Mean field equations, hyperelliptic curves and modular forms: II

A pre-modular form $Z_n(\sigma; \tau)$ of weight $\tfrac{1}{2} n(n + 1)$ is introduced for each $n \in \Bbb N$, where $(\sigma, \tau) \in \Bbb C \times \Bbb H$, such that for $E_\tau = \Bbb C/(\Bbb Z + \Bbb Z \tau)$, every non-trivial zero of $Z_n(\sigma; \tau)$, namely $\sigma \not\in E_\tau[2]$, corresponds to a (scaling family of) solution to the mean field equation \begin{equation} \tag{MFE} \triangle u + e^u = \rho \, \delta_0 \end{equation} on the flat torus $E_\tau$ with singular strength $\rho = 8\pi n$. In Part I (Cambridge J. Math. 3, 2015), a hyperelliptic curve $\bar X_n(\tau) \subset {\rm Sym}^n E_\tau$, the Lam\'e curve, associated to the MFE was constructed. Our construction of $Z_n(\sigma; \tau)$ relies on a detailed study on the correspondence $\Bbb P^1 \leftarrow \bar X_n(\tau) \to E_\tau$ induced from the hyperelliptic projection and the addition map. As an application of the explicit form of the weight 10 pre-modular form $Z_4(\sigma; \tau)$, a counting formula for Lam\'e equations of degree $n = 4$ with finite monodromy is given in the appendix (by Y.-C. Chou).

