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Lattices in filtered $$(\varphi,N)$$-modules. (English) Zbl 1249.14006
Let $$k$$ be a perfect field of characteristic $$p$$, $$W(k)$$ its ring of Witt vectors, $$K_0 = W(k)[1/p]$$, $$K/K_0$$ a finite totally ramified extension and $$G_K:=\mathrm{Gal}(\bar{K}/K)$$. Let $$K'/K$$ be a totally ramified Galois extension and write $$G = G_{K'} = \mathrm{Gal}(\bar{K}/K')$$, $$\Gamma = \mathrm{Gal}(K'/K)$$. Let $$\pi \in K'$$ be a fixed uniformizer with the Eisenstein polynomial $$E(u)$$. For any potentially semi-stable $$p$$-adic representation $$V$$ of $$G_K$$, Fontaine associated a filtered $$(\varphi,N,G_K)$$-module $$D^*_{\mathrm{st}}(V)$$ to $$V$$. Important invariants of $$V$$ can be read from $$D^*_{\mathrm{st}}(V)$$ (such as Weil-Deligne representation attached to $$V$$ if $$K$$ is a finite extension over $$\mathbb{Q}_p$$). In this paper, the author construct integral and $$p^n$$-torsion structures attached to a $$G_K$$-stable $$\mathbb{Z}_p$$-lattice $$T$$ in $$V$$.
Integral Structure: The author defines a functor $$M_{\mathrm{st}}$$ from the category $$\mathrm{Rep}^{\mathrm{pst},r}_{\mathbb{Q}_p}$$ whose objects are representations of $$G_K$$ which are semi-stable over $$K'$$ with Hodge-Tate weights in $$\{0,\dots,r\}$$, to the category $$L^r(\varphi,N,\Gamma)$$, whose objects are lattices in filtered $$(\varphi,N,\Gamma)$$-modules satisfying $$\mathrm{Fil}^0D_{K'} = D_{K'}$$ and $$\mathrm{Fil}^{r+1}_{K'}D_{K'}=0$$. Lattice here means finite free $$W(k)$$-module inside $$D_{\mathrm{st}}(V)$$ which is stable under $$\varphi,N$$ and the $$G_K$$-action. The definition of $$M_{\mathrm{st}}$$ uses the theory of $$(\varphi,\hat{G})$$-modules developed by the author in his paper [Math. Ann. 346, No. 1, 117–138 (2010; Zbl 1208.14017)]. Let $$V$$ be a semi-stable representation of $$G$$ with Hodge-Tate weights in $$\{0,\dots,r\}$$, $$T \subset V$$ a $$G_{K'}$$-stable $$\mathbb{Z}_p$$-lattice. A theorem of loc. cit. associates $$T$$ a $$(\varphi,\hat{G})$$-module $$(\mathfrak{M},\varphi,\hat{G})$$. Put $$M:=\mathfrak{M}/u\mathfrak{M} \subset \mathcal{D}/(I_+S_{K_0})\mathcal{D} =:D$$. It can be shown that there exists a unique isomorphism $$i:D_{\mathrm{st}}(V) \simeq D$$. Set $$M_{\mathrm{st}}(T):=i^{-1}(M) \subset D_{\mathrm{st}}(V)$$. The main result is the following:

Theorem. The functor $$M_{\mathrm{st}}: \mathrm{Rep}^{\mathrm{pst},r}_{\mathbb{Q}_p} \to L^r(\varphi,N,\Gamma)$$ is left exact and faithful. Moreover, the functor $$M_{\mathrm{st}} \otimes_{\mathbb{Z}_p} \mathbb{Q}_p$$ associated to the isogeny categories is naturally isomorphic to $$D_{\mathrm{st}}$$. If $$er<p-1$$, then $$M_{\mathrm{st}}$$ is exact and fully faithful.
