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 $D:=\{x_1,\,...,x_n\}$ for $n\geq 4$ and construct infinitely many geometrically absolutely irreducible $\mathrm{PGL_2}(\mathbb Z_p^{\mathrm{ur}})$-crystalline representations of $\pi_1^\text{et}(\mathbb{P}^1_{{\mathbb{Q}}_p^\text{ur}}\setminus D)$. We find an explicit formula of the self-map for the case $\{0,\,1,\,\infty,\,\lambda\}$ 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 $p$.

The nonabelian Hodge theory established by Hitchin and Simpson associates representations of the topological fundamental group of an algebraic variety X over C to a holomorphic object on X named Higgs bundle. Later, Ogus and Vologodsky established the nonabelian Hodge theory in positive characteristic in their spectacular work [19]. They constructed the Cartier functor and the inverse Cartier functor, which give an equivalence of categories between the category of nilpotent Higgs modules and the category of nilpotent flat modules over a smooth proper and W 2 (k)-liftable variety. This equivalence generalizes the classical Cartier descent theorem. Moreover, it is the starting point of the theory of Higgs-de Rham flows in [12] To attach representations of the fundamental group to these O X -linear objects on X, one still needs an analogue of the classical Riemann-Hilbert correspondence. Unfortunately, there is no direct generalization of Riemann-Hilbert correspondence in the characteristic p case. However, in the p-adic case, a good p-adic analogue of the category of polarized complex variations of Hodge structures, the category MF ∇ [a,b] (X /W ), is introduced by Fontaine and Laffaille [6] for X = Spec W (k) in their study of p-adic Hodge theory. The theory of Fontaine and Laffaile was later generalized by Faltings [4] to the general geometric base case. The objects in this theory are called Fontaine modules and consists of a quadruple (V, ∇, Fil, ϕ), where (V, ∇, Fil) is a filtered de Rham bundle over X and ϕ is a relative Frobenius which is horizontal with respect to ∇ and satisfies the strong p-divisibility condition. The latter condition is a p-adic analogue of the Riemann-Hodge bilinear relations. Then the Fontaine-Laffaille-Faltings correspondence gives a fully faithful functor from MF ∇ [0,w] (X /W ) (w ≤ p − 2) to the category of crystalline representations of π 1 (X K ), where X K is the generic fiber of X . This can be regarded as a p-adic version of the Riemann-Hilbert correspondence.
Faltings [5] has established an equivalence of categories between the category of generalized representations of the geometric fundamental group and the category of Higgs bundles over a p-adic curve, which has generalized the earlier work of Deninger-Werner [2] on a partial p-adic analogue of Narasimhan-Seshadri theory.
In order to establish a p-adic analogue of the Hitchin-Simpson correspondence between the category of representations with small coefficients, namely GL r (W n (F q ))-representations, and the category of Higgs bundles, Lan, Sheng and Zuo introduced the notion of Higgs-de Rham flows, which can be considered as an analogue of the Yang-Mills-Higgs flow attached to a Higgs bundle in the complex analytic situation. The latter is used to solve the Yang-Mills-Higgs equation. The stability of the Higgs bundles guarantees the existence of solutions, the so-called Yang-Mills-Higgs connections. Roughly speaking, a Higgs-de Rham flow is a sequence of graded Higgs bundles and filtered de Rham bundles, connected by the inverse Cartier transform defined by the fundamental work of Ogus and Vologodsky and the grading functor by the attached Hodge filtrations on the de Rham bundles (for details see Section 3 in [12] or Section 3.1 in this paper). The following diagram presents a Higgs-de Rham flow over X 1 ( X 1 is the special fiber of X ): A Higgs-de Rham flow is said to be periodic (of period f ∈ N) if f is the smallest integer such that there exists an isomorphism of Higgs bundles φ : (E f , θ f ) ∼ = (E 0 , θ 0 ). For Higgs bundles over X n /W n (k), the key point of defining the Higgs-de Rham flow is to find the suitable lifting C −1 n of C −1 1 over the truncated Witt ring W n (k). This was done by Lan-Sheng-Zuo in section 4 of [12]. Because of the importance of it in lifting the Higgs-de Rham flow to X /W (k), we recall the construction of functor C −1 n briefly in Section 2.
Theorem 0.1 (Theorem 1.4 in [12]). Let X be a smooth proper scheme over W . For each integer 0 ≤ w ≤ p−2 and each f ∈ N, there is an equivalence of categories between the category of p-torsion free Fontaine-Faltings modules over X of Hodge-Tate weight ≤ m with endomorphism structure W (F p f ) and the category of periodic Higgs-de Rham flows over X of level ≤ w and whose periods are f .
Remark. It is straightforward to generalize Theorem 0.1 to the logarithmic setting. To be more precise, let X be a smooth proper scheme over W and let D ⊂ X be a simple normal crossings divisor relative to W . Then, for each positive integer f , there is an equivalence of categories between the category of strict p n -torsion logarithmic Fontaine modules (with logarithmic structure along D × W n ⊂ X × W n ) with endomorphism structure of W n (F p f ) whose Hodge-Tate weights ≤ p − 2 and the category of periodic logarithmic Higgsde Rham flows over X × W n (with logarithmic structure along D × W n ⊂ X × W n ) whose periods are factors of f and nilpotent exponents are ≤ p − 2.
By Theorem 6.6 in [12], a periodic Higgs bundle must have trivial Chern classes. This fact limits the application of the p-adic Hitchin-Simpson correspondence. For instance, Simpson constructed a canonical Hodge bundle Ω 1 X ⊕ O X on X in his proof of the Miyaoka-Yau inequality (Proposition 9.8 and Proposition 9.9 in [21]), which has nontrivial Chern classes in general. In fact, the classical nonabelian Hodge theorem tells us that the Yang-Mills-Higgs equation is still solvable for a polystable Higgs bundle with nontrivial Chern classes. Instead of getting a flat connection, one can get a projective flat connection in this case, whose monodromy gives a PGL rrepresentation of the fundamental group. This motivates us to find a p-adic Hitchin-Simpson correspondence for graded Higgs bundles with nontrivial Chern classes.
