Projective crystalline representations of étale fundamental groups and twisted periodic Higgs–de Rham flow

Abstract

This paper contains three new results. (1) We introduce new notions of projective crystalline representations and twisted periodic Higgs–de Rham flows. These new notions generalize crystalline representations of étale fundamental groups introduced by Faltings [Algebraic Analysis, Geometry, and Number Theory (1989)] and Fontaine and Laffaille [Ann. Sci. Éc. Norm. Supér. (4) 15 (1983)] and periodic Higgs–de Rham flows introduced by Lan, Sheng and Zuo [J. Eur. Math. Soc. (JEMS) 21 (2019)]. We establish an equivalence between the categories of projective crystalline representations and twisted periodic Higgs–de Rham flows via the category of twisted Fontaine–Faltings module which is also introduced in this paper. (2) We study the base change of these objects over very ramified valuation rings and show that a stable periodic Higgs bundle gives rise to a geometrically absolutely irreducible crystalline representation. (3) We investigate the dynamic of self-maps induced by the Higgs–de Rham flow on the moduli spaces of rank-2 stable Higgs bundles of degree 1 on ${\mathbb{P}}^1$ with logarithmic structure on marked points $D:=\{x_1,\dots ,x_n\}$ for $n\ge 4$ and construct infinitely many geometrically absolutely irreducible ${\mathrm{PGL}}_2({\mathbb{Z}}_p^{ur}$)-crystalline representations of ${\pi }_1^{\text{eˊt}}({\mathbb{P}}_{{\mathbb{Q}}_p^{\mathrm{ur}}}^1\setminus D)$. We find an explicit formula of the self-map for the case $\{0,1,\infty ,\lambda \}$ and conjecture that a Higgs bundle is periodic if and only if the zero of the Higgs field is the image of a torsion point in the associated elliptic curve ${\mathcal{C}}_{\lambda }$ defined by $y^2=x(x-1)(x-\lambda )$ with the order coprime to $p$.