(t) +a2(t) ∈ E is still an one to one map for
and the point~0 always contributes n! to the degree counting. Notice that fkis non-constant on each branch near~0, and every irreducible component of Vn will contain at least one such branch, hence fk is non-constant along
each irreducible component of Vn.
Otherwise, for a6=0 with f2(a) =0, then a1 = −a2and thus a= −a by
Theorem 1.5 (4).
If n = 2k, by Proposition 1.6 (1) the degree contribution from type O points a= {±a1,· · · ,±ak}is given by
(k+1) × (k×2× (n−2)!),
while the degree from the type II points{±a1,· · · ,±ak−1,12ωi,12ωj}is
3×k× ((k−1) ×2× (n−2)!). The sum is 2(4k2−2k)(n−2)!=2n!.
If n = 2k+1, by Proposition 1.6 (2), the degree contribution from type III points{±a1,· · · ,±ak−1,21ω1,12ω2,12ω3}is
k× ((k−1) ×2× (n−2)!), while the type I points{±a1,· · · ,±ak,12ωi}contribute
3× (k+1) × (k×2× (n−2)!).
The sum is again 2(4k2+2k)(n−2)!=2n!.
Thus in both cases we get the total degree n!+2n!=3n! as expected.
During the proof we have actually shown that
Proposition 1.11. The map σ is unramified at the infinity point0n∈ X¯n. We want to emphasize again that both Theorem 1.10 and Proposition 1.11 hold for all Eτ, τ∈H, regardless the smoothness of ¯Xn.
2. PRE-MODULAR FORMS Zn(σ; τ)
When no confusion should arise, we denote the restriction zn|X¯n also by zn. Then znis a rational function on ¯Xnwith poles along the fiber σ−1(0)
under the branch covering (summation) map σ : ¯Xn → E. To avoid trivial
situation we assume that n≥2 (since z1 =0 by definition.)
Theorem 1.10 strongly suggests the following statement:
Theorem 2.1(New pre-modular forms).
(1) There is a (weighted homogeneous) polynomial Wn(z) ∈C[g2, g3,℘(σ),℘0(σ)][z]
of z-degree12n(n+1)such that for σ=∑ ai
Wn(zn(a)) =0.
Indeed, no matter ¯Xnis smooth or not, zn(a)is a primitive generator of
the finite extension of rational function fields K(X¯n)over K(E), which has
(2) The function Zn(σ; τ):=Wn(Z)is “pre-modular” of weight 12n(n+1),
with Z, ℘(σ), ℘0(σ), g2, g3 being of weight 1, 2, 3, 4, 6 respectively.
Here pre-modular means that it is modular in τ forΓ(N)whenever we fix
σ∈ Eτ[N]being an N-torsion point.
Proof. Since z∈K(X¯n), which is algebraic over K(E)with degree12n(n+1)
by Theorem 1.10, its minimal polynomial Wn(z) ∈K(E)[z]certainly exists
with d :=deg Wn| 12n(n+1).
Furthermore, since z has no poles over E×, it is indeed integral over the affine Weierstrass model of E×where
R(E) =C[x, y]/(y2−4x3−g2x−g3)
with x= ℘(σ)and y= ℘0(σ). Thus the major statement in Theorem 2.1 (1)
is the claim that znis essentially a primitive generator.
Notice that for σ0 ∈ E being outside the branch loci of σ : ¯Xn → E,
there are precisely 12n(n+1)different points a = {a1,· · · , an} ∈ X¯n with
σ(a) = ∑ ai = σ0. Thus for the rational function zn = ζ(∑ ai) −∑ ζ(ai) ∈
K(X¯n)to be a primitive generator, it is sufficient to show that znhas exactly 1
2n(n+1) branches over K(E). That is, ∑ ζ(ai)gives different values for
different choices of those a above σ0. Indeed, for any given σ = σ0, the
polynomial Wn(z) = 0 has at most d roots. But now zn(a) with σ(a) =
σ0 gives 12n(n+1) distinct roots of Wn(z), hence we must conclude d = 1
2n(n+1)and znis a primitive generator.
Hence it is sufficient to show the following more precise result:
Theorem 2.2. Let a, b∈ Ynand(a1,· · · , an),(b1,· · · , bn) ∈ Cnbe
representa-tives of a, b such that (2.1) n
∑
i=1 ai = n∑
i=1 bi, n∑
i=1 ζ(ai) = n∑
i=1 ζ(bi).Suppose that ∑℘(ai) 6= ∑℘(bi). Then a, b are branch points of Yn → P1
corresponding to Lam´e functions of the same type.
We emphasize that ¯Xnis not required to be smooth.
Theorem 2.1 (1) follows immediately by choosing σ0outside the branch
loci of ¯Xn → E and σ0 6∈ E[2]. Indeed, let a, b ∈ Ynwith σ(a) = σ(b) = σ0
and zn(a) = zn(b), or more precisely with conditions in (2.1) satisfied. By
Theorem 2.2 we are left with the case∑℘(ai) = ∑℘(bi)but a 6= b. Then
a = −b by Theorem 1.5 (1), and in particular σ(a) = −σ(b). Together with σ(a) =σ(b)we conclude that σ0 =σ(a) =σ(b) ∈ E[2]. This contradicts to
the assumption that σ06∈E[2]. Hence we must have a=b.