A projective flat connection ∇ on a bundle V over C is a (usual) connection whose curvature has the special form Θ = ω ⊗ Id V , where ω is a rational closed (1, 1)-form representing 1 rk(V ) c 1 (V ). Note that, if [ω] ∈ H 2 (X, Z), then by the Lefschetz theorem on (1, 1)-classes one can actually find a line bundle L with a metric connection ∇ L such that (V, ∇) ⊗ (L, ∇ L ) ∨ becomes a flat bundle. Inspired by this we first introduce the 1-periodic twisted Higgs-de Rham flow over X 1 as follows Here L is called a twisting line bundle on X 1 , and φ L : On the Fontaine module side, we also introduce the twisted Fontaine-Faltings module over X 1 . The latter consists of the following data: a filtered de Rham bundle (V, ∇, Fil) together with an isomorphism between de Rham bundles: . We will refer to the isomorphism ϕ L as the twisted ϕ-structure. The general construction of twisted Fontaine-Faltings modules and twisted periodic Higgs-de Rham flows are given in Section 1.5 and Section 3.2 (over X n /W n (k), and multi-periodic case).
Theorem 0.2 (Theorem 3.3). Let X be a smooth proper scheme over W . For each integer 0 ≤ a ≤ p − 2 and each f ∈ N, there is an equivalence of categories between the category of all twisted f -periodic Higgs-de Rham flows over X n of level ≤ a and the category of strict p n -torsion twisted Fontaine-Faltings modules over X n of Hodge-Tate weight ≤ a with an endomorphism structure of W n (F p f ).
Theorem 0.2 can be generalized to the logarithmic case. The precise statement is as follows.
Theorem 0.3 (Theorem 3.4). Let X be a smooth proper scheme over W with a simple normal crossing divisor D ⊂ X relative to W . Then for each natural number f ∈ N, there is an equivalence of categories between the category of strict p n -torsion twisted logarithmic Fontaine-Faltings modules (with pole along D×W n ⊂ X ×W n ) with endomorphism structure of W n (F p f ) whose Hodge-Tate weight ≤ p − 2 and the category of twisted f -periodic logarithmic Higgs-de Rham flows (with pole along D × W n ⊂ X × W n ) over X × W n whose nilpotent exponents are ≤ p − 2.
One of our goals is to associate a PGL n -representation of π 1 to a twisted (logarithmic) Fontaine-Faltings module. To do so, we will need to generalize Faltings's work. Following Faltings [4], we construct a functor D P in section 2.5, which associates to a twisted (logarithmic) Fontaine-Faltings module a PGL n representation of theétale fundamental group.
Theorem 0.4 (Theorem 2.10). Let X be a smooth proper geometrically connected scheme over W with a simple normal crossing divisor D ⊂ X relative to W . Suppose F p f ⊂ k. Let M be a twisted logarithmic Fontaine-Faltings module over X (with pole along D) with endomorphism structure of W (F p f ). Applying D P -functor, one gets a projective representation In Section 3.4, we study several properties of this functor D P . For instance, we prove that a projective sub-representation of D P (M ) corresponds to a sub-object N ⊂ M such that D P (M/N ) is isomorphic to this subrepresentation. Combining this with Theorem 3.3, we infer that a projective representation coming from a stable twisted periodic Higgs bundle (E, θ) with (rank(E), deg H (E)) = 1 must be irreducible.
The next theorem gives a p-adic analogue of the existence of projective flat Yang-Mills-Higgs connection in terms of semistability of Higgs bundles and triviality of discriminant.
Theorem 0.5 (Theorem 3.10). A semistable Higgs bundle over X 1 initials a twisted preperiodic Higgs-de Rham flow if and only if it is semistable and has trivial discriminant.
Consequently we obtain the following theorem on the existence of nontrivial representations ofétale fundamental group in terms of the existence of semistable graded Higgs bundles.
Theorem 0.6 (Theorem 3.14). Let k be a finite field of characteristic p. Let X be a smooth proper geometrically connected scheme over W (k) together with a smooth log structure D/W (k). Assume that there exists a semistable graded logarithmic Higgs bundle Finally we give two applications in Section 4 to show how the general machinery developed in the previous sections works in some concrete situations. Taking the moduli space M of graded stable Higgs bundles of rank-2 and degree 1 over P 1 with logarithmic structure on m(> 3) marked points we show that the self map induced by Higgs-de Rham flow stabilizes the component M (1, 0) of M of maximal dimension (dim = m−3 ) as a rational and dominant map. Hence by Hrushovski's theorem [8] the subset of periodic Higgs bundles is Zariski dense in M (1, 0). In this way we produce infinitely many PGL 2 (F p f )-crystalline representations, which are irreducible in PGL 2 (F p ). By Theorem 3.14, all these representations lift to PGL 2 (Z ur p )crystalline representations. For the case of 4 marked points {0, 1, ∞, λ} we state an explicite formula for the self map and use it to study the dynamic of Higgs-de Rham flows for p = 3 and several values of λ. In the last subsection 4.5, we consider a smooth projective curve X over W (k) of genus g ≥ 2. In the Appendix of [20], de Jong and Osserman have shown that the subset of twisted periodic vector bundles over X 1 in the moduli space of semistable vector bundles over X 1 of any rank and any degree is always Zariski dense. By applying our main theorem for twisted periodic Higgs de Rham flows with zero Higgs fields, which should be regarded as projectiveétale trivializible vector bundles in the projective version of Lange-Stuhe's theorem (see [14]), they all correspond to PGL r (F p f )-representations of π 1 (X 1 ). Once again we show that they all lift to PGL r (Z ur p ) of π 1 (X 1 ). It should be very interesting to make a comparison between the lifting theorem obtained here lifting GL r (F p f )-representations of π 1 (X 1 ) to GL r (Z ur p )-representation of π 1 (X 1F p ) and the lifting theorem developed by Deninger-Werner [2]. In their paper, they have shown that any vector bundle over X /W which isétale trivializible over X 1 lifts to a GL r (C p )representation of π 1 (X K ).