With Appendix C written by You-Cheng Chou. 1

INTRODUCTION
Consider the flat torus E = E τ = C/Λ τ , τ = a + bi, b > 0 and Λ = Λ τ = Zω 1 + Zω 2 with ω 1 = 1 and ω 2 = τ. Let G be the Green function on E: where δ 0 is the Dirac measure at the lattice point 0 ∈ E. In this paper, we continue our study initiated in Part I [4] to study the equations (0. 2) n∇G(a i ) = n ∑ j=1, =i ∇G(a i − a j ), i = 1, . . . , n, with unknown a = (a 1 , · · · , a n ), a i ∈ E × = E\{0} and a i = a j if i = j. The main theme is the connection between (0.2) and the following mean field equation: Equation (0.3) is originated from the prescribed curvature problem in conformal geometry. In general, for any compact Riemann surface (M, g) we may consider the following equation: where K(x) is the Gaussian curvature of the given metric g at x ∈ M, Q j ∈ M are distinct points, and α j > −1 are constants. For any solution u(x) to (0.4), equation (0.4) is equivalent to saying that the Gaussian curvature of the new metricg := e 2v g (where 2v = u − log 2) has constant Gaussian curvatureK = 1 outside those Q j 's. Since (0.4) has singular source at Q j , the conformal metric e u g degenerates at Q j and is called a metric on M with conic singularity at those Q j 's. Equation 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 {u k } is a sequence of bubbling solutions to (0.3) corresponding to ρ = ρ k . Then it was proved in [6] by PDE method that ρ k → 8πn with n ∈ N, and the blow-up set {a 1 , · · · , a n } of {u k } 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 {a 1 , · · · , a n }. For (0.3) with ρ = 8πn, it turn out that a sequence of bubbling solutions could be straightforwardly constructed from a non-trivial solution {a 1 , · · · , a n } to (0.2) through the developing map f and the Liouville formula: where f is a meromorphic function on C which has zeros precisely on z ≡ a i (mod Λ), i = 1, . . . , n, and satisfies the type II constraints: The function f could be explicitly written down in terms of {a 1 , · · · , a n }, see e.g. (5.2) in Proposition 5.1. The purpose of this paper is to develop a theory towards understanding the structure of solutions to (0.2) and how it changes under deformations of E τ in the moduli space. Our starting point is the hyperelliptic geometry discovered in Part I [4] 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 discovery of this paper. To be more precise, the main purpose of this paper is to prove that if (0.2) has a non-trivial solution for E = E τ , then this τ must be a zero of a pre-modular form Z n (σ; τ) of weight 1 2 n(n + 1), with σ ∈ E and τ ∈ H = { τ ∈ C | Im τ > 0 }. That is, Z n (σ; τ) is a modular form of weight 1 2 n(n + 1) with respect to Γ(N) whenever σ ∈ E τ [N] (the N-torsion points). Our function Z n (σ; τ) is holomorphic in τ ∈ H and σ ∈ E τ \E τ [2], therefore it also provides a tool to study the corresponding modular form at any N torsion points via complex deformations in σ. In this paper, we construct the pre-modular forms for all n ∈ N, and apply the idea of deformations to provide a complete solution to (0.3) for the case n = 1 (ρ = 8π). Now we give the detailed descriptions. We start by explaining the notion "non-trivial solutions" to (0.2). In [4] we proved the following result: Theorem A. Let a = (a 1 , · · · , a n ) ∈ E n be a solution to (0.2), then either {a 1 , · · · , a n } ∩ {−a 1 , · · · , −a n } = ∅ or {a 1 , · · · , a n } = {−a 1 , · · · , −a n }. Moreover, (0.2) is equivalent to ∇G(a j ) = 0, and the following holomorphic system (0.9) n ∑ j=1, =i (ζ(a i − a j ) + ζ(a j ) − ζ(a i )) = 0, i = 1, · · · , n.
We remark that the Weierstrass ζ function has singularities only at the lattice points and thus (0.9) is meaningful only if a i = 0 for all i and a i = a j for i = j (as points in E). From the system of equations (0.9) we introduce a hyperelliptic curveȲ n as follows. Let Y n = { (a 1 , · · · , a n ) | a i ∈ E × for all i, a i = a j for all i = j, and (a 1 , · · · , a n ) satisfies (0.9) }/S n , (unordered n-tuples), andȲ n be the closure of Y n in Sym n E = E n /S n , the n-th symmetric product of E. Then Y n = Y n ∪ {(0, · · · , 0)} (c.f. [4] for the proof). Define the map B : Y n → C by and let X n = { {a 1 , · · · , a n } ∈ Y n | {a 1 , · · · , a n } ∩ {−a 1 , · · · , −a n } = ∅ }. Then a classical result says that (i) Y n \X n consists of 2n + 1 points (counting with multiplicities), (ii) B| X n : X n → C is a two to one map (onto its image), (iii)X n =Ȳ n is a hyperelliptic curve which can be parametrized by where n (B) is a polynomial in B of degree 2n + 1. The branch points of the hyperelliptic curve is preciselyX n \X n . In particular, the algebraic curveX n is smooth if n (B) has no multiple roots. A proof of the above statements with rigorous details can be found in [4], which is based on a detailed study of the (integral) Lamé equation on E τ (0.10) w = (n(n + 1)℘ + B)w from both the analytic and algebraic point of views. Indeed, let (0.11) w a (z) := e z ∑ ζ(a i ) be the classical Hermite-Halphen ansatz. Then w a and w −a are independent solutions to (0.10) with B = B a if and only if a ∈ X n . HistoricallyX n is also known as the Lamé 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 curvē X n and then study the remaining single equation on Green functions (0.8) for a ∈X n . Notice that, by the anti-symmetry of ∇G, (0.8) holds automatically if a is a branch point ofX n . Hence the system (0.2) contains all branch points ofX n as its solutions. We say that a solution a to (0.2) is non-trivial if a is not a branch point ofX n .
For n = 1,X 1 ∼ = 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 1 2 ω j , 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 has been answered completely in our previous paper [16]: Theorem B. For any τ ∈ H, the Green function G(z; τ) on the flat torus E τ has at most five critical points.
Since G(z) is an even function, the extra critical points appear in pair ±p. What is the nature of those extra critical points? We answer it in the following theorem, which had been announced in [16] ( §1 Theorem A): Theorem 0.1. Suppose that the pair of non half-period critical points {±p} of G exists, the ±p are the minimal points of G.
We will present a proof of it in §1 based on the mean field equation (0. 3). In fact our proof shows that any solution to (0.3) must be a minimizer of the non-linear functional on u ∈ H 1 (E) ∩ {u | E u = 0}. This completely solves the existence problem on minimizers raised in [25] when the two vortex points collapse into one. The next natural question is to characterize those τ such that G(z; τ) admits extra critical points. Let M 1 = SL(2, Z)\H and Ω 5 = { τ ∈ M 1 | G(z; τ) has five critical points }. From the proof of Theorem B in [16], we know that Ω 5 ⊂ M 1 is open. Similarly we define Ω 3 and we have Ω 3 ∪ Ω 5 = M 1 by Theorem B, and hence Ω 3 is a closed set. We would like to understand the geometry of Ω 3 and Ω 5 .
From (0.14), F = γ(F) is also a fundamental domain for Γ 0 (2) for γ(τ) = (τ − 1)/(2τ − 1), and a fundamental domain for Γ 0 (4) can be taken as is a fundamental quantity in the study of Green functions. Strictly speaking it is not a modular form since e 1 is of weight two while η 1 is of weight one. However we have the following results on its zeros: Corollary 0.5. The function (log ϑ) τ ( 1 2 ; τ) has exactly one zero, which is simple, in any fundamental domain of Γ 0 (4). In particular, (log ϑ) τ ( 1 2 ; τ) has a unique zero τ 0 ∈ F ∪ F . It takes the form From the Weierstrass elliptic curve model y 2 = 4x 3 − g 2 x − g 3 of E τ , we know that the half periods E τ [2] are precisely the branch points. A quantity D(q) defined at any branch point is strongly related to the geometry of E τ at q. In [16] we prove that if u k is a bubbling sequence of solutions to (0.3) with ρ = ρ k → 8π (as k → ∞), ρ k = 8π for large k, and with q the blow-up point, then q must be a half period point. In fact, asymptotically (Here h(z) = e −8πG(z) ,G(z, q) is the regular part of the Green function, and φ(q) =G(q, q). See §4 for more details.) D(q) plays an important role in the construction of bubbling solutions to (0.3), as well as other non-linear PDEs, with ρ k → 8π. The sign of D(q) determines the direction where the bubbling may take place, namely ρ k < 8π or ρ k > 8π. If q is a not halfperiod critical point then D(q) is still defined. But then D(q) = 0 since ρ k = 8π for all k (large).
In general it is difficult to compute D(q) for a given torus. Nevertheless we will prove the following result in §4: Theorem 0.6. For any half period q ∈ E τ , τ = a + bi, we have Together with Corollary 0.4 we have D(q) > 0 if q is a saddle point. In particular if τ ∈ Ω 5 then D(q) > 0 for all half-periods. For any τ ∈ H, D(q) ≤ 0 if and only if q is the minimal point.
In particular, the topological Leray-Schauder degree d ρ , which is 2 for ρ ∈ (8π, 16π) [9], does not reflect the actual number of solutions. Since the proof relies also on some techniques of different flavors from the main text of this paper, we choose to present it the appendix section A.
Remark 0.8. In [17] (see also Part I [4]), it was proved that (0.3) with ρ = 12π has exactly two solutions on E τ for τ = e πi/3 . By Theorem 0.7, we see that when τ ∈ Ω 5 the bifurcation diagram of (0.3) is complicate for ρ ranging from 8π to 12π. It is a natural question whether (0.3) has exactly two solutions for ρ ∈ (8π, 16π) when τ ∈ Ω 3 . Theorem 0.7 also reflects the difficulty in the study the corresponding Lamé equation for the case n ∈ 1 2 N. Now we turn to (0.2) and (0.3) for general n > 1. First we notice that a solution to (0.2) is actually a critical point of the Green function (0.17) We would like to extend the results for n = 1 to general n ≥ 2. In particular, we want to understand the non-trivial solutions to (0.2) and to study their behavior under deformations of the moduli parameter τ. A central issue is to study the following Conjecture 0.9. Under the deformations in τ, equation (0.2) has no bifurcations at any non-trivial solution.
To approach the conjecture, we might want to calculate the Hessian of G n at any non-trivial solution a = (a 1 , · · · , a n ). However, the computation for the 2n × 2n matrix D 2 G n (a) is extremely difficult. In fact we do not know if D 2 G n (a) will vanish, a question not even answered for n = 1.
We remark that the conjecture for n = 1 follows from Theorem B or Corollary 0.3. For n ≥ 2, we do not expect an extension of Theorem B to hold. Instead it is Corollary 0.3 which we would like to extend to n ≥ 2. Our approach to Conjecture 0.9 is to construct a pre-modular form Z n (σ; τ) with σ ∈ E τ such that the non-trivial zeros (σ; τ) of Z n (σ; τ) correspond exactly to the non-trivial solutions to (0.2) for E τ .
Starting from §5, our main purpose is to construct the pre-modular form Z n (σ; τ) similar to the Hecke function Z(σ; τ), which is naturally associated to the family of hyperelliptic curvesX n (τ), τ ∈ H. The goal is to show that any non-trivial solution a = (a 1 , · · · , a n ) comes from the zero of Z n (σ; τ) with σ = ∑ n i=1 a i , and vice versa. Consider the Green function equation In §5 we will show that in fact ∑ n i=1 a i ∈ E τ [2] for any a ∈ X n (τ). Hence (0.8) is equivalent to This motivates us to study the map induced from the addition map E n → E. The algebraic curveX n (τ) might be singular for some τ, but it must be irreducible (c.f. Theorem 5.5 (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 It is well known that deg σ = [K(X n ) : K(E)] = number of points for a general fiber.
The major question is to find a natural primitive element of this covering map. Namely a rational function onX n which has W n as its minimal polynomial. This is achieved by the following fundamental theorem: Theorem 0.11. The rational function z n ∈ 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 z n :X n → P 1 also has degree 1 2 n(n + 1), with its fibration structure over ∞ ∈ P 1 analytically isomorphic to σ :X n → E over 0.
The proof is given in §6, Theorem 6.1 and 6.9. A major tool used in the proof is the notion of tensor product of two Lamé equations w = I 1 w and w = I 2 w, where I = n(n + 1)℘(z), I 1 = I + B a and I 2 = I + B b . Indeed, for given τ withX n not necessarily being smooth, and for a given general point σ 0 ∈ E, we need to show that the 1 2 n(n + 1) points on the fiber ofX n → E above σ 0 has distinct z n values. From the definition of z n , it is enough to . If w 1 = I 1 w 1 and w 2 = I 2 w 2 , then the product q = w 1 w 2 satisfies the fourth order ODE (tensor product) given by We remark that if B a = B b , then I 1 = I 2 and q actually satisfies a third order ODE as the second symmetric product of a Lamé equation. This is a useful tool in Part I [4] in the study of the Lamé curve.
If however a = b, by (0.11) and addition law we find that q = w a w −b + w −a w b is an even elliptic function solution to (0.19), namely a polynomial in x = ℘(z). This leads to strong constraints on (0. 19) in variable x and eventually leads to a contradiction for generic choices of σ 0 . Now we set Then Z n (σ; τ) is pre-modular of weight 1 2 n(n + 1), by which we mean that it is modular with respect to Γ(N) whenever σ ∈ E τ [N] being an N-torsion point. From the construction and (0.18) it is readily seen that Z n (σ; τ) is the generalization of the Hecke function Z = Z 1 we are looking for. In fact, for general n ≥ 1, we establish the following correspondences in §6: (1) Solution u to the mean field equation (0.3) for ρ = 8πn.
(3) Green equation ∇G(a i ) = 0 on the hyperelliptic curve X n .
The construction of Z 2 is based on the addition formula, and the construction of Z 3 is based on the n = 2 case and a technical intermediate step: for holds (c.f. Lemma B.3 in Appendix B). By eliminating terms involving a i 's, a degree six polynomial equation W 3 (z 3 ) = 0 is then achieved. The existence of a modular form Z n of weight 1 2 n(n + 1) is also conjectured in [10], though it is unclear how his method would proceed for n ≥ 4.
For all σ and n, the existence of the (pre)-modular forms Z n (σ; τ) now follows from Theorem 0.11 and (0.20). However the explicit construction is still a challenging problem. In §B.2 we explain why no classical formula like (0.21) with degree n can exist for n ≥ 4, and thus it is not possible to derive the explicit expression of Z n (σ; τ) via a naive induction for n ≥ 4. For this purpose, and for the sake of completeness, we present in §B.1 an alternative treatment of Dahmen's construction for n = 2, 3 in a format that is needed for our discussions in §B.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 :X n → E, c.f. Example B.4).
In a different direction, the Lamé 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 := w − 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 Lamé equations (with I(z) = n(n + 1)℘(z)) provide good (indeed earliest) examples of them. Using this theory, Maier [23] had recently written down an explicit map π :X n → E in terms of coordinates (B, C) onX n (in our notations). It turns out we can prove Theorem 0.12 (c.f. Theorem 7.5). The map π agrees with σ :X n → E.
In principle this allows us to compute the polynomial W n (z) for any n ∈ N by eliminating variables (B, C), though in practice the needed calculations are very demanding and time consuming. §7 is devoted to this explicit construction. In particular the weight 10 pre-modular form Z 4 (σ; τ) is explicitly written down by a couple hours Mathematica calculations (c.f. Example 7.10).
The existence and effective construction of Z n (σ; τ) opens the door to extend our results on equations (0.2) and (0.3) for n = 1 (established in [16] and §1- §4 in this paper) to general n ∈ N. As a related example, the explicit expression of Z 4 (σ; τ) can be used to solve Dahmen's conjecture on a counting formula for integral Lamé equations with finite monodromy for n = 4 (c.f. §8.3 and Appendix C). 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 §8. Proof. Consider the even, normalized, L 1 2 Sobolev space and the non-linear functional It is well known that, as a consequence of the Moser-Trudinger inequality, J ρ attains its minimum for ρ < 8π. Let v ρ be a minimizer of J ρ . Then v ρ is an even solution of By the result of [16], when ρ → 8π, v ρ converges to a smooth function v which satisfies It is then obvious that is an even solution to the Liouville equation Let f be the developing map of u, that is, As before, for λ ∈ R we define u λ and v λ by where the constant c λ is chosen so that E v λ = 0. Thus v λ is also a solution to (1.1) and v λ (z) blows up at z = p as λ → +∞ (i.e. p is a zero of f ).
Next we would like to compute J 8π (v λ ). By differentiation with respect to λ, we have by (1.1) Using (1.3), we shall obtain an upper bound of lim J 8π (v λ ) by a choice of suitable test function ϕ . We fix a half period point q ∈ E and small δ > 0. For any > 0 we define is the regular part of G(z, q) which is defined on z ∈ T(q), the fundamental domain of E centered at q. Notice that the above two expressions for ϕ (z) coincide when |z − q| = δ. SinceG(z, q) depends only on w = z − q, we also denoteG(z, q) =G(z − q) =G(w), which is defined on the fundamental domain T(0) centered at 0. Obviously ϕ is an even function. Since E G = 0, direct integration gives where the notation O is with respect to the limit → 0. Thus ϕ − c ∈ H 1 ev (E) and We will estimate the energy term and the non-linear term separately.
By Green's theorem, we have for w = z − q, To estimate these terms, we first notice that (for δ > 0 fixed) (1.5) Since G = δ 0 − 1/|E|, G = −1/|E|, and E G = 0, it is easy to see that each of three integrals involving G orG is O(δ) and all boundary terms are where γ =G(0) =G(q, q) is a constant independent of q.
Next we compute the non-linear term.
Since both ∇G(q) = 0 and ∇G(z, q)| z=q = ∇G(0) = 0, we have On E\B δ (q), by (1.7) and direct estimate we have where O(1) denotes a bounded number which is independent of δ and . By taking into account of (1.5)-(1.9), we get for 0 < << δ From (1.2), u λ blows up at p as λ → +∞. By using the explicit expression (1.2), a similar calculation as the above shows that Therefore (1.10) implies which finishes the proof. Proof. Since the extra critical point p (reps. −p) is a discrete minimal point, the index of ∇G at p (reps. −p) is 1. By the Hopf-Poincaré index theorem, Since 1 2 ω i is non-degenerate, ∇G has index ±1 at it. Hence the index must be −1 for all i = 1, 2, 3. This implies that 1 2 ω i is a saddle point for all i.
We remark that in §3 we will further prove det D 2 G( 1 2 ω i ) < 0.

