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 $D:=\{x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n\}$ for $n\ge 4$ and construct infinitely many geometrically absolutely irreducible $$\mathrm{P}\mathrm{G}{\mathrm{L}}_2$(\mathbb Z_p^{$\mathrm{u}\mathrm{r}$})$-crystallinerepresentationsof$\pi_1^$\text{et}$($\mathbb{P}$^1_{{$\mathbb{Q}$}_p^$\text{ur}$}\setminus D)$. We find an explicit formula of the self-map for the case $\{0,1,\mathrm{\infty },\lambda \}$ 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 $p$.