Twisted Fontaine-Faltings modules
In this section, we will recall the definition of Fontaine-Faltings modules in [4] and generalize it to the twisted version.
1.1. Fontaine-Faltings modules. Let X n be a smooth and proper variety over W n (k). And (V, ∇) is a de Rham sheaf (i.e. a sheaf with an integrable connection) over X n . In this paper, a filtration Fil on (V, ∇) will be called a Hodge filtration of level in [a, b] if the following conditions hold: and locally on all open subsets U ⊂ X n , the graded factor Fil -Fil satisfies Griffiths transversality with respect to the connection ∇.
In this case, the triple (V, ∇, Fil) is called a filtered de Rham sheaf. One similarly gives the conceptions of (filtered) de Rham modules over a Walgebra.
1.1.1. Fontaine-Faltings modules over a small affine base. Let U = SpecR be a small affine scheme ( which means there exist anétale map W n [T ±1 1 , T ±1 2 , · · · , T ±1 d ] → O Xn (U ), see [4]) over W and Φ : R → R be a lifting of the absolute Frobenius on R/pR, where R is the p-adic completion of R. -ϕ is an R-linear isomorphism V and ∇ on V , i.e. the following diagram commutes: Let M 1 = (V 1 , ∇ 1 , Fil 1 , ϕ 1 ) and M 2 = (V 2 , ∇ 2 , Fil 2 , ϕ 2 ) be two Fontaine-Faltings modules over U of Hodge-Tate weight in [a, b]. The homomorphism set between M 1 and M 2 constitutes by those morphism f : which is parallel with respect to the connection, satisfies the cocycle conditions and induces an equivalent functor of categories Morphisms between Fontaine-Faltings modules are those between sheaves and locally they are morphisms between local Fontaine-Faltings modules. More precisely, for a morphism f of the underlying sheaves of two Fontaine-Faltings modules over X , the map f is called a morphism of Fontaine- 1.2. Inverse Cartier functor. For a Fontaine-Faltings module (V, ∇, Fil, {ϕ i } i∈I ), we call {ϕ i } i the ϕ-structure of the Fontaine-Faltings module. In this section, we first recall a global description of the ϕ-structure via the inverse Cartier functor over truncated Witt rings constructed by Lan, Sheng and Zuo [12]. Note that the inverse Cartier functor C −1 1 (the characteristic p case) is introduced in the seminal work of Ogus-Vologodsky [19]. Here we sketch an explicit construction of C −1 1 presented in [12]. Let (E, θ) be a nilpotent Higgs bundle over X 1 . Locally we have is the homomorphism given by the Deligne-Illusie's Lemma [1]. Those local data (V i , ∇ i )'s can be glued into a global sheave H with integrable connection ∇ via the transition maps {G ij } (Theorem 3 in [13]). The inverse Cartier functor on (E, θ) is . Remark. Note that the inverse Cartier transform C −1 1 also has the logarithmic version. When the log structure is given by a simple normal crossing divisor, an explicit construction of the log inverse Cartier functor is given in the Appendix of [11].
As mentioned in the introduction, we need to generalize C −1 1 to the invers Cartier transform over the truncated Witt ring for Higgs bundles over X n /W m (k). We briefly recall the construction in section 4 of [12].
1.2.1. Inverse Cartier functor over truncated Witt ring. Let S = Spec(W(k)) and F S be the Frobenius map on S. Let X n+1 ⊃ X n be a W n+1 -lifting of smooth proper varieties. Recall that the functor C −1 n is defined as the composition of C −1 n and the base change F S : X ′ n = X n × F S S → X n (by abusing notation, we still denote it by F S ). The functor C −1 n is defined as the composition of two functors T n and F n . In general, we have the following diagram and its commutativity follows easily from the construction of those functors.
These categories appeared in the diagram are explained as following: • MCF a (X n ) is the category of filtered de Rham sheaves over X n of level in [0, a]. • H(X n ) (resp. H(X ′ n )) is the category of tuples (E, θ,V ,∇, F il, ψ), where -(E, θ) is a graded Higgs module over X n (resp. X ′ n = X n ⊗ σ W ) of exponent ≤ p − 2; -(V ,∇, F il) is a filtered de Rham sheaf over X n−1 (resp. over X ′ n−1 ); -and ψ : GrF il (V ,∇) ≃ (E, θ) ⊗ Z/p n−1 Z is an isomorphism of Higgs sheaves over X n (resp. X ′ n ). • MIC(X n ) (resp. MIC(X ′ n )) is the category of sheaves over X n (resp. X ′ n ) with integrable p-connection . • MIC(X n ) (resp. MIC(X ′ n )) is the category of de Rham sheaves over X n (resp. X ′ n ). Functor Gr. For an object (V, ∇, Fil) in MCF p−2 (X n ), the functor Gr is given by where (E, θ) = Gr(V, ∇, Fil) is the graded sheaf with Higgs field, (V , ∇, F il) is the modulo p n−1 -reduction of (V, ∇, Fil) and ψ is the identifying map Faltings tilde functor (·). For an object (V, ∇, Fil) in MCF p−2 (X n ), the (V, ∇, Fil) will be denoted as the quotient Fil i / ∼ with x ∼ py for any The construction of functor T n . Let (E, θ,V ,∇, F il, ψ) be an object in H(X n ) (resp. H(X ′ n )). Locally on an affine open subset U ⊂ X (resp. U ⊂ X ′ ), there exists (V U , ∇ U , Fil U ) (Lemma 4.6 in [12]), a filtered de Rham sheaf, such that The tilde functor associates (V U , ∇ U , Fil U ) to a sheaf with p-connection over U . By gluing those sheaves with p-connections over all U 's (Lemma 4.10 in [12]), one gets a global sheaf with p-connection over X n (resp. X ′ n ). Denote it by T n (E, θ,V ,∇, F il, ψ).
the construction of functor F n . For small affine open subset U of X , there exists endomorphism F U on U which lifts the absolute Frobenius on U k and is compatible with the Frobenius map F S on S = Spec(W (k)). Thus there Locally on U , applying functor F * U /S , we get a de Rham sheaf over U . By Taylor formula, up to a canonical isomorphism, it does not depends on the choice of F U . In particular, on the overlap of two small affine open subsets, there is an canonical isomorphism of two de Rham sheaves. By gluing those isomorphisms, one gets a de Rham sheaf over X n , we denote it by 1.3. Global description of the ϕ-structure in Fontaine-Faltings modules (via the inverse Cartier functor). Let (V, ∇, Fil) ∈ MFC p−2 (X n ) be a filtered de Rham sheaf over X n of level in [0, p − 2]. From the commutativity of diagram (1.3), for any i ∈ I, one has (1.5) As the F n is glued by using the Taylor formula, for any i, j ∈ I, one has the following commutative diagram To give a system of compatible ϕ-structures (for all i ∈ I) . In particular, we have the following results , for some positive integer n; is an isomorphism of de Rham sheaves.