FUNDAMENTAL CORRESPONDENCE
In this and the next sections, our main purpose is to prove Theorem 0.2. Recall the function which is doubly periodic in (t, s) ∈ R 2 . It is related to the Green function on E τ via Z t,s (τ) = −4π∂ z G(t + sτ; τ).
Recall also the q expansion for ζ with q := e 2πiτ : (2.1) (This can be deduced form the Jacobi triple product formula for theta function ϑ and the relation between ϑ and σ, see e.g. [26]. ) We use the Legendre relation η 1 τ − η 2 = 2πi and the above q expansion to compute the q expansion for Z: For fixed s ∈ [0, 1), (2.2) then implies that By the periodicity, the limit is a discontinuous linear function with discontinuity at s ∈ Z.
It is also easy to see that under the translation τ → Tτ := τ + 1, and for t + s ∈ (0, 1), as τ → 1. For t + s = 0, the dominant term is replaced by −π cot π(t + s)/(τ − 1). For general t + s, the value is again determined by periodicity. Let F ⊂ H be the fundamental domain for Γ 0 (2) defined by It is well known that the modular curve X 0 (N) = H/Γ 0 (N) parametrizes the pair (E, C) of an elliptic curve E together with a cyclic subgroup C ⊂ E with |C| = N. For N = p being a prime, [SL(2, Z) : Γ 0 (p)] = p + 1 and a fundamental domain for Γ 0 (p) is given by where F 0 is any fundamental domain for SL(2, Z). For N = p = 2, X 0 (2) parametrizes (E, q) with q a half period. An alternative choice of fundamental domain is F = F 0 ∪ TS(F 0 ) ∪ (TS) 2 (F 0 ) (notice that (TS) 3 = −Id and TS fixes ρ = e πi/3 ).
We will analyze the structure of the solutions τ ∈ F for Z t,s (τ) = 0 by varying (t, s). Since half periods are trivial solutions for all τ, we exclude those cases by assuming that t, s are not half integers in our discussion.
Recall that for τ ∈ ∂F ∩ H, E τ is conformally equivalent to rectangular tori and the only critical points z = t + sτ ∈ E τ of the Green function are given by the half periods. Thus if both t and s are not half integers, then Z t,s (τ) = 0 for τ ∈ ∂F ∩ H. Based on this, the idea of our analysis is to make use of the argument principle along the curve ∂F to analyze the number of zeros of Z t,s in F.
Proof. We separate the proof into three steps.
Thus an extended version of the argument principle shows that the number of zero of Z t,s (τ) is constant in the region Similarly Z t,s (τ) has no solutions for (t, s) ∈ , where := {(t, s) | 1 2 < t < 1 and 0 < s < 1 2 }. This follows from Lemma (2.2) (i) and the fact that ( 3 4 , 1 4 ) ∈ .
Step 2. Z t,s (τ) has no solutions if (t, s) ∈ . Indeed, it follows easily form the argument principle in complex analysis that the points (t, s) such that Z t,s (τ) has only finite solutions form an open set. In particular, by Step 1, for (t, s) ∈ 1 ∪ , the function Z t,s (τ) either has no solutions or has infinite solutions (which corresponds to the trivial case t, s ∈ 1 2 Z and Z t,s ≡ 0).
Step 3. In order to conclude the proof of the theorem, by the same reasoning as in Step 1 we only need to establish the existence and uniqueness of solution Z t,s (τ) = 0 in τ ∈ F for one special point (t, s) ∈ . For this purpose we take (t, s) = ( 1 3 , 1 3 ) ∈ . By an easy symmetry argument (c.f. [16]), Z1 3 , 1 3 (τ) = 0 for τ = ρ := e πi/3 . Conversely we will prove that ρ ∈ F is the unique zero of Z1 3 , 1 3 and it is a simple zero. The following argument motivated by [13,2] is the only place where the theory of modular forms is used. Define where the product is over all pairs (k 1 , k 2 ) with 0 ≤ k 1 , k 2 ≤ 2 and with gcd(k 1 , k 2 , 3) = 1. In this case it simply means (k 1 , k 2 ) = (0, 0). There are 8 factors in the product and in fact Z (3) is a modular function of weight 8 with respect to the full modular group SL(2, Z). The counting formula for the zeros of Z (3) then reads as Since Z1 The counting formula then implies that ν ρ (Z (3) ) = 2 and all the other terms vanish. Hence τ = ρ is a simple (and unique) zero for Z1 Proof. LetΩ 5 be the lifting of Ω 5 in F. The theorem establishes a continuous map φ : (t, s) → τ from ontoΩ 5 . The map φ is one to one due to the uniqueness theorem of extra pair (non half-period points) of critical points of the Green function G proved in [16]. Being the continuous image of a simply connected domain under a one to one continuous function φ on R 2 ,Ω 3 must also be a simply connected domain. (This is the classic result on "Invariance of Domain" proved in algebraic topology. In the current case it follows easily from the inverse function theorem since φ is differentiable.) It is also proven in [16] that the domainΩ 5 contains the vertical line hence it is unbounded. The corresponding statement for Ω 5 follows from the obvious Z 3 identification.

GEOMETRY OF Ω 5
Even thoughΩ 5 , the lifting of Ω 5 in F, is a simply connected domain, its boundary may still possibly be ill-behaved. The purpose in this section is to show that this is not the case. • The general case uses modular forms of weight one. For i = 1, 2, 3 we put It is known that all the half period points 1 2 ω i 's are non-degenerate critical points of G(z; τ) if τ ∈ ∂F . Hence C i (F) ∩ ∂F = ∅ for all i. When no confusion may possibly arise, we will drop the dependence on F and simply write C i .
The main result of this section is We first derive the equation for C i , and then extend the discussion in [16] for rhombus tori to the general cases. To compute the Hessian of G(z; τ) at 1 2 ω i , we recall that for τ = a + bi, z = x + iy, where ϑ denotes the theta function ϑ 1 . Then 2) and the Hessian H is given by The relation to the Weierstrass elliptic functions is linked by . For our discussion, using the SL(2, Z) action we only need to work on the case i = 1. At the critical point z = 1 2 , we have clearly (by (3.1) and (3.4)) (3.6) (log ϑ) z ( 1 2 ; τ) = 0. Recall the heat equation for theta function It allows us to transform the Hessian into deformations in τ. Then That is, the curve C i is the inverse image of the unit circle centered at τ = 1 under the analytic (but not holomorphic) map F → C: To proceed, we need to calculate (log ϑ) ττ at z = 1 2 . By (3.7), (3.6) and .
In particular C 1 is smooth near such τ. .
We compute we conclude again that C 1 is smooth near such τ.
Hence C i are smooth curves for i = 1, 2, 3.
To characterize ∂Ω 5 , we first show that C i ∩Ω 5 = ∅. If not, say C i ∩Ω 5 is a (not necessarily conneted) smooth curve in the open setΩ 5 Thus H(τ) ≤ 0 for all τ ∈ U., this is equivalent to that Im f ≥ 0 over U. But Im f is a harmonic function on U and Im f (τ 0 ) = 0, the maximal principle implies that Im f (τ) ≡ 0 on U and f (τ) is a constant, which leads to a contradiction. Thus C i ∩Ω 5 = ∅ for all i.
Similar argument applies to the open setΩ • 3 , the interior ofΩ 3 , where z = 1 2 is known to be a minimal point and H ≥ 0 (c.f. [16]). Hence we have proved the following result: In particular, for τ ∈Ω 5 ∪Ω • 3 , all the half period points are non-degenerate critical points.
Proof of Theorem 3.1. It remains to show that C i is connected for each i. Since ∂Ω 5 = 3 i=1 C i andΩ 5 is simply connected, C i can not bound any bounded domain. (We note that this can not be proved by the maximal principle as we have done above since f might have singularities on the boundary of this bounded domain. Instead, the contradiction is draw from the unboundedness and simply connectedness ofΩ 5 .) It thus suffices to show that at each cusp (i.e. 0, 1, ∞), C 1 has at most one component near a neighborhood of them.
It is known that as τ → +∞, It is readily checked that the Hessian of G satisfies . Therefore the degeneracy curve C 1 is mapped to the degeneracy curve C 3 and it suffices to show that C 3 ∩ {τ ∈ F | Im τ ≥ R} is a smooth curve for large R.
This implies that near ∞ the curve C 3 is (smooth and) connected. The proof is complete.
In the remainder of this section, we will determine the location of zeros of (log ϑ) τ then F is the domain bounded by 3 half circles: to the real axis. By Lemma 6.1 in [16], 2 ) is the unique zero of the increasing function in b is the unique zero of the increasing function e 1 + η 1 along In particular, maps Ω to the lower half plan {κ | Im κ < 0} in a locally one-to-one manner. Then f is actually one to one overΩ onto respectively. Then f (τ) = ∞ has only one solution τ = 1 2 + ib 0 . Therefore we have proved the following theorem: in E τ and ρ k → 8π. Suppose that ρ k = 8π. In [4], it is proved that u k blow ups at a half period q. Let We recall a result in [9]: The quantity D(q) is well defined for any critical point of G(z, q). However, if q is not a half period then D(q) = 0 since such a blowup can only occur for ρ k = 8π. When q is a half period, D(q) has a geometric interpretation. Indeed, Note that 8π(G(z − q) − G(z)) is a doubly periodic harmonic function in R 2 with singularities −4 log |z − q| at z = q and 4 log |z| at z = 0. Thus and Denote by Λ + (q) be the union of components bounded by γ and covered by T under Σ, and by Λ − (q) the union of components bounded by γ but not covered by T under Σ. Then obviously We will give another characterization of D(q) in terms of the Hessian of G at q, hence establish a correspondence between the geometric interpretation and the degeneracy structure of the Green function: Proof. Without loss of generality, we assume that q = 1 2 ω 1 = 1 2 and denote Λ + (q) and Λ − (q) by Λ + and Λ − respectively. By (4.2), we have To compute the first term, write the Weierstrass zeta function as ζ = u + iv and then ℘ = −ζ = −u x − iv x = −u x + iu y . Hence Using integration by parts, and noticing that the singularity at z = 0 is cancelled out by the second integral, the first limit term then becomes This can be calculated easily as Applying the Legendre relation η 2 = η 1 τ − 2πi, we get Consequently, For the second limit term, we first compute Putting everything together we get (c.f. (3.3) and (3.5)) The proof is completed. (1) q is a half period and a saddle point of G(z; τ) if and only if ρ k > 8π.
(2) q is a half period and a minimal point of G(z; τ) if and only if ρ k < 8π.

