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.

Bio:

左康，武汉大学数学与统计学院教授，德国美因茨大学W3教授（2009-2021）。主要研究领域是代数几何，同时在相关的微分几何和算术几何方面也有重要的学术研究。师从德国代数几何学家（波恩马克斯普朗克数学研究所首任所长 F.Hirzebruch获得博士学位，在学术生涯早期主要研究复代数簇基本群的表示，独立或与（Leipzig马克斯普朗克数学科学研究所首任所长J.Jost在这方面获得多项深刻结果，其后与双有理几何的代表性人物E. Viehweg开始了长达十多年的合作，在模空间的双曲性，Shimura子簇的刻画等重要问题上取得一些重要进展。近年来研究领域扩展到算术几何，与盛茂（清华大学）、杨金榜(中科大）等发展的p-进Higgs-de Rham流可以看作复数域上Yang-Mills-Higgs方程的一个类比，已经成为研究正特征以及混合特征上代数簇基本群的p-进表示的一个重要工具。另一方面, 和陈柯（南京大学）、吕鑫（华东师范大学）合作，在算术几何领域的Coleman-Oort 猜想上也取得了重要进展。近年来同  A.  Javanpeyka, S. Lu 和孙锐然 继续在推进模空间的理论，特别是高维Shafarevich 纲领。

Video：http://archive.ymsc.tsinghua.edu.cn/pacm_lecture?html=Constructing_algebraic_solutions_of_Painleve_VI_equation_from_p-adic_Hodge_theory_and_Langlands_correspondence.html