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Transcript

Finding real roots of Eisenstein series of weight one by Painlevé VI equations Chang-Shou Lin Taida Institute for Mathematical Sciences National Taiwan University Taipei 10617, Taiwan October 22, 2015 Abstract For any (r, s) ∈ C2 \ 21 Z2 , we define pr,s (τ ) ∈ C by ℘(pr,s (τ )|τ ) = ℘(r +sτ |τ )+ ℘0 (r + sτ |τ ) , 2(ζ(r + sτ |τ ) − rη1 (τ ) − sη2 (τ )) τ ∈H The Hitchin theorem says that pr,s (τ ) is a solution of a Painlevé VI equation. Remarkably, the denominator ζ(r + sτ |τ ) − rη1 (τ ) − sη2 (τ ) is the Eisenstein series of weight one if (r, s) is an N -torsion point. Set Zr,s (τ ) = ζ(r + sτ |τ ) − rη1 (τ ) − sη2 (τ ). Then Zr,s (τ ) has another link with geometric and PDE aspects. Any zero τ of Zr,s (τ ) for some real pair (r, s) ∈ R2 \ 12 Z2 , the PDE (mean field equation) ∆u + eu = 8πδ(0) in Eτ , has a solution, where Eτ is the torus C/∧τ , ∧τ is the lattice generated by 1 and τ , and δ(0) is the Dirac measure at lattice points. In this talk, I will give a survey about the story about the connection among such subjects: mean field equations, Painlevé VI, and modular forms. The center of these connections are Green functions and Lame equations. 1