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

Speaker:Prof. Zuo Kang(Wuhan University)
Venue:Zoom Meeting ID: 271 534 5558 Passcode: YMSC


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.


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。