GEOMETRY ON
We would like to extend some of the previous discussions for ρ = 8π to ρ = 8nπ for all n ∈ N. Especially we will generalize Hecke's function Z to Z n whose critical point set again encodes the structure of solutions to the corresponding mean field equation We first recall the structural results from [4].
(1) If solutions exist for ρ = 8nπ, then there is a unique even solution within each type II scaling family.
The solution u is determined by the zeros a 1 , · · · , a n of its developing map f (through expression (0.6) with λ = 0). In fact for g = f / f , subject to the non-degenerate conditions a i ∈ E[2], a i = ±a j for i = j.
The condition ord z=0 g(z) = 2n leads to n − 1 equations for a 1 , . . . , a n : Under the notations (w, x j , y j ) = (℘(z), ℘(p j ), ℘ (p j )), 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 This gives the polynomial system describing the developing maps. Notice that the condition y i = 0 follows automatically.
Indeed the Green function equation (5.4) is equivalent to the type II condition (0.7). From Proposition 5.1 (2), this means that the periods integral L i Ω ∈ √ −1R for L 1 , L 2 being the two fundamental periods of E. Let f be given by (5.2). By evaluating the periods integrals (c.f. Part I [4, (5.4) and (5.7)]), we have where a i = t i ω 1 + s i ω 2 for i = 1, . . . , n. Then by the Liouville theorem, we obtain solutions u λ to (5.1) by (0.6).
Note that the algebraic conditions in (5.3) are equivalent to (See Part I [4, Theorem 0.6].) By the addition formula for ℘, (5.6) is identical with (0.9) provided that a i = ±a j for i = j, or equivalently ℘(a i ) = ℘(a j ) for i = j. Thus, a scaling family of solutions to (5.1) corresponds exactly to a non-trivial solution to (0.2). defines a non-singular open algebraic curve X n ⊂ Sym n E, called the Liouville curve. It has a hyperelliptic structure under the 2 to 1 map a → ∑ ℘(a i ), or algebraically: This is closely related to (integral) Lamé equations: Write is the Hermite-Halphen ansatz for solutions to integral Lamé equations (see [26,4]).
Proposition 5.4 (Explicit map a → B a [4]). a ∈ X n if and only if w a and w −a are independent solutions of the Lamé equation . The full information on the compactified hyperelliptic curveX n → P 1 , especially on the branch points added, is described by Theorem 5.5 (Hyperelliptic geometry onX n , the Lamé curve [4]).
(1) The natural compactificationX n ⊂ Sym n E coincides with the, possibly singular, projective model of the hyperelliptic curve defined by , is a recursively defined polynomial of homogeneous degree k with deg g 2 = 2, deg g 3 = 3, and B = (2n − 1)s 1 .
(3) The curveX n is smooth except for a finite number of τ, namely the discriminant loci of n (B, g 2 , g 3 ) so that n (B) has multiple roots.X n is an irreducible curve which is smooth at infinity. (4) The 2n + 2 branch points a ∈X n \X n are characterized by −a = a. In fact {−a i } ∩ {a i } = ∅ ⇒ −a = a. Also 0 ∈ {a i } ⇒ a = (0, 0, · · · , 0). (5) The limiting system of (5.7) at a = 0 n is given by  Moreover, the system (5.11) and (5.12) has a unique solution in P n−1 up to permutations.
The point a = 0 n ∈X n is referred as the point at infinity. For the other 2n + 1 finite branch points with a = −a, w a = w −a which is still a solution to the Lamé equation. In the literature, these 2n + 1 functions are known as the Lamé functions. Indeed, there are four species of them, depending on the number of half periods contained in {a i }. 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: (2) For n = 2k + 1, l 0 (B) consists of type III roots with deg l 0 (B) = 1 2 (n − 1) = k. For i = 1, 2, 3, l i (B) consists of type I roots a which contains 1 2 ω i . Moreover, deg l i (B) = 1 2 (n + 1) = k + 1. We remark that both Proposition 5.6 and Theorem 5.5 [5] will be used in the proof of Theorem 0.10 (= Theorem 5.10 later in this section). Here are some examples to illustrate Proposition 5.6: (2) n = 2, k = 1, (notice that e 1 + e 2 + e 3 = 0) (3) n = 3, k = 1, deg l i (B) = 2 for i = 1, 2, 3, .
Now we study the last equation (5.4) in Theorem 5.2, namely the Green function equation onX n : ∇G(a i ) = 0.

Definition 5.8 (Fundamental rational function). Consider the function on E n :
z n (a 1 , . . . , a n ) := ζ(a 1 + · · · + a n ) − z n is a rational function on E n since it is meromorphic and periodic in each variable a i , and E n is an abelian variety.
It satisfies the reduction property along half periods: If a = a ∪ { 1 2 ω i } then z n (a) = z n−1 (a ).
Hence the Green equation (5.13) is equivalent to (5.14) This motivates us to study the branched covering map induced from the addition map E n → E. Now we may move to the discussions concerning new modular functions. We start by determining deg σ. A standard reference is [14, II.6, Proposition 6.9], where nonsingular curves are treated. The irreducible case is reduced to the nonsingular case through normalizationsX → X andỸ → Y, since it is clear that the induced finite morphismf :X →Ỹ has the same degree as f . Furthermore, the definition also extends to the case f : X → Y where X = k i=1 X i has a finite number of irreducible components. We require that f | X i is a finite morphism for each i and then deg f := ∑ k i=1 deg f | X i . Since all curves considered here are proper (projective), it is enough to require f | X i to be non-constant to ensure that it is a finite morphism.
Theorem 5.10. The map σ :X n → E has degree 1 2 n(n + 1). Proof. The idea is to apply Theorem of the Cube [24, p.58, Corollary 2] for morphisms 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 We will apply it to the algebraic curve V = V n ⊂ E n which consists of the ordered n-tuples a's so that V n /S n =X n .
Take L be any line bundle with deg L = 0. Using the fact that deg F * L = deg F deg L for any finite morphism F : V → E, (5.16) implies that We prove inductively that for j = 1, . . . , n the branched covering map f j : V n → E defined by f j (a) := a 1 + · · · + a j has degree 1 2 j(j + 1)n!. The case j = n then gives the result since f n descends to σ under the S n action. (Notice that the map f j can not descend to a map onX n for all j < n.) Since the curve V n is not necessarily irreducible, we need to make sure that f j is non-constant on each irreducible component to guarantee that f j is 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) = f j−1 (a), g(a) = a j , and h(a) = a j+1 . Then by (5.17), f j+1 has degree n! times It remains to investigate the case j = 1 and j = 2. For j = 1, by Theorem 5.5 (4), the inverse image of 0 ∈ E under f 1 : V n → E consists of a single point 0. By Theorem 5.5 (5), the limiting system of equations (5.11) and (5.12) has a unique non-degenerate solution in P n−1 up to permutations. From this, we conclude that there are precisely n! branches of V n → E near 0. For a point b ∈ E close to 0, each branch will contribute a point a with a 1 = b. Thus the degree of f 1 is n!.
For j = 2, we consider the inverse image of 0 ∈ E under f 2 : V n → E. Namely V n a → a 1 + a 2 = 0.
The point a = 0 again contributes degree n! by a similar branch argument: Indeed, over each branch near 0 we may represent a = (a i (t)) as an analytic curve in t. Then t → a 1 (t) + a 2 (t) ∈ E is still an one to one map for t close to 0. It is also clear that the same reasoning applies to any f k : V n → E and the point 0 always contributes n! to the degree counting. Notice that f k is non-constant on each branch near 0, and every irreducible component of V n will contain at least one such branch, hence f k is non-constant along each irreducible component of V n .
Thus in both cases we get the total degree n! + 2n! = 3n! as expected.
During the proof we have actually shown that Proposition 5.11. The map σ is unramified at the infinity point 0 n ∈X n .
We want to emphasize again that both Theorem 5.10 and Proposition 5.11 hold for all E τ , τ ∈ H, regardless the smoothness ofX n .