(2) follows from (1) and Hecke’s theorem that Z(σ; τ)is a modular
func-tion of weight one whenever σ = (k1ω1+k2ω2)/N being an N-torsion
point [9].
We will give two proofs of Theorem 2.2. The first proof is longer but contains more informations.
Recall that the Hermite–Halphen ansatz w±a(z) =e±z∑ ζ(ai) n
∏
i=1 σ(z∓ai) σ(z)are solutions to w00 = (n(n+1)℘(z) +Ba)w=: I1w, and
w±b(z) =e±z∑ ζ(bi) n
∏
i=1 σ(z∓bi) σ(z)are solutions to w00 = (n(n+1)℘(z) +Bb)w =: I2w. Then qa,−b := waw−b
and q−a,b := w−awb are solutions to the fourth order ODE formed by the
tensor product of the two Lam´e equations. By assumption. (2.2) qa,−b(z) = n
∏
i=1 σ(z−ai)σ(z+bi) σ2(z)is an elliptic function since ∑ ai = ∑ bi. Similarly q−a,b(z) = qa,−b(−z)is
elliptic. In particular there exists an even elliptic function solution
Q := 12(qa,−b+q−a,b) = (−1)n∏
n
i=1σ(ai)σ(bi)
z2n +higher order terms. Lemma 2.3. The fourth order ODE is given by
(2.3) q0000−2(I1+I2)q00−6I0q0+ ((Ba−Bb)2−2I00)q=0.
Here I =n(n+1)℘(z), I1 = I+Baand I2 = I+Bb.
Proof. This follows from a straightforward computation. Indeed, q0 =w10w2+w1w02,
q00 = (I1+I2)q+2w01w20,
q000=2I0q+ (I1+I2)q0+2(I1w1w02+I2w01w2).
Notice that if a = b (or just Ba = Bb) then I1 = I2 and we stop here to get
the third order ODE as the symmetric product of the Lam´e equation. In general, we take one more differentiation to get
q0000 =2I00q+4I0q0+ (I1+I2)q00+2I0q0+2(I1+I2)w01w02+4I1I2q
=2(I1+I2)q00+6I0q0+ (2I00− (I1−I2)2)q.
This proves the lemma.
Now we investigate the equation in variable x = ℘(z). To avoid confu-sion, we denote ˙f=∂ f/∂x and f0 = ∂ f/∂z.
Let y2 = p(x) = 4x3−g 2x−g3. Then℘0 =y,℘00 = 6℘2− 12g2 = 12˙p(x). ℘000 =12℘℘0 =12xy,℘0000=12℘02+12℘℘00 =12p(x) +6x ˙p(x). Also q0 = ˙q℘0 =y˙q, q00= ¨q℘02+ ˙q℘00= p(x)¨q+ 12˙p(x)˙q, q000=...q℘03+3 ¨q℘0℘00+ ˙q℘000, q0000=....q℘04+6...q℘02℘00+3 ¨q(℘00)2+4 ¨q℘0℘000+ ˙q℘0000 = p(x)2....q +3p(x)˙p(x)...q+ 43 ˙p(x)2+48xp(x) ¨q+ 12p(x) +6x ˙p(x) ˙q. By substituting these into (2.3) and get the ODE in x:
L4q := p2 .... q +3p ˙p...q + 34 ˙p2−2(2(n2+n−12)x+β)p ¨q − (2(n2+n−3)x+β)˙p+6(n2+n−2)p ˙q + α2−n(n+1)˙pq =0. (2.4) where (2.5) α:=Ba−Bb and β:=Ba+Bb.
For the rest of the proof, we want to discuss when L4q=0 with α6=0 has
a polynomial solution. Here g2 and g3 could be arbitrary, not necessarily
satisfy the non-degenerate condition g32−27g23 6=0. Suppose that q(x)is a polynomial in x of degree m ≥1:
q(x) =xm−s1xm−1+s2xm−2− · · · + (−1)msm,
(2.6)
which satisfies
(2.7) degxL4q(x) ≤1.
Then we can solve sjrecursively in terms of α2, β and g2, g3.
Indeed, the top degree xm+2in (2.4) has coefficient
16m(m−1)(m−2)(m−3) +144m(m−1)(m−2) +108m(m−1)
−16(n2+n−12)m(m−1) −24(n2+n−3)m
−24(n2+n−2)m−12n(n+1)
= (m−n)4m3+ (4n+68)m2+ (8n−101)m+3(n+1),
which vanishes precisely when m=n. This we may assume that m=n. The next order term xn+1without the s1factor has coefficient
and the coefficient of−s1xn+1is given by 16(n−1)(n−2)(n−3)(n−4) +144(n−1)(n−2)(n−3) +108(n−1)(n−2) −16(n2+n−12)(n−1)(n−2) −24(n2+n−3)(n−1) −24(n2+n−2)(n−1) −12n(n+1) = −8n(2n−1)(2n+1). Hence (2.8) s1= β 2(2n−1).