1.4.
Fontaine-Faltings modules with endomorphism structure. Let f be a positive integer. We call (V, ∇, Fil, ϕ, ι) a Fontaine-Faltings module over X with endomorphism structure of W (F p f ) whose Hodge-Tate weights is a continuous ring homomorphism. We call ι an endomorphism structure (for some n ∈ N) together with isomorphisms of de Rham sheaves and Comparing σ i+1 (ξ)-eigenspaces of ι(ξ) on both side of . Conversely, we can construct the Fontaine-Faltings module with endomorphism structure in an obvious way.
1.5. Twisted Fontaine-Faltings modules with endomorphism structure. Let L n be a line bundle over X n . Then there is a natural connection ∇ can on L p n n by 5.1.1 in [10]. Tensoring with (L p n n , ∇ can ) induces a self equivalence functor on the category of de Rham bundles over X n . Definition 1.3. An L n -twisted Fontaine-Faltings module over X n with endomorphism structure of W n (F p f ) whose Hodge-Tate weights lie in [a, b] is a tuple consisting the following data: to denote the category of all twisted Fontaine-Faltings modules over X n with endomorphism structure of W n (F p f ) whose Hodge-Tate weights lie in [a, b].

A morphism between two objects (V
is equivalent to give a strict p n -torsion Fontaine-Faltings module over X n with endomorphism structure of W n (F p f ) and whose Hodge- It induces a trivialization of flat bundle τ p n j : be an L n -twisted Fontaine-Faltings module over X n with endomorphism structure of W n (F p f ) whose Hodge-Tate weights lie in [a, b]. Then one gets a local Fontaine-Faltings module over R j with endomorphism structure of W n (F p f ) whose Hodge-Tate weights lie in [a, b] We call M (τ j ) the trivialization of of M on U j via τ j . Logarithmic version. Finally, let us mention that everything in this section extends to the logarithmic context. Let X be a smooth and proper scheme over W and X o is the complement of a simple normal crossing divisor D ⊂ X relative to W . Similarly, one constructs the category T MF ∇ [a,b],f (X o n+1 /W n+1 ) of strict p n -torsion twisted logarithmic Fontaine modules (with pole along D × W n ⊂ X × W n ) with endomorphism structure of W n (F p f ) whose Hodge-Tate weights lie in [a, b].

Projective Fontaine-Laffie-Faltings functor
The functor D Φ . Let R be a small affine algebra over W = W (k) with a σ-linear map Φ : R → R which lifts the absolute Frobenius of R/pR. If Φ happens to beétale in characteristic 0, Faltings (page 36 of [4]) constructed a map κ Φ : R → B + ( R) which respects Frobenius-lifts. Thus the following diagram commutes which is equipped with the natural ϕ-structure and filtration.
where the homomorphisms are B + ( R)-linear and respect filtrations and the is defined via the connection on V , which commutes with the ϕ's and hence induces an For each i ∈ I, the functor D Φ i associates to any Fontaine-Faltings module over X a compatible system ofétale sheaves on U i,K (the generic fiber of U i ). By gluing and using the results in EGA3, one obtains a locally constant sheaf on X K and a globally defined functor D.
In the following, we give a slightly different way to construct the functor D. Let J be a finite subset of the index set I, such that {U j } j∈J forms a covering of X .
Let (V, ∇, Fil, {ϕ i } i∈I ) be a Fontaine-Faltings module over X . For each j ∈ J, the functor D Φ j gives us a finite Z p -representation of π 1 ( U j , x). Recall that the functor D Φ does not depends on the choice of Φ, up to a canonical isomorphism. In particular, for all j 1 , j 2 ∈ J, there is a natural isomorphism of By Theorem 2.6, all representations D(V (U j ), ∇, Fil, ϕ j )'s descend to a Z prepresentations of π 1 (X K , x). Up to a canonical isomorphism, this representation does not depend on the choice of J and s. This representation is just D(V, ∇, Fil, {ϕ i } i∈I ) and we construct the Fontaine-Laffaille-Faltings' D-functor in this way.
Theorem 2.1 (Faltings). The functor D induces an equivalence of the ]modules whose objects are dual-crystalline representations. This subcategory is closed under sub-objects and quotients. In this case, we call the pair  Thus, there is a natural restriction functor res from the category of π 1 (S, s)sets to the category of π 1 (U, s)-sets, which is given by  The second one follows the very definition, one can find the proof in 5.1 of [18] As a consequence, one has the following result, which should be well-known for the experts. We still give a proof for the reader's convenience. Proof. According to Proposition 2.4, we have the following commutative diagram, with two bijective horizontal maps F s .
The fully faithful of the restriction functor follows from Corollary 2.3. Under a fully faithful functor, a morphism is an isomorphism if and only if its image under this functor is an isomorphism. So the corollary 2.5 follows.
In the following, we fix a finite index set J and an open covering {U j } j∈J of S with s ∈ j U j . Then for any j ∈ J, the inclusion map U j → S induces a surjective group morphism of fundamental groups τ j : π 1 (U j , s) ։ π 1 (S, s).
Theorem 2.6. Let (Σ j , ρ j ) be a finite π 1 (U j , s)-set for each j ∈ J. Suppose for each pair i, j ∈ J, there exists an isomorphism of π 1 (U ij , s)-sets η ij : Σ i ≃ Σ j . Then every Σ j descends to a π 1 (S, s)-set (Σ j , ρ j ) uniquely. Moreover, the image of ρ j equals that of ρ j .
Proof. Fix j 0 ∈ J. One has an isomorphism of π 1 (U jj 0 , s)-sets η jj 0 : Σ j ≃ Σ j 0 . As F s is an equivalent functor, there is a covering of U j of U j with isomorphism η j : F s ( U j ) → Σ j of π 1 (U j , s)-sets for each j ∈ J. Denote The η j and η jj 0 are π 1 (U J , s)-isomorphisms, so do f j . Denote The equivalence of F s over U J induces the following commutative diagram of finiteétale coverings of U J .
, , In particular, for all j 1 , j 2 , j 3 ∈ J, By Corollary 2.5, there is a unique isomorphism of finiteétale coverings of Using Corollary 2.5 once again, one has So one can glue { U j } j∈J by isomorphisms {f ij } i,j∈J into a finiteétale covering S of S. Applying the fiber functor F s on the structure isomorphisms The bijections F s (f j ) and η j give us isomorphisms of permutation groups Since the F s (f j ) and η j are isomorphisms of π 1 (U j , s)-sets, the following diagram commutes Let ρ j denote the composition The commutativity of diagram (2.4) means that ρ i descends to ρ j . Other statements can be easily deduced from the surjectivity of τ j and τ ij .

2.4.
Comparing representations associated to local Fantaine-Faltings modules underlying isomorphic filtered de Rham sheaves. In this section we compare several representations associated to local Fontaine-Faltings modules underlying isomorphic filtered de Rham sheaves. To do so, we first introduce a local Fontaine-Faltings module, which corresponds to a W n (F p f )-character of the local fundamental group. We will then use this character to measure the difference of the associated representations.
Let R be a small affine algebra over W (k) and denote R n = R/p n R for all n ≥ 1. Fix a lifting Φ : R → R of the absolute Frobenius on R/pR. Recall that κ Φ : Under such a lifting, the Frobenius Φ B on B + ( R) extends to Φ on R. Element a n,r . Let f be an positive integer. For any r ∈ R × , we construct a Fontaine-Faltings module of rank f as following. Let be a free R n -module of rank f . The integrable connection ∇ on V is defined by formula ∇(e i ) = 0, and the filtration Fil on V is the trivial one. Applying the tilde functor and twisting by the map Φ, one gets The ϕ is parallel due to d(r p n ) ≡ 0 (mod p n ). By lemma 1.1, the tuple (V, ∇, Fil, ϕ) forms a Fontaine-Faltings module. Applying Fontaine-Laffaille-Faltings' functor D Φ , one gets a finite Z p -representation of Gal( R/ R), which is a free Z/p n Z-module of rank f .

