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## Constructing algebraic solutions of Painleve VI equation from p-adic Hodge theory and Langlands correspondence

Abstract:

For  the projective line  over complex numbers with 4 punctures 0, 1, \infity, \lambda, where lambda  is a parameter running in P^-{0, 1, \infty} we construct infinitely many rank-2 local systems with the prescribed local monodromy around those 4 punctures, which come from geometry origin. Consequently, they all are algebraic solutions of the Painleve VI equation. The method  is totally different to the traditional approach. It relies on p-adic Higgs-de Rham flow developed by Lan-Sheng-Zuo, Fontaine-Faltings theorem on crystalline local systems and Drinfeld’s work on Langlands correspondence over function field of characteristic-p.

We  have predicted that those solutions are exactly parametrized by torsion points on the elliptic curves E_\lambda as double cover of P^1 ramified on 0,1,\infty, \lambda, which is now affirmatively answered by the recent work of Mao Sheng joint with  his student Xiaojin Lin and postdoc Jianping Wang.

This is a joint work with Raju  Krishnamoorthy and Jinbang Yang.

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