Inductively the xn+2−icoefficient in (2.4) gives recursive relations to solve
siin terms of β, α2and g2, g3for i=1, . . . , n. It implies that Lemma 2.4. For i=1, . . . , n, there is a polynomial expression
si =si(α2, β, g2, g3) =Ciβi+ · · ·
which is homogeneous of degree i with deg α = deg β = 1 and deg g2 = 2,
deg g3 =3. Moreover, Ci is a non-zero rational number.
A much detailed description will be given in the proof of Lemma 2.6 and the precise value of Cican be determined (c.f. from (2.11)).
There are still two remaining terms in (2.7), that is, (2.9) L4q= F1(α, β, g2, g3)x+F0(α, β, g2, g3).
The basic structure of the consistency equations is described by the fol-lowing two lemmas:
Lemma 2.5. We have
F1(α, β) =α2G1(α, β) =α2((−1)n−1sn−1(α2, β, g2, g3) + · · · ),
F0(α, β) =α2G0(α, β) =α2((−1)nsn(α2, β, g2, g3) + · · · ).
The remaining terms have either g2or g3as a factor, hence with lower α, β degree.
Proof. Equation (2.9) gives
F1(α, β) = (−1)n−1α2sn−1+terms in s1,· · · , sn−2,
F0(α, β) = (−1)nα2sn+terms in s1,· · · , sn−1.
We note that if α = 0, then for any β there is a solution q(x)to L4(q) = 0
which is a polynomial in x of degree n.
Indeed q(x) = ∏ni=1(x−xi), with β = 2(2n−1)∑ni=1xi, which comes
from the Lam´e equation (see [3, 16]). Thus F1(0, β) =0= F0(0, β). Since Fi
depends on α2, we have Fi(α, β) = α2Gi(α, β), i = 0, 1, for some
homoge-neous polynomials G0, G1in α2, β, g2, g3of degree n and n−1 respectively,
and Gi’s can be written as
G1(α, β) = (−1)n−1sn−1+ · · · ,
To see the dependence of the remaining terms on g2and g3, we let g2 =
0 = g3, and then L4(q) ≡ α2((−1)n−1sn−1x+ (−1)nsn) (mod x2)because
both p(x) = 4x3 and ˙p(x) = 12x2 vanish modulo x2. Thus we have
F1(α, β) = (−1)n−1α2sn−1and F0(α, β) = (−1)nα2snwhenever g2=0= g3.
This proves the lemma.
Lemma 2.6. The polynomials G1and G0have no common factors for any g2, g3.
Proof. We consider first the special case g2= g3=0. Then (2.7) becomes
16x6....q +144x5...q+ 108x4−8x3(2(n2+n−12)x+β) ¨q
− 12x2(2(n2+n−3)x+β) +24x3(n2+n−2) ˙q
+ α2−12n(n+1)x2q≡0 (modC⊕Cx).
(2.10)
The coefficient of xn−k, k=0,· · · , n−2, gives recursive equation
(2.11) (−1)k(mksk+2+nkβ sk+1+α2sk) =0,
where the constants mk and nkare given by
mk =16(n− (k+2))(n− (k+3))(n− (k+4))(n− (k+5)) +144(n− (k+2))(n− (k+3))(n− (k+4)) + (108−16(n2+n−12))(n− (k+2))(n− (k+3)) −24(2n2+2n−5)(n− (k+2)) −12n(n+1) = −4(k+2)(2n− (k+1))(2n− (2k+1))(2n− (2k+3)), nk = (8(n− (k+1))(n− (k+2)) +12(n− (k+1))) =4(n− (k−1))(n− (k+1)).
Since k≤n−2, we have mk 6=0 and nk 6=0.
Let γ(α, β)be a non-trivial common factor of both G1and G0.
In the case g2 = g3 = 0 we have G1 = (−1)n−1sn−1and G0 = (−1)nsn.
Then γ and α are co-prime, because if α = 0 then sn−1(0, β) = cn−1βn−1
and sn(0, β) = cnβn for some non-zero constants cn−1 and cn. By (2.11)
for k = n−2, we have γ | sn−2(α2, β, 0, 0)too. By induction on k for k =
n−3,· · · , 0 in decreasing order we conclude that γ | s0 = 1, which leads
to a contradiction.
For g2, g3 ∈C, we see by Lemma 2.5 that the leading terms of G1, G0, as
polynomials of α and β, are(−1)n−1sn−1(α2, β, 0, 0)and(−1)nsn(α2, β, 0, 0)
respectively. Since sn−1(α2, β, 0, 0)and sn(α2, β, 0, 0)are co-prime, we
con-clude that G1(α, β, g2, g3)and G0(α, β, g2, g3)are also co-prime. The proof
is complete.
