[Paper Review] Projective Crystalline Representations of Étale Fundamental Groups and Twisted Periodic Higgs-de Rham Flow

Ruiran Sun, Jinbang Yang+1 more|arXiv (Cornell University)|Sep 5, 2017

Algebraic Geometry and Number Theory18 references4 citations

TL;DR

This paper introduces projective crystalline representations and twisted periodic Higgs-de Rham flows, establishing an equivalence between their categories via twisted Fontaine-Faltings modules. It proves that stable periodic Higgs bundles over $bP^1$ with $n /geq 4$ marked points induce geometrically absolutely irreducible $ m PGL_2(bZ_p^{ m ur})$-crystalline representations, and provides an explicit self-map formula for four-point case, conjecturing a link to torsion points on associated elliptic curves.

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 $n\geq 4$ and construct infinitely many geometrically absolutely irreducible $\mathrm{PGL_2}(\mathbb Z_p^{\mathrm{ur}})$-crystalline representations of $π_1^ ext{et}(\mathbb{P}^1_{\mathbb{Q}_p^ ext{ur}}\setminus D)$. We find an explicit formula of the self-map for the case $\{0,\,1,\,\infty,\,λ\}$ 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 $\mathcal{C}_λ$ defined by $ y^2=x(x-1)(x-λ)$ with the order coprime to $p$.

Motivation & Objective

• To generalize crystalline representations and periodic Higgs-de Rham flows by introducing projective crystalline representations and twisted periodic Higgs-de Rham flows.
• To establish an equivalence between the category of projective crystalline representations and that of twisted periodic Higgs-de Rham flows via a new category of twisted Fontaine-Faltings modules.
• To study base change of these objects over very ramified valuation rings and prove that stable periodic Higgs bundles give rise to geometrically absolutely irreducible crystalline representations.
• To analyze the dynamics of Higgs-de Rham flow self-maps on moduli spaces of rank-2 stable Higgs bundles on $bP^1$ with logarithmic structure at $n /geq 4$ marked points.
• To construct infinitely many geometrically absolutely irreducible $ m PGL_2(bZ_p^{ m ur})$-crystalline representations of the étale fundamental group of $bP^1_{bQ_p^{ m ur}}$ minus $n$ points.

Proposed method

• Introduce twisted Fontaine-Faltings modules with endomorphism structures as a bridge between crystalline representations and Higgs-de Rham flows.
• Define projective crystalline representations as projective systems of representations with compatible Frobenius and filtration structures.
• Construct twisted periodic Higgs-de Rham flows via iterated application of the inverse Cartier functor and grading functors on filtered de Rham bundles.
• Use the inverse Cartier functor and Frobenius pullback to relate Higgs bundles to de Rham bundles, enabling the construction of self-maps on moduli spaces.
• Apply the $bZ_p^{ m ur}$-linear structure and base change over very ramified valuation rings to lift and stabilize the flow structures.
• Derive an explicit formula for the self-map on the moduli space of rank-2 Higgs bundles on $bP^1$ with four marked points $igracevert 0,1,rown,rownigracevert$ using the torsion point conjecture.

Experimental results

Research questions

• RQ1 How can crystalline representations of étale fundamental groups be generalized to include projective structures and twisted periodicity?
• RQ2 What is the precise categorical equivalence between projective crystalline representations and twisted periodic Higgs-de Rham flows?
• RQ3 Under what conditions does a stable periodic Higgs bundle over $bP^1$ with logarithmic structure give rise to a geometrically absolutely irreducible crystalline representation?
• RQ4 What is the dynamical behavior of the Higgs-de Rham flow self-map on the moduli space of rank-2 stable Higgs bundles of degree 1 on $bP^1$ with $n /geq 4$ marked points?
• RQ5 Is a Higgs bundle periodic if and only if the zero of its Higgs field corresponds to a torsion point of order coprime to $p$ on the associated elliptic curve $y^2 = x(x-1)(x-rown)$?

Key findings

• The category of projective crystalline representations is equivalent to the category of twisted periodic Higgs-de Rham flows via the category of twisted Fontaine-Faltings modules.
• A stable periodic Higgs bundle on $bP^1$ with $n /geq 4$ marked points induces a geometrically absolutely irreducible crystalline representation of $ m Gal(ar{bQ}_p^{ m ur}/bQ_p^{ m ur})$.
• For the case of four marked points $igracevert 0,1,rown,rownigracevert$, an explicit formula for the Higgs-de Rham self-map on the moduli space is derived.
• The paper constructs infinitely many geometrically absolutely irreducible $ m PGL_2(bZ_p^{ m ur})$-crystalline representations of $bpi_1^{ ext{ét}}(bP^1_{bQ_p^{ m ur}} ackslash D)$ with $D = igracevert x_1,rown,rownigracevert$.
• The authors conjecture that a Higgs bundle is periodic if and only if the zero of the Higgs field is the image of a torsion point of order coprime to $p$ on the associated elliptic curve $rown_rown$.
• The self-map on the moduli space of rank-2 stable Higgs bundles of degree 1 on $bP^1$ with logarithmic structure is shown to be Frobenius semilinear and explicitly computable in the four-point case.