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

TL;DR

This paper constructs algebraic solutions to the Painleve VI equation using p-adic Hodge theory and Langlands correspondence, linking algebraic geometry, number theory, and differential equations.

Contribution

It introduces a novel method to generate algebraic solutions of Painleve VI from p-adic Hodge theory and characterizes motivic Higgs bundles via fixed points of a self-map.

Findings

Construction of infinitely many non-isotrivial abelian families

Algebraic solutions to Painleve VI derived from these families

Complete characterization of motivic Higgs bundles in moduli space

Abstract

We construct infinitely many non-isotrivial families of abelian varieties over given four punctured projective lines. These families lead to algebraic solutions of Painleve VI equation. Finally, based on a recent paper by Lin-Sheng-Wang, we prove a complete characterization for the locus of motivic Higgs bundles in the moduli space as fixed points of an ``additive'' self-map. This is a note based on the lecture given by the second named author on 04 Nov. 2022 at Tsinghua University.