Mean field equations, hyperelliptic curves and modular forms II
Resource
submitted, preprint 2014: arXiv:1502.03295
Journal
Journal de l
Pages
557-593
Date Issued
2014
Date
2014
Author(s)
Abstract
A pre-modular form Zn(σ; τ) of weight 12 n(n + 1) is introduced for each n ∈ N, where (σ, τ) ∈ C × H, such that for Eτ = C/(Z + Zτ), every non-trivial zero of Zn(σ; τ), i.e. σ is not a 2-torsion of Eτ , corresponds to a (scaling family of) solution to the equation (MFE) u + eu = ρ δ0, on the flat torus Eτ with singular strength ρ = 8πn. In Part I [1], a hyperelliptic curve Xn(τ) ⊂ SymnEτ , the Lamé curve, associated to the MFE was constructed. Our construction of Zn(σ; τ) relies on a detailed study of the correspondence P1(C) ← Xn(τ) → Eτ induced from the hyperelliptic projection and the addition map.
Type
journal article
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