Lemma 2.7.
Let n and f be two positive integers and let r be an invertible element in R. Then there exists an a n,r ∈ B + ( R) × such that Φ f B (a n,r ) ≡ κ Φ (r) p n · a n,r (mod p n ). (2.5) Proof. Since D Φ (V, ∇, Fil, ϕ) is free over Z/p n Z of rank f . one can find an element g with order p n . Recall that D Φ (V, ∇, Fil, ϕ) is the sub-Z p -module of Hom B + ( R) (V ⊗ κ Φ B + ( R), D) consisted by elements respecting the filtration and ϕ. In particular, the following diagram commutes Since the image of g is p n -torsion, Im(g) is contained in D[p n ] = 1 p n B + ( R)/B + ( R), the p n -torsion part of D. Choose a lifting a n,r of g(e 0 ⊗ κ Φ 1) under the sur- Then the equation (2.5) follows. Similarly, one can define a n,r −1 for r −1 . By equation (2.5), we have Φ f (a n,r · a n,r −1 ) = a n,r · a n,r −1 .
Thus a n,r · a n,r −1 ∈ W (F p f ). Since both a n,r and a n,r −1 are not divided by p (by the choice of g), we know that a n,r · a n,r −1 ∈ W (F p f ) × . The invertibility of a n,r follows.
Comparing representations. Let n and f be two positive integers. For all be isomorphisms of de Rham R-modules. Let r be an element in R × . Since d(r p n ) = 0 (mod p n ), the map r p n ϕ f −1 is also an isomorphism of de Rham ii). The multiplication of a n,r on Hom B + ( R) V ⊗ κ Φ B + ( R), D induces a W n (F p f )-linear map between these two submodules Proof. i). We only give the W n (F p f )-linear structure on D Φ (M ). Let g : One checks that a.g is also contained in D Φ j (M (τ j )). Let δ be an element in Gal( R/ R). Then In this way, D Φ j (M (τ j )) forms a W n (F p f )-module with a continuous semilinear action of π 1 (U K ). ii). Recall that D Φ (M ) (resp. D Φ (M ′ )) is defined to be the set of all morphisms in Hom B + ( R) V ⊗ κ Φ B + ( R), D compatible with the filtration and ϕ (resp. ϕ ′ ). Comparing the rank of D Φ (M ) and D Φ (M ′ ), we only need to show that a n, , which means that f satisfies the following two conditions: 1). f is strict for the filtrations. i.e.
is an isomorphism of projective W n (F p f )-representations of Gal( R/ R). In particular, we have an bijection of Gal( R/ R)-sets 2.5. The functor D P . In this section, we assume f to be a positive integer with F p f ⊂ k. Let {U j } j∈J be a finite small affine open covering of X . Let U j = (U j ) K . For every j ∈ J, fix Φ j as a lifting of the absolute Frobenius on U j ⊗ W k. Fix x as a geometric point in U J = j∈J U j and fix j 0 an element in J. Let (V, ∇, Fil, ϕ, ι) be a Fontaine-Faltings module over X n with an endomorphism structure of W (F p f ) whose Hodge-Tate weights lie in [0, p−2]. Locally, Applying Fontaine-Laffaille-Faltings' functor D Φ j , one gets a finite W n (F p f )representation ̺ j of π 1 (U j , x). Faltings shows that there is an isomorphism ̺ j 1 ≃ ̺ j 2 of Z/p n Z-representations of π 1 (U j 1 j 2 , x). By Lan-Sheng-Zuo [12], this isomorphism is W n (F p f )-linear. By Theorem 2.6, these ̺ j 's uniquely descend to a W n (F p f )-representation of π 1 (X K , x). Thus one reconstructs the W n (F p f )-representation D(V, ∇, Fil, ϕ, ι) in this way. Now we construct functor D P for twisted Fontaine-Faltings modules, in a similar way. Let ,f (X n+1 /W n+1 ) be an L n -twisted Fontaine-Faltings module over X n with endomorphism structure of W n (F p f ) whose Hodge-Tate weights lie in [0, p−2]. For each j ∈ J, choosing a trivialization M (τ j ) and applying Fontaine-Laffaille-Faltings' functor D Φ j , we get a W n (F p f )-module together with a linear action of π 1 (U j , x). Denote its projectification by ̺ j . By Corollary 2.9, there is an isomorphism ̺ j 1 ≃ ̺ j 2 as projective W n (F p f )-representations of π 1 (U j 1 j 2 , x). In what follows, we will show that these ̺ j 's uniquely descend to a projective W n (F p f )-representation of π 1 (X K , x) by using Theorem 2.6. In order to use Theorem 2.6, set Σ j to be the quotient π 1 (U j , x)-set Obviously the kernel of the canonical group morphism Let's denote by ρ j the composition of ̺ j and GL D Φ j (M (τ j )) → Aut(Σ j ) for all j ∈ J.
By Corollary 2.9, the restrictions of (Σ j 1 , ρ j 1 ) and (Σ j 2 , ρ j 2 ) on π 1 (U j 1 j 2 , x) are isomorphic for all j 1 , j 2 ∈ J. Hence by Theorem 2.6, the map ρ j 0 descends to some ρ j 0 and the image of Up to a canonical isomorphism, this projective representation does not depends on the choices of the covering {U j } j∈J , the liftings Φ j 's and j 0 . And we denote this projective W n (F p f )-representation of π 1 (X K , x) by Similarly as Faltings's functor D in [4], our construction of the D P functor can also be extended to the logarithmic version. More precisely, let X be a smooth and proper scheme over W and let X o be the complement of a simple normal crossing divisor D ⊂ X relative to W . Similarly, by replacing X K and U j with X o K = X K and U o j , we construct the functor from the category of strict p n -torsion twisted logarithmic Fontaine modules (with pole along D × W n ⊂ X × W n ) with endomorphism structure of W n (F p f ) whose Hodge-Tate weights lie in [0, p − 2] to the category of free W n (F p f )-modules with projective actions of π 1 (X o K ). Summarizing this section, we get the following result.
Theorem 2.10. Let M be a twisted logarithmic Fontaine-Faltings module over X (with pole along D) with endomorphism structure of W (F p f ). Te D P -functor associates to M and its endomorphism structure n a projective representation ρ :

Twisted periodic Higgs-de Rham flows
In this section, we will recall the definition of periodic Higgs-de Rham flows and generalize it to the twisted version.
3.1. Higgs-de Rham flow over X n ⊂ X n+1 . Recall [12] that a Higgsde Rham flow over X n ⊂ X n+1 is a sequence consisting of infinitely many alternating terms of filtered de Rham bundles and Higgs bundles which are related to each other by the following diagram inductively where -(V, ∇, Fil) Remark. In case n = 1, the data of (V , ∇, Fil) (n−1) −1 is empty. The Higgs-de Rham flow can be rewritten in the following form In this way, the diagram becomes In the rest of this section, we will give the definition of twisted periodic Higgs-de Rham flow (section 3.2), which generalizes the periodic Higgs-de Rham flow in [12].