Proposition 2.7. The common zeros of G1=0 and G0 =0 are precisely given by
the pair of branch points(a, b)corresponding to Lame functions of the same type. If ¯Xnis non-singular, there are exactly n(n−1)such ordered pairs(a, b)’s.
Proof. It suffices to prove the (generic) case that ¯Xnis non-singular, namely
the case that all the Lam´e functions are distinct. The general case follows from the non-singular case by a limiting argument.
For any two Lam´e functions wa, wbof the same type, it is easy to see that
we may arrange the representatives of a and b so that (2.1) holds. It follows that q :=qa,−b=q−a,b(see (2.2)) is an even elliptic function solution to (2.3),
or equivalently q(x)is a polynomial solution to L4q(x) =0.
From the above discussion,(α, β)must be a common root of G1 and G0
(where α=Ba−Bb, β= Ba+Bb). By Lemma 2.4 and 2.5, we have deg G1 =
n−1 and deg G0 =n and G1, G0are co-prime to each other by Lemma 2.6.
Hence by Bezout theorem there are at most n(n−1)common roots. On the other hand, the number of such ordered pairs can be determined by Proposition 1.6. Indeed, if n=2k is even, then we have
(k+1)k+3k(k−1) =4k2−2k= n(n−1)
such pairs. If n=2k+1 is odd, the number of pairs is given by k(k−1) +3(k+1)k=4k2+2k=n(n−1).
Hence in all cases the number of ordered pairs coming from the Lam´e func-tions of the same type agrees with the Bezout degree of the polynomial sys-tem defined by G1=0=G0. Thus these n(n−1)pairs form the zero locus
as expected (and there is no infinity contribution).
The above discussions from Lemma 2.3 to Proposition 2.7 constitute a complete proof of Theorem 2.2. Here is a summary: We already know that Q is an even elliptic function with singularity only at 0∈E. Thus
Q(x) =C n
∏
i=1 (℘(z) − ℘(ci)) =: C n∏
i=1 (x−xi)is a polynomial solution to the ODE (2.4) with α=Ba−Bb, β= Ba+Bb.
Since α = Ba−Bb 6=0, by Lemma 2.5 (α, β)must be a common root of
G1(α, β) = 0 = G0(α, β). Then Proposition 2.7 says that (α, β)is pair of
Lam´e functions of the same type. This proves Theorem 2.2.
For future reference, we combine Theorem 2.2 and Proposition 2.7 into the following statement on a fourth order ODE which arises from the tensor product of two different (integral) Lam´e equations with the same parameter n.
Due to its importance, we will give a second (shorter and more direct) proof of the part corresponding to Theorem 2.2.
Theorem 2.8. Let I(z) =n(n+1)℘(z). The fourth order ODE
(2.12) q0000(z) −2(I+β)q00(z) −6I0q0(z) + (α2−2I00)q(z) =0
with α6=0 has an elliptic function solution if and only if(α, β)is a pair of common
Second Proof to Theorem 2.2. Following the definition of qa,−b(z)in (2.2), we
now consider the odd elliptic solution to (2.12) (=(2.3)) instead: q(z) = 12(qa,−b(z) −q−a,b(z)),
which has a pole of order 3+2l at 0 ∈ E with l ≤ n−2. Thus q(z)/℘0(z)
is an even elliptic function with the only pole at 0 since q(12ωi) = 0 for
1≤i≤3. If q(z)does not vanish completely, then q(z) =C℘0(z) l
∏
i=1 (℘(z) − ℘(ci)) =: C℘0(z)f(℘(z)), where f(x) =∏li=1(x− ℘(ci)) =xl−s1xl−1+ · · · + (−1)lsl. By Lemma 2.3, q(z)satisfies q0000(z) −2(β+2n(n+1)℘(z))q00(z) −6n(n+1)℘0(z)q0(z) + (α2−2n(n+1)℘00(z))q(z) =0. (2.13)By straightforward calculations, we can compute all derivatives of q in terms of derivatives of℘(z)and f0(x). For example,
q0(z) = ℘00(z)f(x) + ℘0(z)2f0(x), q00(z) = ℘000(z)f(x) +3℘00(z)℘0(z)f0(x) + ℘0(z)3f00(x), etc. Then (2.13) is equivalent to f(x)(360−96n(n+1))x2−24βx+ (4n(n+1) −18)g2+α2 +f0(x)(1320−96n(n+1))x3−36βx2 + (12n(n+1) −150)g2x+ (6n(n+1) −60)g3+3βg2 +f00(x)(1020−16n(n+1))x4−8βx3+ (4n(n+1) −210)g2x2 + (2βg2+ (4n(n+1) −120)g3)x+2βg3+154g22 +f000(x)(60x2−30g2)(4x3−g2x−g3) +f0000(x)(4x3−g2x−g3)2 =0.
By comparing the coefficients of xl+2, we obtain
(360−96n(n+1)) +l(1320−96n(n+1)) +l(l−1)(1020−16n(n+1))
+240l(l−1)(l−2) +16l(l−1)(l−2)(l−3) =0.