PRE-MODULAR FORMS Z n (σ; τ)
When no confusion should arise, we denote the restriction z n |X n also by z n . Then z n is a rational function onX n with poles along the fiber σ −1 (0) under the branch covering (summation) map σ :X n → E. To avoid trivial situation we assume that n ≥ 2 (since z 1 = 0 by definition.) Theorem 5.10 strongly suggests the following statement: Theorem 6.1 (New pre-modular forms).
Indeed, no matterX n is smooth or not, z n (a) is a primitive generator of the finite extension of rational function fields K(X n ) over K(E), which has degree 1 2 n(n + 1), and with W n (z) being its minimal polynomial.
Proof. Since z ∈ K(X n ), which is algebraic over K(E) with degree 1 2 n(n + 1) by Theorem 5.10, its minimal polynomial W n (z) ∈ K(E)[z] certainly exists with d := deg W n | 1 2 n(n + 1). Furthermore, since z has no poles over E × , it is indeed integral over the affine Weierstrass model of E × where with x = ℘(σ) and y = ℘ (σ). Thus the major statement in Theorem 6.1 (1) is the claim that z n is essentially a primitive generator.
Notice that for σ 0 ∈ E being outside the branch loci of σ :X n → E, there are precisely 1 2 n(n + 1) different points a = {a 1 , · · · , a n } ∈X n with σ(a) = ∑ a i = σ 0 . Thus for the rational function z n = ζ(∑ a i ) − ∑ ζ(a i ) ∈ K(X n ) to be a primitive generator, it is sufficient to show that z n has exactly 1 2 n(n + 1) branches over K(E). That is, ∑ ζ(a i ) gives different values for different choices of those a above σ 0 . Indeed, for any given σ = σ 0 , the polynomial W n (z) = 0 has at most d roots. But now z n (a) with σ(a) = σ 0 gives 1 2 n(n + 1) distinct roots of W n (z), hence we must conclude d = 1 2 n(n + 1) and z n is a primitive generator.
Hence it is sufficient to show the following more precise result: Theorem 6.2. Let a, b ∈ Y n and (a 1 , · · · , a n ), (b 1 , · · · , b n ) ∈ C n be representatives of a, b such that Suppose that ∑ ℘(a i ) = ∑ ℘(b i ). Then a, b are branch points of Y n → P 1 corresponding to Lamé functions of the same type.
We will give two proofs of Theorem 6.2. The first proof is longer but contains more informations.

Recall that the Hermite-Halphen ansatz
are solutions to w = (n(n + 1)℘(z) + B a )w =: I 1 w, and are solutions to w = (n(n + 1)℘(z) + B b )w =: I 2 w. Then q a,−b := w a w −b and q −a,b := w −a w b are solutions to the fourth order ODE formed by the tensor product of the two Lamé equations. By assumption.
In particular there exists an even elliptic function solution Lemma 6.3. The fourth order ODE is given by Here I = n(n + 1)℘(z), I 1 = I + B a and I 2 = I + B b .
Notice that if a = b (or just B a = B b ) then I 1 = I 2 and we stop here to get the third order ODE as the symmetric product of the Lamé equation.
This proves the lemma.
Now we investigate the equation in variable x = ℘(z). To avoid confusion, we denoteḟ = ∂ f /∂x and f = ∂ f /∂z.
By substituting these into (6.3) and get the ODE in x: For the rest of the proof, we want to discuss when L 4 q = 0 with α = 0 has a polynomial solution. Here g 2 and g 3 could be arbitrary, not necessarily satisfy the non-degenerate condition g 3 2 − 27g 2 3 = 0. Suppose that q(x) is a polynomial in x of degree m ≥ 1: q(x) = x m − s 1 x m−1 + s 2 x m−2 − · · · + (−1) m s m , (6.6) which satisfies (6.7) deg x L 4 q(x) ≤ 1.
Then we can solve s j recursively in terms of α 2 , β and g 2 , g 3 . Indeed, the top degree x m+2 in (6.4) has coefficient which vanishes precisely when m = n. This we may assume that m = n. The next order term x n+1 without the s 1 factor has coefficient −8n(n − 1)β − 12nβ = −4n(2n + 1)β, and the coefficient of −s 1 x n+1 is given by Hence (6.8) .
Inductively the x n+2−i coefficient in (6.4) gives recursive relations to solve s i in terms of β, α 2 and g 2 , g 3 for i = 1, . . . , n. It implies that Lemma 6.4. For i = 1, . . . , n, there is a polynomial expression which is homogeneous of degree i with deg α = deg β = 1 and deg g 2 = 2, deg g 3 = 3. Moreover, C i is a non-zero rational number.
A much detailed description will be given in the proof of Lemma 6.6 and the precise value of C i can be determined (c.f. from (6.11)).
The remaining terms have either g 2 or g 3 as a factor, hence with lower α, β degree.
We note that if α = 0, then for any β there is a solution q(x) to L 4 (q) = 0 which is a polynomial in x of degree n.
Proof. It suffices to prove the (generic) case thatX n is non-singular, namely the case that all the Lamé functions are distinct. The general case follows from the non-singular case by a limiting argument.
For any two Lamé functions w a , w b of the same type, it is easy to see that we may arrange the representatives of a and b so that (6.1) holds. It follows that q := q a,−b = q −a,b (see (6.2)) is an even elliptic function solution to (6.3), or equivalently q(x) is a polynomial solution to L 4 q(x) = 0.
From the above discussion, (α, β) must be a common root of G 1 and G 0 (where α = B a − B b , β = B a + B b ). By Lemma 6.4 and 6.5, we have deg G 1 = n − 1 and deg G 0 = n and G 1 , G 0 are co-prime to each other by Lemma 6.6. 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 5.6. Indeed, if n = 2k is even, then we have such pairs. If n = 2k + 1 is odd, the number of pairs is given by Hence in all cases the number of ordered pairs coming from the Lamé functions of the same type agrees with the Bezout degree of the polynomial system defined by G 1 = 0 = G 0 . Thus these n(n − 1) pairs form the zero locus as expected (and there is no infinity contribution).
The above discussions from Lemma 6.3 to Proposition 6.7 constitute a complete proof of Theorem 6.2. Here is a summary: We already know that Q is an even elliptic function with singularity only at 0 ∈ E. Thus Since α = B a − B b = 0, by Lemma 6.5 (α, β) must be a common root of G 1 (α, β) = 0 = G 0 (α, β). Then Proposition 6.7 says that (α, β) is pair of Lamé functions of the same type. This proves Theorem 6.2.
For future reference, we combine Theorem 6.2 and Proposition 6.7 into the following statement on a fourth order ODE which arises from the tensor product of two different (integral) Lamé 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 6.2. Theorem 6.8. Let I(z) = n(n + 1)℘(z). The fourth order ODE with α = 0 has an elliptic function solution if and only if (α, β) is a pair of common root to G 0 (α, β) = 0 and G 1 (α, β) = 0. Moreover, this solution must be even.
Proof. By definition, z −1 n (∞) = σ −1 (0) as sets. So the crucial point is to compare the fibration (ramification) structures ofX n → E at 0 ∈ E and X n → P 1 at ∞ ∈ P 1 . Let a ∈X n with σ(a) = 0. Then for b = {b i } n i=1 ∈X n in a small analytic neighborhood of a we have b i = 0 all i. Moreover, In the local coordinate of P 1 at ∞, the map z n is then represented by which is clearly analytically equivalent to σ(b).
We summarize the key points of our discussions: We start with n = 1, Z 1 ≡ 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 (5.1) 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 (5.1) for ρ = 8πn.
(3) Green equation ∇G(a i ) = 0 on the hyperelliptic curve X n .
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 \X n 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 6.12. Let a = (a 1 , · · · , a n ) ∈ Y n be a solution to the Green equation ∑ n i=1 ∇G(a i ) = 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 ∈ X n , by Proposition 5.1 and (5.5), it gives rise to a type II developing map f with Here a i = t i ω 1 + s i ω 2 for i = 1, . . . , n.
If σ(a) ∈ E [2], then both exponential factors reduce to one and we conclude that f (z) is an elliptic function on E. Notice that the only zero of f (z) is at z = 0 which has order 2n, and the only poles of f (z) are at −a i of order 2, i = 1, . . . , n. This forces that σ(a) ≡ 0 (mod Λ) and for some constants E 1 , . . . , E n and C 1 , since f is residue free. Then But then ℘(a i ) = ℘(a j ) for i = j forces that E j = 0 for all j. This is a contradiction and so we must have σ(a) ∈ E [2]. Now the following theorem completes the analogy with the case n = 1.
Theorem 6.13 (Extra critical points vs zeros of pre-modular forms).
Proof. (i) For any given σ 0 , by substituting σ by σ 0 in W n (z), we get a polynomial W n,σ 0 (z) of degree 1 2 n(n + 1). Since W n (z) is the minimal polynomial of the rational function z n ∈ K(X n ) over K(E), those z n (a) with a ∈X n and σ(a) = σ 0 give precisely all the roots of W n,σ 0 (a), counted with multiplicities.
, hence there is a point a ∈ X n corresponds to it, i.e. Z(σ 0 ) = z n (a) with σ(a) = σ 0 , which is unique by Theorem 6.2. Notice that if a ∈X n \X n then a = −a and then σ(a) ∈ E [2]. So in fact we must have a ∈ X n .