3.2.
Twisted periodic Higgs-de Rham flow and equivalent categories. Let L n be a line bundle over X n . For all 1 ≤ ℓ < n, denote L ℓ = L n ⊗ O Xn O X ℓ the reduction of L n on X ℓ . In this subsection, let a ≤ p − 2 be a positive integer. We will give the definition of L n -twisted Higgs-de Rham flow of level in [0, a].
Definition 3.1. Let f be a positive integer. An f -periodic L 1 -twisted Higgsde Rham flow over X 1 ⊂ X 2 of level in [0, a], is a Higgs-de Rham flow over And for any i ≥ 0 the isomorphism

strictly respects filtrations Fil
(1) f +i and Fil (1) i . Those φ (1) f +i 's are relative to each other by formula f +i ). Denote the category of all twisted f -periodic Higgs-de Rham flow over X 1 of level in [0, a] by HDF a,f (X 2 /W 2 ).

3.2.2.
Twisted periodic Higgs-de Rham flow X n ⊂ X n+1 . Let n ≥ 2 be an integer and f be a positive integer. And L n is a line bundle over X n . Denote by L ℓ the reduction of L n modulo p ℓ . We define the category T HDF a,f (X n+1 /W n+1 ) of all f -periodic twisted Higgs-de Rham flow over X n ⊂ X n+1 of level in [0, a] in the following inductive way.
Definition 3.2. An L n -twisted f -periodic Higgs-de Rham flow over X n ⊂ X n+1 is a Higgs-de Rham flow which is a lifting of an L n−1 -twisted f -periodic Higgs-de Rham flow It is constructed by the following diagram for 2 ≤ ℓ ≤ n, inductively ?
ℓ−1 is a lifting of the Hodge filtration Fil • Repeating the process above, one gets the data Fil i+f . And these morphisms are related to each other by formula φ i+f ). Denote the twisted periodic Higgs-de Rham flow by The category of all periodic twisted Higgs-de Rham flow over X n ⊂ X n+1 of level in [0, a] is denoted by T HDF a,f (X n+1 /W n+1 ).
Remark. For the trivial line bundle L n , the definition above is equivalent to the original definition of periodic Higgs-de Rham flow in [12] by using the identification φ : (E, θ) 0 = (E, θ) f .
Note that we can also define the logarithmic version of the twisted periodic Higgs-de Rham flow, since we already have the log version of inverse Cartier transform. X is a smooth proper scheme over W and X o is the complement of a simple normal crossing divisor D ⊂ X relative to W . Similarly, one constructs the category T HDF a,f (X o n+1 /W n+1 ) of twisted f -periodic logarithmic Higgs-de Rham flows (with pole along D × W n ⊂ X × W n ) over X × W n whose nilpotent exponents are ≤ p − 2 .

Equivalence of categories.
We establish an equivalence of categories between T HDF a,f (X n+1 /W n+1 ) and T MF ∇ [0,a],f (X n+1 /W n+1 ). Theorem 3.3. Let a ≤ p − 1 be a natural number and f be an positive integer. Then there is an equivalence of categories between T HDF a,f (X n+1 /W n+1 ) and be an f -periodic L n -twisted Higgs-de Rham flow over X n with level in [0, a].
Taking out f terms of filtered de Rham bundles together with f − 1 terms of identities maps , one gets a tuple This tuple forms an L n -twisted Fontaine-Faltings module by definition. It gives us the functor IC from T HDF a, We construct the corresponding flow by induction on n. In case n = 1, we already have following diagram By this isomorphism, we identify (V 0 , ∇ 0 ) with C −1 1 (E 0 , θ 0 ). Under this isomorphism, the Hodge filtration Fil 0 induces a Hodge filtration Fil f on (V f , ∇ f ). Take Grading and denote Inductively, for i > f , we denote (V i , ∇ i ) = C −1 1 (E i , θ i ). By the isomorphism the Hodge filtration Fil i−f induces a Hodge filtration Fil i on (V i , ∇ i ). Denote Then we extend above diagram into the following twisted periodic Higgs-de Rham flow over X 1 For n ≥ 2, denote This gives us a L n−1 -twisted Fontaine-Faltings module over X n−1 By induction, we have a twisted periodic Higgs-de Rham flow over X n−1 where the first f -terms of filtered de Rham bundles over X n−1 are those appeared in the twisted Fontaine-Faltings module over X n−1 . Based on this flow over X n−1 , we extend the diagram similarly as the n = 1 case, Now it is a twisted periodic Higgs-de Rham flow over X n . Denote this flow by It is straightforward to verify GR • IC ≃ id and IC • GR ≃ id.
This Theorem can be straightforwardly generalized to the logarithmic case and the proof is similar as that of Theorem 3.3.
Theorem 3.4. Let X be a smooth proper scheme over W with a simple normal crossings divisor D ⊂ X relative to W . Then for each natural number f ∈ N, there is an equivalence of categories between T HDF a,f (X o n+1 /W n+1 ) and T MF ∇ [0,a],f (X o n+1 /W n+1 ) 3.2.4. A sufficient condition for lifting the twisted periodic Higgs-de Rham flow. We suppose that the field k is finite in this section. Let X be a smooth proper variety over W (k) and denote X n = X × W (k) W n (k). Let D 1 ⊂ X 1 be a W (k)-liftable normal crossing divisor over k. Let D ⊂ X be a lifting of D 1 .
Proposition 3.5. Let n be an positive integer and let L n+1 be a line bundle over X n+1 . Denote by L ℓ the reduction of L n+1 on X ℓ . Let be an L n -twisted periodic Higgs-de Rham flow over X n ⊂ X n+1 . Suppose -Lifting of the graded Higgs bundle (E, θ) (n) i is unobstructed. i.e. there exist a logarithmic graded Higgs bundle (E, θ) (n+1) i over X n+1 , whose reduction on X n is isomorphic to (E, θ) (n) i .