After simplification, this is reduced to
4n(n+1) = (2l+3)(2l+5),
which obviously leads to a contradiction since the RHS is odd. Therefore we must have q≡0 from the beginning. That is,{ai,−bi} = {−ai, bi}.
If one of a, b does not correspond to a Lam´e function, say a ∈ Xn, then
{a1,· · · , an} ∩ {−a1,· · · ,−an} = ∅ and we conclude that {ai} = {bi}.
Otherwise a and b correspond to Lam´e functions of the same type.
Theorem 2.9. The map zn : ¯Xn → P1has degree 12n(n+1), with its fibration
structure over∞∈ P1analytically isomorphic to σ : ¯X
n→E over 0.
Proof. By definition, z−1
n (∞) = σ−1(0)as sets. So the crucial point is to
compare the fibration (ramification) structures of ¯Xn → E at 0 ∈ E and
¯
Xn → P1at∞ ∈P1. Let a ∈ X¯nwith σ(a) = 0. Then for b= {bi}ni=1 ∈ X¯n
in a small analytic neighborhood of a we have bi 6=0 all i. Moreover,
σ(b) =b1+ · · ·bn, and zn(b) =ζ(σ(b)) − n
∑
i=1 ζ(bi) = 1 b1+ · · · +bn (1+O(σ(b))).In the local coordinate ofP1at∞, the map znis then represented by
(zn(b))−1= (b1+ · · · +bn)(1+O(σ(b))),
which is clearly analytically equivalent to σ(b).
In particular, deg zn=deg σ= 12n(n+1)by Theorem 2.1.
Example 2.10. For n=2, β= Ba+Bb, α= Ba−Bb, we have
s1= 16β, s2= 361 β2+721α2− 14g2.
The first compatibility equation from x1is s1(α2+36g2) −6βg2 =0.
After substituting s1we get
(2.14) 16α2β=0.
The second compatibility equation from x0is
s2(α2+6g2) −s1(βg2+24g3) +4βg3+32g22=0.
By substituting s1, s2and noticing the (expected) cancellations we get
(2.15) α2(361β2+721α2−16g2) =0.
If Ba 6=Bbthen (2.14) implies that Bb = −Baand then (2.15) leads to
B2a =3g2 =⇒ ℘(a1) + ℘(a2) = ±p g2/3.
By Example 1.7 (2), such a ∈ X¯2 lies in the branch loci of the hyperelliptic (Lam´e) curve. In particular, a, b ∈ σ−1(0) and they are excluded by the
assumption in Proposition 2.2. Denote by ℘(±q±) = ±p g2/12. Then
a := {q+,−q+} 6= b := {q−,−q−}unless g2 = 0. When g2 6=0, z2fails to
distinguish the two points a and b. When g2 = 0 (equivalently τ = eπi/3),
Example 2.11. For n=3, β= Ba+Bb, α= Ba−Bb. Then
s1= 101β,
s2= 6001 (4β2+α2−150g2),
s3= 36001 (2β3+3α2β−120βg2+900g3).
The two compatibility equations from x1and x0are 0= 6001 α2(4β2+α2+60g2),
0= 36001 α2(2β3+3α2β−90βg2+540g3).
If α6=0 then α2= −4β2−60g2and the second equation becomes
β3+27g2β−54g3=0.
It is clear that there are only finite solutions(Ba, Bb)’s to this, though it may
not be so straightforward to see that these 6 solution pairs (for generic tori) come from the branch loci as proved in Proposition 2.7.
We summarize the key points of our discussions:
We start with n = 1, Z1 ≡ Z = −4π∇G which is essentially the Hecke
modular function. The non-trivial solutions, i.e. non half periods, to the critical point equation (⇐⇒ solutions to the mean field equation (1.1) for
ρ=8π) is transformed into zeros of pre-modular form Z (which is modular
for σ∈ E being an N torsion point).
For general n≥1, we consider the following statements: (1) Solution u to the mean field equation (1.1) for ρ=8πn. (2) Periods integral Z Lj g∈ √−1R (c.f. (1.5)). (3) Green equation n
∑
i=1∇G(ai) =0 on the hyperelliptic curve Xn.
(4) Coincidence equation zn(a) =Z(σ(a))for a∈Xn.
(5) Zero of pre-modular form Zn(σ; τ):=Wn(Z)with σ6∈Eτ[2].
We have proved the equivalence of (1), (2), (3) and (4).
As in the case n = 1, we call the 2n+1 finite branch points a ∈ X¯n\Xn
the trivial critical points (since a = −a and the Green equation (3) holds trivially). They satisfy a nice compatibility condition with the case n = 1 under the addition map:
Lemma 2.12. Let a = (a1,· · · , an) ∈ Yn be a solution to the Green equation
∑n
i=1∇G(ai) =0. Then a is trivial, i.e. a= −a, if and only if σ(a) ∈E[2].