EXPLICIT DETERMINATION OF Z n
We have studied extensively the Hermite-Halphen ansatz with a ∈ Y n to represent solutions to the integral Lamé equation There is another ansatz, the Hermite-Krichever ansatz, which can also be used to construct solutions to (7.2). It takes the form where U(x) and V(x) are polynomials in x, a 0 ∈ E × , and κ ∈ C is a constant. As usual, we set (x, y) = (℘(z), ℘ (z)) and (x 0 , y 0 ) = (℘(a 0 ), ℘ (a 0 )) to be the corresponding algebraic coordinates. Notice that (7.3) makes sense since φ only has poles at z = 0 (the one at z = a 0 from (℘(z) − ℘(a 0 )) −1 cancels with the zero from σ(z − a 0 )). Moreover, in order for ord z=0 φ(z) = −n, we must have By an obvious normalization, in case (i) we may assume that In both cases, the requirement that φ(z) satisfies (7.2) leads to recursive relations on u i 's and v i 's. In doing so, it is more convenient to work on the algebraic coordinates. This had been carried out by Maier in [23, §4]. The following is a summary: In case (i) the recursion determines v i (v m−1 = 1) and then u i for i = m − 1, m − 2, · · · in decreasing order. In case (ii) it starts with u m = 1 and determines v i and then u i for i = m − 1, m − 2, · · · . There are two compatibility equations coming from u −1 (B, κ, x 0 , y 0 ) = 0 and v −1 (B, κ, x 0 , y 0 ) = 0. The two parameters x 0 , y 0 satisfy y 2 0 = 4x 3 0 − g 2 x 0 − g 3 . Hence there are four variables (B, κ, x 0 , y 0 ) ∈ C 4 which are subject to three polynomial equations. By taking in to account the limiting cases with (x 0 , y 0 ) = (∞, ∞), this recovers the Lame curveȲ n , which was denoted by Γ in [23] There are four natural coordinate projections (rational functions)Ȳ n → P 1 , namely B, κ, x 0 and y 0 respectively. The first one B :Ȳ n → P 1 is simply the hyperelliptic structure map. The main result in [23] is an explicit description of the other 3 maps in terms of the coordinates (B, C) onȲ n : Theorem 7.2 ([23, Theorem 4.1]). For all n ∈ N and i ∈ {1, 2, 3}, The formula for x 0 (B) is independent of the choices of i.
Example 7.4. For later usage, we recall Maier's formulas for lt j (B) and l θ (B) for n ≤ 4.
Let a ∈ Y n . The two expressions (7.1) and (7.3), which correspond to the same solution to the Lamé equation (7.2), must be proportional to each other by a constant. Hence we get As a well defined meromorphic function onȲ n , we conclude that a 0 (a) = σ(a) + c for some constant c ∈ C. Consider a point a ∈ Y n \X n with σ(a) = 1 2 ω 1 , i.e. l 1 (B a ) = 0. Such a exists by Proposition 5.6. Then z n (a) = 0 trivially. We also have κ(a) = 0 by Theorem 7.4 since C 2 a = c 2 n l 0 (B a )l 1 (B a )l 2 (B a )l 3 (B a ) = 0 (again by Proposition 5.6). So (7.5) implies 0 = 1 2 η 1 − ζ( 1 2 ω 1 + c), and hence c = 0. This proves σ(a) = a 0 , which represents π(a) in E, and also κ(a) = −z n (a). The proof is complete. Now we may describe the explicit construction of the polynomial W n (z) in Theorem 6.1 based on Theorem 7.2. It is indeed merely an application of the elimination theory using resultant. By Theorem 7.2 and 7.5, we may eliminate C to get (7.6) y 0 z n = 16 n 2 (n + 1) 2 (n − 1)(n + 2) which leads to a polynomial equation g = 0 for (7.7) g := z n On the other hand, the 3 rational expressions of x 0 lead to f = 0 for (7.8) Notice that f , g are polynomials in g 2 , g 3 (and B, x 0 , y 0 ) instead of e i 's.
Let R( f , g; B) be the resultant of the two polynomials f and g arising from the elimination of the variable B.
Proof. From the explicit rational expressions (7.4) in Theorem 7.2, the resultant of f , g, defined in (7.8), (7.7) gives rise to the equation of the branched covering map σ :Ȳ n → E over the loci outside E [2]\{0}. More precisely, if C = 0 then the formulas for y 0 and κ = −z n are equivalent to g = 0. However, if C = 0 we have l i (B) = 0 for some 0 ≤ i ≤ 3 by Proposition 5.6. From (7.8), we get 1 ≤ i ≤ 3 in order to have f = 0. Hence l 0 (B)lt 0 (B) = 0 and then z n = 0 = y 0 in (7.4). But the equation g = 0 in (7.7) gives extra solutions. Indeed by (7.7) we conclude only that y 0 = 0 and z n could be arbitrary. In conclusion, the additional solutions consist of (x 0 , y 0 ) = (e i , 0), i = 1, 2, 3, with z = z n being arbitrary. Thus the additional solutions contribute a factor to R( f , g; B)(z) which divides ∏ 3 i=1 (x 0 − e i ) = y 2 0 /4 and is independent of z. In particular R( f , g; B)(z) has degree 1 2 n(n + 1) in z. Also by construction R( f , g; B)(z n ) = 0, hence it must be a multiple of the minimal polynomial W n (z) of z n . Since deg W n = 1 2 n(n + 1), the full multiplier λ n is independent of z, and with y 2 0 | λ n . Proposition 7.7. The pre-modular form Z n (σ; τ) = W n (Z) can be explicitly computed for any n ∈ N.
In practice, such a computation is very time consuming even using computer. In the following, we apply it to the initial cases up to n = 4. As before we denote x 0 = ℘(σ) =: ℘ and y 0 = ℘ (σ) =: ℘ . Example 7.8. For n = 2, it is easy to see that The resultant R( f , g; B) is calculated by the 6 × 6 Sylvester determinant: A direct evaluation gives Example 7.9. For n = 3, we have It takes a couple seconds to evaluate the corresponding 12 × 12 Sylvester determinant (e.g. using Mathematica) to get where W 3 (z) is given by It seems impractical to evaluate this resultant by hand. .
In Appendix B we will discuss a classical approach, namely a "twosteps" procedure, to construct Z n (σ; τ). It is essentially the method used in [10] which works well for n = 2, 3 and can allow computations by hand. The above two examples agree with Example B.1 and Example B.2 in Appendix B respectively, which appeared in [10] already. Unfortunately this classical approach fails for all n ≥ 4 (see Proposition B.5). Hence Proposition 7.7 is the only general method to construct Z n (σ; τ).
Applications of these explicit pre-modular forms will be discussed in a subsequent work.
We end this section with a brief discussion on the rationality property. We have constructed two affine curves fromX n . One is the hyperelliptic model Y n = {(B, C) | C 2 = n (B)}, another one is Y n := {(x 0 , y 0 , z n ) | y 2 0 = 4x 2 0 − g 2 x 0 − g 3 , W n (x 0 , y 0 ; z n ) = 0} which is understood as a degree 1 2 n(n + 1) branched cover of the original curve Y n is birational to Y n over E, namely the addition map σ : Y n → E is compatible with σ : Y n → E. Notice that both n and W n have coefficients in Q[g 2 , g 3 ]. The explicit birational map φ : (B, C) (x 0 , y 0 , z n ) (given in Theorem 7.2 and 7.5 via z n = −κ) also has coefficients in Q[g 2 , g 3 ]. This implies that φ is defined over Q. Moreover φ extends to a birational mor-phismȲ n (∞) via Theorem 6.9. From (the proof of) Proposition 7.6, the morphism φ is an isomorphism outside those branch points for Y n → P 1 lying over E [2]\{0}. In particular, the non-isomorphic loci lie in z n = 0 by (7.4) and Theorem 7.5.
Remark 7.11. In contrast to the smoothness of Y n (τ) for general τ, for all n ≥ 3 the model Y n (τ) is singular at points z n = 0 = y 0 (and hence x 0 = e i for some i). Indeed from (7.4) this is equivalent to C = 0 and l i (B)lt i (B) 2 = 0 for some 1 ≤ i ≤ 3. For n = 2, there is only one solution B for each fixed i (c.f. Example 7.4). However, for n ≥ 3 there are more than one solutions B. These points (B, 0) ∈ Y n are collapsed to the same point (x 0 , y 0 , z n ) = (e i , 0, 0) ∈ Y n under φ, thus (e i , 0, 0) is a singular point of Y n .
For n = 3, 4 this is easily seen from the equation W n (z) = 0 given above since it contains a quadratic polynomial in (z, ℘ ) (i.e. in (z n , y 0 )) as its lowest degree terms.
In particular, the birational map φ −1 is also represented by rational functions B = B(x 0 , y 0 , z n ) and C = C(x 0 , y 0 , z n ) with coefficients in Q[g 2 , g 3 ] and with at most poles along z n = 0. In principle such an explicit inverse can be obtained by a Groebner basis calculation associated to the ideal of the graph Γ φ .
Example 7.12. For small n, these formulas can be obtained from Example 7.4. For n = 2, 3 they can also be obtained from the intermediate step of the classical construction of W n (z) in §B.1. For n = 2, from (B.1) we have (7.10) B = 3(z 2 2 − x 0 ), and then C is obtained from z 2 (B, C) by substituting B. We get Proposition 7. 13. Let E be defined over Q, i.e. g 2 , g 3 ∈ Q. Then the Lamé curvē Y n is also defined over Q for all n ∈ N. Moreover,Ȳ n and all the morphisms σ, σ , φ are also defined over Q.
A rational point (B, C) ∈Ȳ n is mapped to a rational point (x 0 , y 0 , z n ) ∈Ȳ n by φ. For the converse, given (x 0 , y 0 ) ∈ E(Q), a point (x 0 , y 0 , z n ) in the σ fiber gives a unique (B, C) ∈Ȳ n (Q) if z n ∈ Q and (x 0 , y 0 , z n ) = (e i , 0, 0) for any i.
Remark 7.14. It is well known that there are only few (i.e. at most finite) rational points on a non-elliptic hyperelliptic curve. This phenomenon is consistent with the irreducibility of the polynomial W n (z) over K(E) in light of Hilbert's irreducibility theorem that there is a infinite (Zariski dense) set of (g 2 , g 3 , x 0 , y 0 ) ∈ Q 4 so that the specialization of W n (z) is still irreducible. Nevertheless, it might be interesting to see if z n plays any role in the study of rational points.

