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\geq 4 and construct infinitely many geometrically absolutely irreducible \operatorname{PGL}_2(\mathbb{Z}_{p}^{ur} )-crystalline representations of {\pi_1^{\textup{\'{e}t}}(\mathbb{P}^{1}_{\mathbb{Q}_p^{\mathrm{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 .

Higgs Bundles in Geometry and Arithmetic

“…We confirmed this conjecture via explicit computation for cases that the characteristic p ≤ 50, see [SYZ17] Note that self-maps on P 1 descended from n-multiplication by maps on an elliptic curve π : E → P 1 (n ∈ Z) are called Lattès maps ([Mil06]). If we vary λ ∈ P 1 \ {0, 1, ∞}, we obtain a family of self-maps ψ λ .…”

“…In [SYZ17] one considers rank-2 stable logarithmic Higgs bundles over the projective line (P 1 , S) with n punctures and with the parabolic structure attached to one puncture in S of weight {1/2, 1/2}. The moduli space HIG contains irreducible components of dimensions n − 3, n − 5, n − 7, • • • .…”

“…p-adic Nonabelian Hodge Theory and Dynamical Systems over F p and Mixed Characteristic. Conjecture 11 is motivated by Sun-Yang-Zuo's work [SYZ17]. The Higgs-de Rham flow on (P 1 , {0, 1, ∞, λ}) induces a self map ψ λ on the moduli space HIG (≃ P 1 Fq ) of rank-2 logarithmic Higgs bundles with the parabolic structure of the weight (1/2, 1/2) attched to a puncture.…”

On irreducible symplectic varieties of K3[]-type in positive characteristic

“…In [SYZ,Section 2] the authors extend Fontaine-Laffaille-Faltings' functor D so that they are able to obtain some projective representations of π ét 1 (X ⊗ W K)…”

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

“…Conjecture 0.6 (Sun-Yang-Zuo [SYZ17]). The self-map induced by Higgs-de Rham flow on M gr 1 2 Higλ over F q comes from multiplication map by p on the associated elliptic curve over…”