Mean-field equations, Hyperelliptic curves and Modular forms

Abstract

We develop a theory connecting the following three areas: (a) the mean field equation (MFE) 4u + eu = ρ δ0, ρ ∈ R>0 on flat tori Eτ = C/(Z + Zτ), (b) the classical Lame equations and (c) modular forms. A major theme in part I is a classification of developing maps f attached to solutions u of the mean field equation according to the type of transformation laws (or monodromy) with respect to Λ satisfied by f . We are especially interested in the case when the parameter ρ in the mean field equation is an integer multiple of 4π. In the case when ρ = 4π(2n + 1) for a non-negative integer n, we prove that the number of solutions is n + 1 except for a finite number of conformal isomorphism classes of flat tori, and we give a family of polynomials which characterizes the developing maps for solutions of mean field equations through the configuration of their zeros and poles. Modular forms appear naturally already in the simplest situation when ρ = 4π In the case when ρ = 8πn for a positive integer n, the solvability of the MFE depends on the moduli of the flat tori Eτ and leads naturally to a hyperelliptic curve Xn = Xn(τ) arising from the Hermite-Halphen ansatz solutions of Lame’s differential equation d2w dz2 − (n(n + 1)℘(z; Λτ) + B)w = 0. We analyse the curve Xn from both the analytic and the algebraic perspective, including its local coordinate near the point at infinity, which turns out to be a smooth point of Xn. We also specify the role of the branch points of the hyperelliptic projection Xn → P1 when the parameter ρ varies in a neighborhood of ρ = 8πn. In part II, we study a “premodular form” Zn(σ; τ), a real-analytic function in two variables associated to Xn(τ), which has many symmetries and also the property that the τ-coordinates of zeros of Zn(σ; τ) correspond exactly to those flat tori where the MFE with parameter ρ = 8πn has a solution.