FUTURE PLANS
As discussed in the introduction, one of our purposes in this series of papers is to study the structure of non-trivial solutions to (0.2), and to understand how non-trivial solutions change during the deformation of torus in the moduli space. Several important issues are listed below.

Does Conjecture 0.9 hold?
If we could show that (8.1) Z n (σ; τ) has only simple zeros in τ for any σ ∈ E τ [2], then Conjecture 0.9 follows from Lemma 6.12 and Theorem 6.13 immediately. The reason is as follows: Let a(τ 0 ) be a non-trivial solution to (0.2) with E = E τ 0 . By Lemma 6.12 and Theorem 6.13, we have σ 0 := σ(a(τ 0 )) ∈ E τ 0 [2] and Z n (σ 0 ; τ 0 ) = 0. If (8.1) holds, then near (σ 0 , τ 0 ) the equation Z n (σ; τ) = 0 for fixed σ has only one solution (σ, τ). This implies that a(τ 0 ) is not a bifurcation point of equation (0.2). In fact, it is easy to see that (8.1) is equivalent to Conjecture 0.9. For n = 1, (8.1) is proved in Corollary 0.3 (as corollary of Theorem 2.3). For n = 1, 2, 3 with σ being an N-torison point, the simple zero property of Z n (σ; τ) had been shown by Dahmen in [10,11] (see also [2]), by counting the number of integral Lamé equations with finite monodromy group by means of dessin d'enfants (see §8.3 for more explanations). However their results could not provide a complete answer to the question whether (8.1) holds (for 1 ≤ n ≤ 3). As in our proof to the case n = 1, we still need the following: 8.2. Non-existence of solutions to (5.1) for rectangular tori.
More precisely, we pose Conjecture 8.1. There is no solutions to u + e u = 8πnδ 0 for τ = ib, b > 0.
For n = 1, Conjecture 8.1 was proved by the authors in [16]. By applying the theory developed in this paper, we are able to prove it for n = 2 in the forthcoming Part III [18] of this series of papers. Thus, together with the result of Dahmen, we could prove (8.1) in its fully generality for n = 2. The affirmative answers to both (8.1) and Conjecture 8.1 are the first important steps towards the complete picture of (0.2), or equivalently (5.1) (= (0.3) with ρ = 8πn), over the moduli space of flat tori. As in the case n = 1 treated in §2 and §3, we work on the fundamental domain F. When n = 2, there are 5 trivial critical points coming from the branch points of the hyperelliptic curve Y 2 . We could deform the pre-modular form Z 2 (σ; τ) in σ and study the geometry of the degeneracy curves in F corresponding to these 5 trivial solutions. Our numerical study of these curves (using Mathematica) is shown in Figure 4 below.
In Part III, we will show that the number of non-trivial solutions to (0.2) is constant in τ when τ moves in a connected region in F separated by these degenerate curves. Moreover we shall be able to determine the number of solutions explicitly in each connected region.

Dahmen's conjecture.
Although we have shown the existence of Z n (σ; τ) for all n in Theorem 6.1 and provided the constructions of them in Proposition 7.7, in practice we still need to find the explicit expressions of them in order to get useful informations, e.g. to attack (8.1).
Indeed, in [10,11] Dahmen developed a theory to count the number of equivalence classes of the algebraic form of integral Lamé equations where a 2m = a 2m+1 := m(m + 1)/2 and and n (N) is 0 or 1 to make the expression in (8.3) an integer. Under the assumption of the existence of suitable modular forms (which is now provided by our Z n (σ; τ) with σ ∈ E τ [N]), Dahmen proved that if (8.3) holds for some n, N then has only simple zeros. Furthermore, by the method of dessin d'enfants, he established (8.3) for any N and n = 1, 2, 3. In general, we may reduced his conjecture for all n, N to the counting of vanishing order of (8.4) at ∞.
With the explicit expression of W 4 (z) in (7.9) and hence the weight 10 pre-modular form Z 4 (σ; τ), we are able to determine the precise vanishing order at ∞. It turns out agrees with the expected one, hence confirms Dahmen's conjecture for n = 4 (and for all N). The detailed verification is written up by Y.-C. Chou, under the advise of the second author, and is included as Appendix C.
8.4. The geometry ofX n at its branch points over P 1 .
gives us the first motivation to study the geometry ofX n as follows. It was proved in Part I [4, Theorem 0.8] that if {u k } ∞ k=1 is a sequence of blowup solutions to (0.3) with ρ k → 8πn and ρ k = 8πn for large k, then the blowup set p = {p 1 , · · · , p n } of {u k } is a trivial solution to (0.2). Here {u k } is said to blowup if max E u k → +∞ as k → ∞. Given such a blowup sequence, it was proved in [3] for n = 1 and in [19] for n ≥ 2 that We had seen the invariant D(p) in §4 for the case n = 1. To define the invariant D(p) for general n, we first recall the regular part of the Green functionG(z, q) := G(z − q) + 1 2π log |z − q|, and define where Ω i is any open neighborhood of p i in E such that Ω i ∩ Ω j = ∅ for i = j, and n i=1 Ω i = E. We note that in a neighborhood of p i , e f p i (z)−1 is a harmonic function with quadratic terms in (x, y) plus O(|z − p i | 3 ), where z = x + iy, hence the limit in (8.6) is finite and well-defined.
It is clear that when D(p) = 0 its sign provides important information when we study bubbling solutions (blow-up sequence) with ρ = 8πn (e.g. if D(p) > 0 then the bubbling may only occur from the right hand side). Also in the case ρ k = 8πn for all k, if the blow-up family u λ exists then D(p) = 0 trivially, provided that p is a non-trivial solution to (0.2).
From the analytic point of view, the sign of D(p) plays many important roles in the concentration of bubbling solutions of a class of non-linear PDE in two dimensions. For example, in the mean field equations, Chern-Simons-Higgs equations, and even a system of equations like SU(3) Chern-Simons system etc.. Recently, it is also related to the so-called local uniqueness of bubbling solutions, which states that for any two sequence of bubbling solutions under some non-degenerate conditions (with D(p) = 0 being one of them) and with the same ρ k , if their blowup sets are identical then the two sequence of solutions are equal. For recent applications of these quantities, we refer the readers to [19,20,22].
However, D(p) or even its sign is in general very difficult to compute. Thus the question how the analytic invariant D(p) would be related to the geometry ofX n is one of the main issues we are going to pursue. Recall the conjecture we posed in Part I: Conjecture 8.2 ([4]). For rectangular tori, there are n branch points onX n with D < 0 and n + 1 branch points with D > 0.
Conjecture 8.2 was proved in [9] for n = 1 and in [19] for n = 2. We are speculating that it is closely related to Conjecture 8.1 on non-existence of solutions to the mean field equations on rectangular tori.

