Mean field equations, hyperelliptic curves and modular forms: I

The geometry of generalized Lamé equation, II: Existence of pre-modular forms and application

“…For the Lamé case n 1 = n 2 = n 3 = 0, Theorem 3.1 was proved in [2]. For general case n k ∈ Z ≥0 as considered here, the necessary part of the first assertion in Theorem 3.1 was already proved in [7,9].…”

“…Note that if a is a branch point, then it follows from (2.9) that σ n (a) ∈ E τ [2]. To prove d n = deg σ n , i.e.…”

“…Theorem 2.1 in the Lamé case n 1 = n 2 = n 3 = 0 was proved in [2]. The proof of Theorem 2.1 for general n k is technical and long, and will be given…”

“…To prove d n = deg σ n , i.e. z n (a) is a primitive generator, we take σ 0 ∈ E τ \ E τ [2] outside the branch loci of σ n : X n (τ) → E τ . Then there are precisely deg σ n different points a ∈ X n (τ) with σ n (a) = σ 0 , and Lemma 2.3 shows that these deg σ n different points a give deg σ n different values z n (a).…”

New existence results for the mean field equation on compact surfaces via degree theory

“…More recently, equation (1.1) has been studied in the context of hyperelliptic curves and of the Painlevé equations, see [9] and [11], respectively. Equation (1.1) plays also an important role in mathematical physics.…”