[論文レビュー] 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 Theory参考文献 18被引用数 4

ひとこと要約

本稿では、射影的クリスタリン表現とねじれ周期的ヒッグス＝ド・ラーム流れを導入し、ねじれフォンテーヌン＝ファルティングス加群を介してそれらのカテゴリの同値性を確立する。$\mathbf{P}^1$ 上の $n \geq 4$ 個のマークド点を持つ安定的周期的ヒッグス bundle が、幾何的に絶対的単純な $\mathrm{PGL}_2(\mathbf{Z}_p^{\mathrm{ur}})$-クリスタリン表現を誘導することを証明し、4点の場合の明示的な自己写像公式を提示する。また、関連する楕円曲線上の torsion 点との関係を予想する。

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 $p$.

研究の動機と目的

• 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 $\bbP^1$ with logarithmic structure at $n \geq 4$ marked points.
• To construct infinitely many geometrically absolutely irreducible $\rm PGL_2(\bbZ_p^{\rm ur})$-crystalline representations of the étale fundamental group of $\bbP^1_{\bbQ_p^{\rm ur}}$ minus $n$ points.

提案手法

• 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 $\bbZ_p^{\rm ur}$-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 $\bbP^1$ with four marked points $\big\bracevert 0,1,\frown,\frown\big\bracevert$ using the torsion point conjecture.

実験結果

リサーチクエスチョン

• 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 $\bbP^1$ 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 $\bbP^1$ 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 $y^2 = x(x-1)(x-\frown)$?

主な発見

• 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 $\bbP^1$ with $n \geq 4$ marked points induces a geometrically absolutely irreducible crystalline representation of $\rm Gal(\bar{\bbQ}_p^{\rm ur}/\bbQ_p^{\rm ur})$.
• For the case of four marked points $\big\bracevert 0,1,\frown,\frown\big\bracevert$, an explicit formula for the Higgs-de Rham self-map on the moduli space is derived.
• The paper constructs infinitely many geometrically absolutely irreducible $\rm PGL_2(\bbZ_p^{\rm ur})$-crystalline representations of $\bbpi_1^{\text{ét}}(\bbP^1_{\bbQ_p^{\rm ur}} \backslash D)$ with $D = \big\bracevert x_1,\frown,\frown\big\bracevert$.
• 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 $\frown_\frown$.
• The self-map on the moduli space of rank-2 stable Higgs bundles of degree 1 on $\bbP^1$ with logarithmic structure is shown to be Frobenius semilinear and explicitly computable in the four-point case.