No abstract

We study the problem: How many singular points of a solution λ(t) to the Painlevé VI equation with parameter ( 1 8 , −1 8 , 1 8 , 3Based on Hitchin's formula, we explore the connection of this problem with Green function and the Eisenstein series of weight one. Among other things, we prove:(i) There are only three solutions which have no singular points in C \ {0, 1}. (ii) For a special type of solutions (called real solutions here), any branch of a solution has at most two singular points (in particular, at most one pole) in C \ {0, 1}. (iii) Any Riccati solution has singular points in C \ {0, 1}. (iv) For each N ≥ 5 and N = 6, we calculate the number of the real j-values of zeros of the Eisenstein series E N 1 (τ; k 1 , k 2 ) of weight one, where (k 1 , k 2 ) runs over [0, N − 1] 2 with gcd(k 1 , k 2 , N) = 1.The geometry of the critical points of the Green function on a flat torus E τ , as τ varies in the moduli M 1 , plays a fundamental role in our analysis of the Painlevé IV equation. In particular, the conjectures raised in [22] on the shape of the domain Ω 5 ⊂ M 1 , which consists of tori whose Green function has extra pair of critical points, are completely solved here. 8 , −1 8 , 1 8 , 3 8 ) . We say that two solutions λ r,s (t) and λ r ,s (t) give (or belong to) the same solution if λ r ,s (t) is the analytic continuation of λ r,s (t) along some closed loop in C \ {0, 1}. In §4, we will prove that λ r,s and λ r ,s give the same solution to PVI ( 1 8 , −1 8 , 1 8 , 3 8 ) if and only if (s , r ) ≡ (s, r) · γ mod Z 2 for some matrix γ ∈ Γ(2) = {γ ∈ SL(2, Z)

In literature, it is known that any solution of Painlevé VI equation governs the isomonodromic deformation of a second order linear Fuchsian ODE on CP 1 . In this paper, we extend this isomonodromy theory on CP 1 to the moduli space of elliptic curves by studying the isomonodromic deformation of the generalized Lamé equation. Among other things, we prove that the isomonodromic equation is a new Hamiltonian system, which is equivalent to the elliptic form of Painlevé VI equation for generic parameters. For Painlevé VI equation with some special parameters, the isomonodromy theory of the generalized Lamé equation greatly simplifies the computation of the monodromy group in CP 1 . This is one of the advantages of the elliptic form. u zz − 1 2 u 2 z z = (u zz ) z − u z u zz = − 1 4 e u z + 1 4 e u u z = 0.

In this paper, we prove that the spectral curve Γn of the generalized Lamé equation with the Treibich-Verdier potentialcan be embedded into the symmetric space Sym N Eτ of the N -th copy of the torus Eτ , where N = n k . This embedding induces an addition map σn(·|τ ) from Γn onto Eτ . The main result is to prove that the degree of σn(·|τ ) is equal to 3 k=0 n k (n k + 1)/2. This is the first step toward constructing the premodular form associated with this generalized Lamé equation.

Let M be a compact Riemann surface, α j > −1, and h (x) a positive C 2 function of M . In this paper, we consider the following mean field equation:We prove that for α j ∈ N and any ρ > ρ 0 , the equation has one solution at least if the Euler characteristic χ (M ) ≤ 0, whereis the Gaussian curvature, and N * = 4π d j=1 α j . This result was proved in [10] when α j = 0. Our proof relies on the bubbling analysis if one of the blowup points is at the vortex q j . In the case where α j / ∈ N, the sharp estimate of solutions near q j has been obtained in [11]. However, if α j ∈ N, then the phenomena of non-simple blowup might occur. One of our contributions in part 1 is to obtain the sharp estimate for the non-simple blowup phenomena.