Proof. If a is trivial, then σ(a) ∈ E[2]clearly. If a is non-trivial, i.e. a ∈ Xn,
by Proposition 1.1 and (1.5), it gives rise to a type II developing map f with f(z+ω1) =e−4πi∑isif(z), f(z+ω2) =e4πi∑itif(z).
Here ai =tiω1+siω2for i=1, . . . , n.
If σ(a) ∈ E[2], then both exponential factors reduce to one and we con-clude that f(z)is an elliptic function on E. Notice that the only zero of f0(z)
is at z=0 which has order 2n, and the only poles of f0(z)are at−aiof order 2, i=1, . . . , n. This forces that σ(a) ≡0 (mod Λ)and
f0(z) =
∑
nj=1Ej℘(z+aj) +C1
for some constants E1, . . . , Enand C1, since f0is residue free. Then
f(z) = −
∑
nj=1Ejζ(z+ai) +C1z+C2
for some constant C2. But f(z)is elliptic, which implies that C1 = 0 and
∑n
j=1Ej = 0. Now f2k−1(0) =0 for k = 1, . . . , n leads to a system of linear
equations in Ej’s (c.f. [3, Lemma 2.5]):
∑
nj=1℘ k(a
j)Ej =0, k=1, . . . , n.
But then ℘(ai) 6= ℘(aj) for i 6= j forces that Ej = 0 for all j. This is a
contradiction and so we must have σ(a) 6∈ E[2].
Now the following theorem completes the analogy with the case n=1.
Theorem 2.13(Extra critical points vs zeros of pre-modular forms).
(i) Given σ0 ∈ Eτ\Eτ[2]with Zn(σ0; τ) =0, there is a unique a ∈ Xnsuch
that σ(a) =σ0and zn(a) =Z(σ0).
(ii) Conversely, if a ∈ Xnand zn(a) = Z(σ(a)), then Zn(σ(a); τ) = 0 and
σ(a) 6∈Eτ[2].
Proof. (i) For any given σ0, by substituting σ by σ0in Wn(z), we get a
poly-nomial Wn,σ0(z)of degree
1
2n(n+1). Since Wn(z)is the minimal
polyno-mial of the rational function zn∈K(X¯n)over K(E), those zn(a)with a∈X¯n
and σ(a) = σ0give precisely all the roots of Wn,σ0(a), counted with
multi-plicities.
Now Z(σ0)is a root of Wn,σ0(z)with σ0 6∈ E[2], hence there is a point
a ∈ Xn corresponds to it, i.e. Z(σ0) = zn(a) with σ(a) = σ0, which is
unique by Theorem 2.2. Notice that if a ∈ X¯n\Xn then a = −a and then
σ(a) ∈E[2]. So in fact we must have a∈Xn.
(ii) It is clear that Zn(σ(a)) ≡ Wn(Z(σ(a)) = Wn(zn(a)) = 0. Since
a ∈Xn, we know that∑ni=1∇G(ai) =0 by the above equivalence of (4) and
(3). But since a is non-trivial (a ∈ Xnby assumption), Lemma 2.12 implies
that σ(a) 6∈ E[2].
Remark 2.14. For σ = k1
Nω1+ kN2ω2, ZnisΓ(N)modular. This corresponds
to the finite monodromy problem of integral Lam´e equations (c.f. [1, 6], see also §5.2 and Appendix A).
3. EXPLICIT DETERMINATION OFZn
We have studied extensively the Hermite–Halphen ansatz (3.1) wa(z):=ez∑ ζ(ai) n
∏
i=1 σ(z−ai) σ(z)with a∈Ynto represent solutions to the integral Lam´e equation
(3.2) w00 = (n(n+1)℘(z) +B)w.
There is another ansatz, the Hermite–Krichever ansatz, which can also be used to construct solutions to (3.2). It takes the form
(3.3) φ(z):= U(℘(z)) +V(℘(z))℘ 0(z) + ℘0(a 0) ℘(z) − ℘(a0) σ(z−a0) σ(z) e (ζ(a0)+κ)z,
where U(x)and V(x)are polynomials in x, a0 ∈ E×, and κ ∈ C is a
con-stant. As usual, we set(x, y) = (℘(z),℘0(z))and(x0, y0) = (℘(a0),℘0(a0))
to be the corresponding algebraic coordinates.
Notice that (3.3) makes sense since φ only has poles at z = 0 (the one
at z = a0 from (℘(z) − ℘(a0))−1 cancels with the zero from σ(z−a0)).
Moreover, in order for ordz=0φ(z) = −n, we must have
Lemma 3.1(Degree constraints).
