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

Abstract

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 étale 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 $n\ge 4$ and construct infinitely many geometrically absolutely irreducible $\mathrm{PG}{\mathrm{L}}_2({\mathbb{Z}}_p^{\mathrm{ur}})$-crystalline representations of ${\pi }_1^{\text{et}}({\mathbb{P}}_{{\mathbb{Q}}_p^{\text{ur}}}^1\setminus D)$. 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 $p$.

84 pages

References in corpus (1)

• Logarithmic nonabelian Hodge theory in characteristic p

Cited by in corpus (3)

• Nearby cycles and semipositivity in positive characteristic
• Base change of twisted Fontaine-Faltings modules and Twisted Higgs-de Rham flows over very ramified valuation rings
• Finiteness of logarithmic crystalline representations II