PROJECTIVE REPRESENTATIONS AND TWISTED HIGGS-DE RHAM FLOWS 31
-Lifting of the Hodge filtration Fil (n) i is unobstructed. i.e. for any lift- , whose reduction on X n is Fil Then every twisted periodic Higgs-de Rham flow over X n can be lifted to a twisted periodic Higgs-de Rham flow over X n+1 .
Proof. By assumption, we choose (E ′ , θ ′ ) which is a lifting of (V, ∇) (n) i . By assumption, we choose a lifting Fil which is a lifting of (E, θ) i+1 . From the φ-structure of the Higgs-de Rham flow, for all m ≥ 0 there is an isomorphism , one gets a lifting of (E, θ) By deformation theory, the lifting space of (E, θ) n is a torsor space modeled by H 1 Hig X 1 , End (E, θ) (1) n . Therefore, the torsor space of lifting n as a graded Higgs bundle should be modeled by a subspace of H 1 Hig . We give a description of this subspace as follows. For simplicity of notations, we shall replace (E, θ) (1) n by (E, θ) in this paragraph. The decomposition of E = p+q=n E p,q induces a decomposition of End(E): (End(E)) k,−k := p+q=n (E p,q ) ∨ ⊗ E p+k,q−k Furthermore, it also induces a decomposition of the Higgs complex End(E, θ). One can prove that the hypercohomology of the following Higgs subcomplex gives the subspace corresponding to the lifting space of graded Higgs bundles.
Thus by the finiteness of the torsor space, there are two integers m > m ′ ≥ 0, such that By twisting suitable power of the line bundle L n+1 we may assume m ′ = 0. By replacing the period f with mf , we may assume m = 1. For integer i ∈ [n, n + f − 1] we denote Then (3.5) can be rewritten as and φ n+1 i+1 as follows.
According to the isomorphism the Hodge filtration Fil Taking grading on equation (3.7), one gets a lifting of φ and a twisted Higgs-de Rham flow over X n+1 ⊂ X n+2 which lifts the given twisted periodic flow over X n ⊂ X n+1 .
Remark. In the proof we see that one needs to enlarge the period for lifting the twisted periodic Higgs-de Rham flow.
3.3. The choice of the twisting line bundle and semi-stable Higgs bundle with trivial discriminant. Let X 1 be a smooth proper W 2liftable variety over k, with dim X 1 = n. Let H be a polarization of X 1 . Let r < p be a positive integer and (E, θ) 0 be a nilpotent semistable Higgs bundle over X 1 of rank r. Recall the main result in the Appendix of [12]: There is a Higgs-de Rham flow over X 1 with initial term (E, θ) 0 .
In the construction of the Higgs-de Rham flow given by Theorem 3.6, the key step is to prove the existence of Simpson's graded semistable Hodge filtration Fil (Theorem A.4 in [12] and Theorem 5.12 in [16]), which is the most coarse Griffiths transverse filtration on a semi-stable de Rham module such that the associated graded Higgs module is still semi-stable. Denote (V, ∇) 0 := C −1 1 (E 0 , θ 0 ) and Fil 0 the Simpson's graded semistable Hodge filtration on (V, ∇) 0 . Denote (V, ∇) 1 := C −1 1 (E 1 , θ 1 ) and Fil 1 the Simpson's graded semistable Hodge filtration on (V, ∇) 1 . Repeating this process, we construct a Higgs-de Rham flow over X 1 with initial term (E, θ) Since the Simpson's graded semistable Hodge filtration is unique, this flow is also uniquely determined by (E, θ) 0 . The purpose of this subsection is to find a canonical choice of the twisting line bundle L such that this Higgs-de Rham flow is twisted preperiodic. Firstly, we want to find a positive integer f 1 and a suitable twisting line bundle Under these two condition, both (E, θ) 0 and (E, θ) f 1 are contained in the moduli scheme M ss Hig (X 1 /k, r, a 1 , a 2 ), which is constructed by Langer in [15] and classifies all the semistable Higgs bundles over X 1 with some fixed topological invariants (which will be explained later). Following [15], we introduce S ′ X 1 /k (d; r, a 1 , a 2 , µ max ) the family of Higgs sheaves over X 1 such that (E, θ) is a member of the family if E is reflexive of dimension d, µ max (E, θ) ≤ µ max ,a 0 (E) = r,a 1 (E) = a 1 and a 2 (E) ≥ a 2 . Here µ max (E, θ) is the slope of the maximal destabilizing sub sheaf of (E, θ), and a i (E) are defined by By the results of Langer, the family S ′ X 1 /k (d; r, a 1 , a 2 , µ max ) is bounded (see Theorem 4.4 of [15]). So M ss Hig (X 1 /k, r, a 1 , a 2 ) is the moduli scheme which corepresents this family. Note that a i (E) = χ(E| j≤d−i H j ) where H 1 , . . . , H d ∈ |O(H)| is an E-regular sequence (see [9]). Using Hirzebruch-Riemann-Roch theorem, one finds that a 1 (E),a 2 (E) will be fixed if c 1 (E) and c 2 (E) · [H] n−2 are fixed. Proof. Since c 1 (C −1 1 (E 0 , θ 0 )) = pc 1 (E 0 ) and c 1 (L 1 ) = 1−p f 1 r ·c 1 (E 0 ), we have Theorem 3.10. A semistable Higgs bundle over X 1 with trivial discriminant is preperiodic after twisting. Conversely, a twisted preperiodic Higgs bundle is semistable with trivial discriminant.
Proof. For a Higgs bundle (E, θ) in M ss Hig (X 1 /k, r, a 1 , a 2 ), we consider the iteration of the self-map Υ. Since M ss Hig (X 1 /k, r, a 1 , a 2 ) is of finite type over k and has only finitely many k-points, there must exist a pair of integers (e, f 2 ) such that Υ e (E, θ) ∼ = Υ e+f 2 (E, θ). By Proposition 3.9, we know that (E, θ) is preperiodic after twisting. Conversely, let (E, θ) be the initial term of a twisted f -perperiodic Higgs-de Rham flows. We show that it is semistable. Let (F, θ) ⊂ (E, θ) be a proper sub bundle. Denote (F (1) i , θ So µ(F e ) ≤ µ(E e ) (otherwise there are subsheaves of E e with unbounded slopes, but this is impossible). So we have This shows that (E, θ) is semistable. The discriminant equals zero follows from the fact that ∆(C −1 1 (E, θ)) = p 2 ∆(E). Corollary 3.11. Let (E, θ) ⊃ (F, θ) be the initial terms of a twisted periodic Higgs-de Rham flow and a sub twisted periodic Higgs-de Rham flow. Then µ(F ) = µ(E).