(i) If n=2m with m∈ N then deg U≤m−1 and deg V =m−1. (ii) If n=2m+1 with m∈N∪ {0}then deg U =m and deg V ≤m−1. By an obvious normalization, in case (i) we may assume that U(x) =
∑m−1
i=0 uixi, V(x) = ∑ m−1
i=0 vixi with vm−1 = 1, and in case (ii) U(x) =
∑m
i=0uixi with um = 1 and V(x) = ∑mi=−01vixi. In both cases, the
require-ment that φ(z)satisfies (3.2) leads to recursive relations on ui’s and vi’s. In
doing so, it is more convenient to work on the algebraic coordinates. This had been carried out by Maier in [14, §4]. The following is a summary:
In case (i) the recursion determines vi (vm−1 = 1) and then ui for i =
m−1, m−2,· · · in decreasing order. In case (ii) it starts with um = 1 and
determines viand then uifor i=m−1, m−2,· · ·. There are two
compati-bility equations coming from u−1(B, κ, x0, y0) =0 and v−1(B, κ, x0, y0) =0.
The two parameters x0, y0satisfy y20 =4x03−g2x0−g3. Hence there are four
variables (B, κ, x0, y0) ∈ C4 which are subject to three polynomial
equa-tions. By taking in to account the limiting cases with(x0, y0) = (∞, ∞), this
recovers the Lame curve ¯Yn, which was denoted byΓ` in [14] with` =n.
There are four natural coordinate projections (rational functions) ¯Yn → P1, namely B, κ, x
0 and y0respectively. The first one B : ¯Yn → P1 is
sim-ply the hyperelliptic structure map. The main result in [14] is an explicit description of the other 3 maps in terms of the coordinates(B, C)on ¯Yn:
Theorem 3.2([14, Theorem 4.1]). For all n∈N and i∈ {1, 2, 3}, x0(B) =ei+ 4 n2(n+1)2 li(B)lti(B)2 l0(B)lt0(B)2 , y0(B, C) = 16 n3(n+1)3 C cn lt1(B)lt2(B)lt3(B) l0(B)2lt0(B)3 , κ(B, C) = −(n−1)(n+2) n(n+1) C cn lθ(B) l0(B)lt0(B) . (3.4)
The formula for x0(B)is independent of the choices of i.
In the above formulas, ltj(B), j = 0, 1, 2, 3, are the twisted Lam´e
polyno-mials whose zeros correspond to solutions to the Lam´e equation given by the Hermite–Krichever ansatz with κ6=0 and a0=0,12ω1,12ω2,12ω3
respec-tively, i.e.(x0, y0) = (∞, ∞),(e1, 0),(e2, 0),(e3, 0)respectively. The
polyno-mial lθ(B)is the theta-twisted polynomial whose roots correspond to the case
κ = 0 and a0 6∈ E[2]. (For κ = 0 and a0 ∈ E[2]they correspond to the
ordi-nary Lam´e polynomials li(B)’s.) All the polynomials are taken to be monic in B. They are indeed homogeneous by setting the weights of B, ei, g2, g3to
be 1, 1, 2, 3 respectively.
Remark 3.3. In [14] ν=C/cnis used instead. Also l0(B), li(B), lt0(B), lti(B),
and lθ(B) (i = 1, 2, 3) are written there as LI`(B; g2, g3), L`I I(B; ei, g2, g3),
Lt`I(B; g2, g3), LtI I` (B; ei, g2, g3), and Lθ`(B; g2, g3)respectively, where` = n.
The compatibility equations from the recursive formulas for these special cases give rise to explicit formulas for ltj(B)’s and lθ(B)’s. Tables for lt0(B),
lθ(B)up to n =8, and for lti(B)up to n=6, are given in [14, Table 5, 6].
Example 3.4. For later usage, we recall Maier’s formulas for ltj(B)and lθ(B)
for n≤4.
(1) First of all, lθ(B) =1 for n≤3. For n=4,
lθ(B) =B2− 1933 g2.
Also for n=1, ltj(B) =1 for all j.
(2) n=2: lt0(B) =1, lti(B) =B−6eifor i=1, 2, 3.
(3) n=3: lt0(B) =B2− 754g2, and for i=1, 2, 3,
lti(B) =B2−15eiB+ 754g2−225e2i.
(4) n=4: lt0(B) =B3− 3434 g2B−17152 g3. For i=1, 2, 3,
lti(B) =B4−55eiB3+ (5394 g2−945e2i)B2
+ (1960eig2+2450g3)B+61740ei2g2−68600eig3−9261g22.
To apply Theorem 3.4, we need to compare the projection map π : ¯Yn →
E defined by a 7→ a0, or equivalently (℘(a0),℘0(a0)) = (x0(Ba), y0(Ba)),
Theorem 3.5. π(a) =σ(a) =∑in=1ai. Moreover, κ(a) = −zn(a).
Proof. During the proof we view ai ∈C instead of its image[ai] ∈E.
Let a ∈ Yn. The two expressions (3.1) and (3.3), which correspond to
the same solution to the Lam´e equation (3.2), must be proportional to each other by a constant. Hence we get
κ(a) =
n
∑
i=1
ζ(ai) −ζ(a0).
Recall that zn(a) =ζ(σ(a)) −∑ni=1ζ(ai). Then
(3.5) zn(a) +κ(a) =ζ(σ(a)) −ζ(a0).
