Green function, Painlevé VI equation, and Eisenstein series of weight one

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)

Critical points of the classical Eisenstein series of weight two

“…The reason why we choose the fundamental domain F 0 will be clear from the proof, particularly Lemma 2.3. The basic strategy is similar to that of proving Theorem A in [6]. However, the argument is more involved and new techniques are needed.…”

Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions

“…Remark 2.3. Recently, Chen, Kuo, Lin and Wang in [36] showed that G(z) might have an extra pair of "non-trivial" critical points other than the three half periods points for a class of flat tori. Moreover, the "non-trivial" critical points are always non-degenerate.…”

The geometry of generalized Lamé equation, I

“…all solutions of (1.11) are free of logarithmic singularity at any singular point. See [4,5,7,18] for recent developments of (1.11). Like (1.2), we could associate a hyperelliptic curve Γ n,p (τ ) := {(A, W )|W 2 = Q n,p (A; τ )} and an addition map σ n,p with (1.11).…”