Sub-representations and sub periodic Higgs-de Rham flows.
In this section, we assume F p f is contained in k. Recall that the functor D P is contravariant and sends quotient object to subobject, i.e. for any sub twisted Fontaine-Faltings module N ⊂ M with endomorphism structure, the projective representation D P (M/N ) is a sub-projective representation of D P (M ). Conversely, we will show that every sub-projective representation comes from this way. By the equivalence of the category of twisted Fontaine-Faltings modules and the category of twisted periodic Higgs-de Rham flows, we construct a twisted periodic sub Higgs-de Rham flow for each sub-projective representation. Let X be a smooth proper W (k)-variety. Denote by X n the reduction of X on W n (k) . Let {U i } i∈I be a finite covering of small affine open subsets and we choose a geometric point x in i∈I U i,K . Proof. Recall that the functor D P is defined by gluing representations of ∆ i = π 1 (U i,K , x) into a projective representation of ∆ = π 1 (X K , x). Firstly, we show that the sub-projective representation V is actually corresponding to some local sub-representations. Secondly, since the Fontaine-Laffaille-Faltings' functor D is fully faithful, there exists local Fontaine-Faltings modules corresponding to those sub-representations. Thirdly, we glue those local Fontaine-Faltings modules into a global twisted Fontaine-Faltings module. For i ∈ I, we choose a trivialization M i = M (τ i ) of M on U i , which gives a local Fontaine-Faltings module with endomorphism structure on U i . By definition of D P , those representations D U i (M i ) of ∆ i are glued into the projective representation D P (M ). In other words, we have the following On the other hand, one has D(M j /N j ) = V j = a 1,r V i by diagram (3.12). Thus one has D(M j /N ′ i ) = D(M j /N j ). Since D is fully faithful and contravariant, N ′ i = N j . In particular, on the overlap U i ∩ U j the local Fontaine-Faltings modules N i and N j have the same underlying subbundle. By gluing those local subbundle together, we get a subbundle of the underlying bundle M . The connection, filtration and the ϕ-structure can be restricted locally on this subbundle, so does it globally. And we get the desired sub-Fontaine-Faltings module.
Let E be a twisted f -periodic Higgs-de Rham flow. Denote by M = IC(E) the Fontaine module with the endomorphism structure corresponding to E . By the equivalence of the category of twisted Fontaine-Faltings modules and the category of periodic Higgs-de Rham flow, one get the following result. Finally we arrive at the main theorem of our paper: Theorem 3.14. Let k be a finite field of characteristic p. Let X be a smooth proper scheme over W (k) together with a smooth log structure D/W (k). Assume that there exists a semistable graded logarithmic Higgs bundle (E, θ)/(X , D) 1 with discriminant ∆ H (E) = 0, rank(E) < p and (rank(E), deg H (E)) = 1. Then there exists a positive integer f and an absolutely irreducible projective We only show the result for D = ∅, as the proof of the general case is similar. By Theorem 3.10, there is a twisted preperiodic Higgs-de Rham flow with initial term (E, θ). Removing finitely many terms if necessary, we may assume that it is twisted f -periodic, for some positive integer f . By using Theorem 3.3 and applying functor D P , one gets a PGL rank(E) (F p f )representation ρ of π 1 (X o K ′ ). Since (rank(E), deg H (E)) = 1, the semi-stable bundle E is actually stable. According to Corollary 3.11, there is no non-trivial sub twisted periodic Higgs-de Rham flow. By Corollary 3.13, there is no non-trivial sub projective representation of ρ, so that ρ is irreducible.
Remark. For simplicity, we only consider results on X 1 . Actually, all results in this section can be extended to truncated level.

Constructing crystalline representations ofétale fundamental groups of p-adic curves via Higgs bundles
As an application of the main theorem (Theorem 3.14), we construct irreducible PGL 2 crystalline representations of π 1 of the projective line removing m (m ≥ 4) marked points. Let M be the moduli space of semistable graded Higgs bundles of rank 2 degree 1 over P 1 /W (k), with logarithmic Higgs fields which have m poles {x 1 , x 2 , . . . , x m } (actually stable, since the rank and degree are coprime to each other). The main object of this section is to study the self map Υ (Corollary-Definition 3.
If we consider the lifting problem over an extension k ′ of k, which contains Σ, then there are exactly p liftings of the twisted 1-periodic Higgs-de Rham flow over P 1 W 2 (k ′ ) .

4.4.
Examples of dynamic of Higgs-de Rham flow on P 1 with fourmarked points. In the following, we give some examples in case k = F 3 4 . For any λ ∈ k \ {0, 1}, the map ϕ λ,3 is a self k-morphism on P 1 k . So it can be restricted as a self map on the set of all k-points ϕ λ,3 : k ∪ {∞} → k ∪ {∞}.
In the following diagrams, the arrow β → γ means γ = ϕ λ,3 (β). And an mlength loop in the following diagrams just stands for a twisted m-periodic Higgs-de Rham flow, which corresponds to PGL 2 (F 3 m )-representation by Theorem 3.4 and Theorem 3.14.

4.5.
Projective F -units crystalline on smooth projective curves. Let X be a smooth proper scheme over W (k). In [12] an equivalence between the category of f -periodic vector bundles (E, 0) of rank-r over X n (i.e. (E, 0) initials an f -periodic Higgs-de Rham flow with zero Higgs fields in all Higgs terms) and the category of GL r (W n (F p f ))-representations of π 1 (X 1 ) has been established. This result generalizes Katz's original theorem for X being an affine variety. As an application of our main theorem, we show that Theorem 4.6. The D P functor is faithful from the category of rank-r twisted f -periodic vector bundles (E, 0) over X n to the the category of projective W n (F p f )-representations of π 1 (X 1,k ′ ) of rank r, where k ′ is the minimal extension of k containing F p f .
Remark. For n = 1 the above theorem is just a projective version of Lange-Stuhe's theorem.
Theorem 4.7 (lifting twisted periodic vector bundles). Let (E, 0)/X 1 be an f -periodic vector bundle after twisting line bundle. Assume H 2 (X 1 , End(E)) = 0. Then for any n ∈ N there exists some positive integer f n with f | f n such that (E, 0) lifts to a twisted f n -periodic vector bundle over X n .
Translate the above theorem in of representations: Theorem 4.8 (lifting projective representations of π 1 (X 1 )). Let ρ be a projective F p f -representation of π 1 (X 1 ). Assume H 2 (X 1 , End(ρ)) = 0, then there exist an positive integer f n divided by f such that ρ lifts to a projective W n (F p fn )-representation of π 1 (X 1,k ′ ) for any n ∈ N, where k ′ is the minimal extension of k containing F p fn .
Assume X is a smooth proper curve over W (k), de Jong and Osserman (see Appendix A in [20]) have shown that the subset of periodic vector bundles over X 1,k is Zariski dense in the moduli space of semistable vector bundles over X 1 (Laszlo and Pauly have also studied some special case, see [17]). Hence by Lange-Stuhe's theorem (see [14]) every periodic vector bundle corresponds to a (P )GL r (F p f )-representations of π 1 (X 1,k ′ ), where f is the period and k ′ is a definition field of the periodic vector bundle containing F p f . Corollary 4.9. Every (P)GL r (F p f )-representation of π 1 (X 1,k ′ ) lifts to a (P)GL r (W n (F p fn ))-representation of π 1 (X 1,k ′′ ) for some positive integer f n divided by f , where k ′′ is a definition field of the periodic vector bundle containing F p fn .
Remark. It shall be very interesting to compare this result with Deninger-Werner's theorem (see [2]), they have shown that any vector bundle over X , which is preperiodic over X 1 , lifts to a GL r (C p )-representation of π 1 (XK).