The geometry of generalized Lam? equation, II: Existence of pre-modular forms and application
Journal
Journal des Mathematiques Pures et Appliquees
Journal Volume
132
Pages
251-272
Date Issued
2019
Author(s)
Abstract
In this paper, the second in a series, we continue to study the generalized Lamé equation with the Treibich-Verdier potential y″(z)=[∑k=03nk(nk+1)℘(z+ [Formula presented] |τ)+B]y(z),nk∈Z≥0 from the monodromy aspect. We prove the existence of a pre-modular form Zr,s n(τ) of weight [Formula presented] ∑nk(nk+1) such that the monodromy data (r,s) is characterized by Zr,s n(τ)=0. This generalizes the result in [17], where the Lamé case (i.e. n1=n2=n3=0) was studied by Wang and the third author. As applications, we prove among other things that the following two mean field equations Δu+eu=16πδ0andΔu+eu=8π∑k=13δ [Formula presented] on a flat torus has the same number of even solutions. This result is quite surprising from the PDE point of view.
Type
journal article