Projective Crystalline Representations of Etale Fundamental Groups and Twisted Periodic Higgs-de Rham Flow

This paper contains three new results. {bf 1}.We introduce new notions of projective crystalline representations and twisted periodic Higgs-de Rham flows. These new notions generalize crystalline representations of etale fundamental groups introduced in [7,10] and periodic Higgs-de Rham flows introduced in [19]. 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. {bf 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. {bf 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 $D:={x_1,,...,x_n}$ for $ngeq 4$ and construct infinitely many geometrically absolutely irreducible $mathrm{PGL_2}(mathbb Z_p^{mathrm{ur}})$-crystalline representations of $pi_1^text{et}(mathbb{P}^1_{{mathbb{Q}}_p^text{ur}}setminus D)$. 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 $p$.