As a well defined meromorphic function on ¯Yn, we conclude that
a0(a) =σ(a) +c
for some constant c ∈ C. Consider a point a ∈ Yn\Xn with σ(a) = 12ω1,
i.e. l1(Ba) = 0. Such a exists by Proposition 1.6. Then zn(a) = 0 trivially.
We also have κ(a) =0 by Theorem 3.4 since
C2a =c2nl0(Ba)l1(Ba)l2(Ba)l3(Ba) =0
(again by Proposition 1.6). So (3.5) implies 0= 12η1−ζ(12ω1+c), and hence
c =0. This proves σ(a) =a0, which represents π(a)in E, and also κ(a) =
−zn(a). The proof is complete.
Now we may describe the explicit construction of the polynomial Wn(z)
in Theorem 2.1 based on Theorem 3.2. It is indeed merely an application of the elimination theory using resultant.
By Theorem 3.2 and 3.5, we may eliminate C to get (3.6) y0 zn = 16 n2(n+1)2(n−1)(n+2) lt1(B)lt2(B)lt3(B) l0(B)lt0(B)2lθ(B) , which leads to a polynomial equation g=0 for
(3.7) g :=zn 3
∏
i=1 lti(B) −y0 n2(n+1)2(n−1)(n+2) 16 l0(B)lt0(B) 2l θ(B).On the other hand, the 3 rational expressions of x0lead to f =0 for
f :=li(B)lti(B)2− (x0−ei) n2(n+1)2 4 l0(B)lt0(B) 2 = 1 3 3
∑
i=1 li(B)lti(B)2−x0 n2(n+1)2 4 l0(B)lt0(B) 2. (3.8)Notice that f , g are polynomials in g2, g3(and B, x0, y0) instead of ei’s.
Let R(f , g; B) be the resultant of the two polynomials f and g arising from the elimination of the variable B.
Proposition 3.6. R(f , g; B)(z) = λnWn(z) ∈ C[g2, g3, x0, y0][z], where λn =
Proof. From the explicit rational expressions (3.4) in Theorem 3.2, the resul-tant of f , g, defined in (3.8), (3.7) gives rise to the equation of the branched covering map σ : ¯Yn→E over the loci outside E[2]\{0}.
More precisely, if C6=0 then the formulas for y0and κ= −znare
equiv-alent to g = 0. However, if C = 0 we have li(B) = 0 for some 0 ≤ i ≤ 3
by Proposition 1.6. From (3.8), we get 1 ≤ i ≤ 3 in order to have f = 0. Hence l0(B)lt0(B) 6= 0 and then zn = 0 = y0 in (3.4). But the equation
g = 0 in (3.7) gives extra solutions. Indeed by (3.7) we conclude only that y0 = 0 and zn could be arbitrary. In conclusion, the additional solutions
consist of (x0, y0) = (ei, 0), i = 1, 2, 3, with z = zn being arbitrary. Thus
the additional solutions contribute a factor to R(f , g; B)(z)which divides ∏3
i=1(x0−ei) =y20/4 and is independent of z.
In particular R(f , g; B)(z)has degree12n(n+1)in z. Also by construction R(f , g; B)(zn) =0, hence it must be a multiple of the minimal polynomial
Wn(z)of zn. Since deg Wn = 12n(n+1), the full multiplier λn is
indepen-dent of z, and with y2
0|λn.
Proposition 3.7. The pre-modular form Zn(σ; τ) = Wn(Z) can be explicitly
computed for any n∈N.
In practice, such a computation is very time consuming even using com-puter. In the following, we apply it to the initial cases up to n = 4. As before we denote x0 = ℘(σ) =:℘and y0= ℘0(σ) =:℘0.
Example 3.8. For n=2, it is easy to see that
f =B3−9℘B2+27(g2℘ +g3),
g=zB3−9℘0B2−9zg2B+27(g2℘0−2zg3).
The resultant R(f , g; B)is calculated by the 6×6 Sylvester determinant:
1 −9℘ 0 27(g2℘ +g3) 0 0 0 1 −9℘ 0 27(g2℘ +g3) 0 0 0 1 −9℘ 0 27(g2℘ +g3) z −9℘0 −9zg2 27(g2℘0−2zg3) 0 0 0 z −9℘0 −9zg2 27(g2℘0−2zg3) 0 0 0 z −9℘0 −9zg2 27(g2℘0−2zg3) .
A direct evaluation gives
R(f , g; B) = −39∆(℘0)2(z3−3℘z− ℘0).
Here∆ = g32−27g23is the discriminant. This gives W2(z) =z3−3℘z− ℘0
and Z2(σ; τ) =W2(Z) =Z3−3℘Z− ℘0.
Example 3.9. For n=3, we have
f =16B6−576B5℘ +360B4g2+5400B3(5g3+4g2℘)
−3375B2g22−84375∆−101250Bg2(3g3+2g2℘),
g=16B6z−1440B5℘0−1800B4g2z+54000B3(g2℘0−g3z)