APPENDIX A. UNIQUENESS OF SOLUTIONS
In this appendix we want to classify all solutions to (A. 1) u + e u = ρ δ 0 on E for ρ ≤ 8π + 0 where 0 is a small positive number. Recall in [16] we showed that equation (A.1) has a unique solution for ρ = 4π, and a unique even solution for 4π ≤ ρ ≤ 8π. Here we want to prove the uniqueness result without the evenness assumption. Proof. We first show that for any solution u to (A.1) with ρ ≤ 4π, the linearized equation Suppose that φ is a solution to (A.2). Then a straightforward computation shows that (φ zz − u z φ z )z = 0. Since u(z) ∼ ρ 2π log |z|, φ zz − u z φ z is an elliptic function on E whose only singularity is a pole of order one at 0. This forces that φ z (0) = 0 and φ zz − u z φ z = c 1 on E for some constant c 1 , or equivalenly Notice that and if ρ = 4π the above limit is finite. If c 1 = 0, this implies that which leads to a contradiction. So we have c 1 = 0 and e −u φz is an elliptic function. By (A.3) this again implies that e −u φz = c 3 is a constant.
If φ ≡ 0 then φ has a maximum point p and a minimum point q with p = q. One of p, q is not a lattice point where φz = 0. This implies that c 3 = 0 and hence φz ≡ 0. This leads to φ ≡ 0 which is a contradiction to φ ≡ 0. Hence we must have φ ≡ 0. Now the uniqueness follows from the fact that (A.1) has only one solution at ρ = 4π.
Remark A.2. In [16] we showed that the unique even solution to (A.1) with ρ ∈ [4π, 8π] is non-degenerate in the class of H 1 ev = {u ∈ H 1 | u(−z) = u(z)}. Now the proof of Lemma A.1 allows us to remove the evenness assumption: u is non-degenerate in the whole space H 1 , provided that 0 < ρ < 8π.
To see this, we may assume that the solution φ is odd. Therefore φ zz − u z φ z is odd and by exactly the same calculation we have This implies that e −u φz is an elliptic function on E, with 0 being its only pole. However, since φ is odd, φz is even and the estimate (A.3) can be improved to |e −u φ z (z)| ≤ c 2 |z| 2−ρ/2π .
If ρ < 8π, we find 2 − ρ/2π > −2. This implies that e −u φz is a constant. If φ ≡ 0, by evaluating it at a maximum or minimum point, with one of it not a lattice point, we conclude that e −u φz ≡ 0, and then φ ≡ 0 follows. (Notice that if ρ = 8π then e −u φz = c℘(z) for some constant c = 0.) Now we may conclude that the unique even solution u is always a minimum point of the non-linear functional J ρ in §1 for 0 < ρ ≤ 8π. But in fact we can prove a stronger result, namely Theorem 0.7. Lemma A.3. Let u be a solution to (A.1) with ρ ∈ 8πN. Then u is even.
Proof. This was proved in Part I [4] for ρ ∈ 4πl with l being an odd integer, so we assume that ρ ∈ 4πN.
Let f (z) be a multi-valued developing map of u. The readers are referred to [4], §8 for the details to treat these multi-valued functions as global analytic functions f(ξ), which are defined on ξ ∈ H → E × . In particular the cusp ξ = 0 is mapped to the cusp z = 0 in E × .
As in [4], we have S( f ) = 2(η(η + 1)℘ + B) for η = ρ/8π for some B ∈ C. Thus f = w 1 /w 2 and f = w 1 /w 2 for two linearly independent solutions w 1 and w 2 to the Lamé equation Sincew i (z) := w i (−z) are also two linearly independent solutions to (A.4), f :=w 1 /w 2 also defines a global analytic functionf and we havẽ Consider the covering transformations on H: g 1 , g 2 ∈ SL(2, R) determined by the two free generators of π 1 (E × ) ∼ = Z * Z. Let Γ < SL(2, R) be the rank two free subgroup generated by g 1 and g 2 , and r : Γ → PSU(2) be the unitary representation associated to the solution u. The mapping (−1) : z → −z on E × lifts to a map ι on H which is not a covering map for H → E × . Nevertheless the composition ι • ι, namely we apply (−1) twice, does give a covering map for H → E × . That is, the matrix S 2 can be represented as an element generated by S 1 := r(g 1 ) and S 2 := r(g 2 ).
By considering the action of (−1) in a simply connected neighborhood U of 0 ∈ E, we see that S 2 f = f (e 2πi z) = β f (z) for some β ∈ PSU (2, C).
Proof of Theorem 0.7. By Lemma A.1 and A.3, the uniqueness of solutions holds for 0 < ρ < 8π. The statements for ρ = 8π was proved in [16]. For τ ∈ Ω 3 , the unique solutions u ρ blows up as ρ 8π (since equation (A.1) has no solutions at ρ = 8π). The blow-up point of u ρ must be the minimum point which is one of the half periods. The other two half periods q 1 and q 2 are saddle critical point of G. By Corollary 0.4 and Theorem 0.6, we have det D 2 G(q i ) < 0 and then D(q i ) > 0. Under these conditions, by the method in [7] we can construct a bubbling sequence of solutions u ρ,i to (A.1), for each i = 1, 2, with ρ > 8π which blows up at q i .
Remark A.4. In [7] the non-degenerate condition D(q i ) = 0 was replaced by some other non-degenerate condition. Nevertheless the similar process as there still works in our current case (see e.g. the remark in [9] concerning with the degree counting formula).
Indeed, for the Chern-Simons-Higgs equation, the same non-degenerate conditions D(q) < 0 and det D 2 G(q) = 0 were recently used to construct such kind of bubbling solutions [20]. Now we need the following uniqueness theorem: Theorem A.5. Suppose that u k andũ k are two sequences of solutions to (A.1) with ρ k → 8π, and both sequences have the same blow-up point q.
If D(q) = 0, i.e. q is a non-degenerate critical point of G by Theorem 0.6, then u k =ũ k for large k.
This is recently proved in [22] for the Chern-Simons-Higgs equation but the proof given there also works for (A.1). By Theorem A.5, u ρ,i are exactly all the solutions to equation (A.1) for 8π < ρ < 8π + 0 . This proves (i).
For τ ∈ Ω 5 , all the three half periods are saddle points of G. By Theorem A.5 again, we must have three bubbling solutions. On the other hand, (A.1) has a unique even solution u for ρ = 8π whose linearized equation in the class of even functions is non-degenerate. Therefore for 8π < ρ < 8π + 0 there is a unique even solution u ρ which converges to to u as ρ 8π. By Lemma A.3, (A.1) has only even solutions for 8π < ρ < 8π + 0 , we conclude that (A.1) has the only one even solution u ρ which converges to u as ρ 8π. Hence there are four solutions in total. This proves (ii) and thus completes the proof Theorem 0.7.

APPENDIX B. A REMARK ON THE CLASSICAL APPROACH TO Z n
A "two stpes" approach to the determination of Z 2 and Z 3 based on the addition law (B.1) and a classical cubic identity (B.3) of elliptic functions was developed in [10]. One might hope that a more sophisticated application of the Frobenius-Stickelberger type identities (e.g. [26, p.458]) may lead to a construction of Z n for n ≥ 4. The purpose of this appendix is to show that this classical approach fails for all n ≥ 4 (see Proposition B.5).
B.1. Explicit constructions for n = 2, 3. We describe the first two cases n = 2, 3 in this subsection using a "two steps" procedure.
Example B.2 (n = 3). For z = z 3 (a), we have on X 3 : The derivation of W 3 (z) is more involved. It consists of two steps. The first step is the following classical identity. We supply a detailed proof of it since a variant of the proof will be used for our later discussions on the general cases n ≥ 4.
Proof. We will prove it by viewing both sides as functions of s = a 3 and by comparing the principal parts on both sides. In doing so we emphasize that the case n = 2 is used in an essential way.
Then we get W 3 (z) by a straightforward manipulation with (B.9).
Example B.4. The branched coverX n → E is in general not a Galois cover. Namely, W n (z) does not split into product of linear factors in K(X n ).
We will show that w ± are not single valued onX 2 for general tori. The rational function h(a) := 4℘(σ(a)) − z 2 2 (a) has poles of total order six by Theorem 6.9. By Example 6.10, if g 2 = 0, they consists of 3 poles with each of order 2, and for g 2 = 0, 0 2 ∈X 2 is of order 2 and (q, −q) ∈X 2 with ℘(±q) = 0 is of order 4. In order for w ± being single valued, the zeros of h must also be of even order.
If h(a) = 0 but ℘(σ(a)) = 0 for some a ∈X 2 , then it is easy to see that there must be some zeros of h with odd order. Thus we only need to consider the case that all zeros of h are also zeros of ℘(σ(a)). Since h(a) = 3℘(a 1 + a 2 ) − (℘(a 1 ) + ℘(a 2 )), we have ℘(a 1 ) + ℘(a 2 ) = 0 too. The addition theorem then implies that ℘ (a 1 ) = ℘ (a 2 ). But the equation for X 2 is given by ℘ (a 1 ) + ℘ (a 2 ) = 0, hence ℘ (a 1 ) = 0 = ℘ (a 2 ). That is, a 1 = 1 2 ω i , a 2 = 1 2 ω j and a 1 + a 2 = 1 2 ω k is the third half period. Then we get the non-trivial equation e k = 0. Thus for general E = E τ , W 2 (z) does not split into product of linear factors. B.2. A remark on n ≥ 4. A closer look at the proof of Example B.2 shows that the overall important equation to start with is not the classical polynomial identity (B.3) in z. Instead, the polynomial equation (B.7) on z and δz with admissible coefficients (depending only on σ) is what we really need for the proof. (For simplicity we use the notation A(δ * z) for it.) To the authors' knowledge, historically a reasonably clean polynomial identity of degree n with symmetric coefficients in a i 's like (B.1) and (B.3) was unknown for n ≥ 4. For n = 4, naive generalization of the proof of Lemma B.3 to get a degree 4 polynomial fails immediately. Thus we try to get an admissible equation in z and δ i z's instead. (These two type of expressions are equivalent by (B.10) below.) As a result, we will be able to prove that the degree 4 polynomial indeed does not exist.
Notice that while z = ζ(∑ a i ) − ∑ ζ(a i ) is symmetric in the last variable s = a n under s → −σ n = −σ n−1 − s, δz = 1 n ∑ n i=1 ℘(a i ) − ℘(σ n ) breaks such a symmetry. It thus distinguishes the two poles s = 0 and s = −σ n−1 which is a key property we shall use.
This shows that for n = 4 no monic admissible polynomial equations of degree n for z n may exist. The proof shows also that if A n (δ * z) exists for some n ∈ N, then A n−1 (δ * z) must also exist by looking at the residue term in the Laurent expansion of z n n = A n (δ * z n ). By induction this implies A n (δ * z) does not exists for all n ≥ 4.
That is, L n (N) ≤ U n (N). Moreover, the equality holds if and only if each factor Z n ((k 1 + k 2 τ)/N; τ) has only simple zeros. We will show the equality holds by comparing it with the counting formula for the projective monodormy group PL n (N) (c.f. [10], Lemma 74).
We recall the relation between L n (N) and PL n (N): If n is even and N is odd, we have PL n (N) = L n (N) + L n (2N) On the other hand, using the method of dessin d'enfants, Dahmen showed directly that the equality holds [11]. Thus all the intermediate inequalities are indeed equalities, and in particular L n (N) = U n (N) holds. Moreover, Z 4 (σ; τ) with σ ∈ E τ [N] has only simple zeros in τ ∈ H.