$$p^n$$-torsion Structure: By $$\mathrm{Rep}_{\mathrm{tor}}^{\mathrm{pst},r}$$, the author means the category whose objects are torsion potentially semi-stable representations with Hodge-Tate weights in $$\{0,\dots,r\}$$, in the sense that, for any $$T \in \mathrm{Rep}_{\mathrm{tor}}^{\mathrm{pst},r}$$, there exists $$G_K$$-stable $$\mathbb{Z}_p$$-lattices $$L\subset L'$$ in a $$V \in \mathrm{Rep}_{\mathrm{tor}}^{\mathrm{pst},r}$$ such that $$T \simeq L'/L$$ as $$\mathbb{Z}_p[G_K]$$-modules. The pair $$L\subset L'$$ is called a lift of $$T$$, and denoted by $$\mathcal{L}: L \hookrightarrow L'$$. By the theorem above, there exists a morphism $$\tilde{j}:M_{\mathrm{st}}(L') \to M_{\mathrm{st}}(L)$$ in $$L^r(\varphi,N,\Gamma)$$. Set $$M_{\mathrm{st},\mathcal{L}}(T) := M_{\mathrm{st}}(L)/\tilde{j}(M_{\mathrm{st}}(L'))$$. It has $$G_K$$-action, Frobenius $$\varphi$$ and monodromy $$N$$ induced from $$M_{\mathrm{st}}(L)$$. These form a category $$M_{\mathrm{tor}}(\varphi,N,\Gamma)$$ whose objects are finite $$W(k)$$-modules $$M$$ killed by some $$p$$-power and endowed with a Frobenius semi-linear map $$\varphi: D \to D$$, a $$W(k)$$-linear map $$N:D \to D$$ such that $$N\varphi = p\varphi N$$, and a $$W(k)$$-linear $$\Gamma$$-action on $$D$$ such that $$\Gamma$$ commutes with $$\varphi$$ and $$N$$. The following main result says that the definition of $$M_{\mathrm{st},\mathcal{L}}(T)$$ (does not totally depend on the lifts) is ‘unique’ up to $$p^{\mathfrak{c}}$$ in the following sense:
Theorem. There exists a constant $$\mathfrak{c}$$ only depending on $$e$$ and $$r$$ such that the following statement holds: for any morphism $$f:T' \to T$$ in $$\mathrm{Rep}_{\mathrm{tor}}^{\mathrm{pst},r}$$ and any lift $$\mathcal{L}', \mathcal{L}$$ of $$T'$$, $$T$$ respectively, there exists a morphism $$g:M_{\mathrm{st},\mathcal{L}}(T) \to M_{\mathrm{st},\mathcal{L}'}(T')$$ in $$M_{\mathrm{tor}}(\varphi,N,\Gamma)$$ such that
(1) if there exists a morphism of lifts $$\hat{f}: \mathcal{L}' \to \mathcal{L}$$ which lifts $$f$$ then $$g = p^{\mathfrak{c}}M_{\mathrm{st},\hat{f}}(f)$$;
(2) let $$f':T''\to T'$$ be a morphism in $$\mathrm{Rep}_{\mathrm{tor}}^{\mathrm{pst},r}$$ with $$\mathcal{L}''$$ the lift of $$T''$$ and $$g':M_{\mathrm{st},\mathcal{L}'}(T') \to M_{\mathrm{st},\mathcal{L}''}(T'')$$ the morphism in $$M_{\mathrm{tor}}(\varphi,N,\Gamma)$$ attached to $$f'$$, $$\mathcal{L}'$$ and $$\mathcal{L}''$$; if there exists a morphism of lifts $$\hat{h}:\mathcal{L}'' \to \mathcal{L}$$ which lifts $$f \circ f'$$, then $$g'\circ g = p^{2\mathfrak{c}}M_{\mathrm{st},\hat{h}}(f\circ f')$$.
Based on the above constructions, the paper provides two applications:
Application 1: Let $$V$$ be a potentially semi-stable representation of $$G_K$$ with Hodge-Tate weights in $$\{0, \dots, r\}$$, and $$T \subset V$$ a $$G_K$$-stable $$\mathbb{Z}_p$$-lattice. There exists a constant $$\alpha$$ depending on the dimension $$d = \text{dim}_{\mathbb{Q}_p}(V)$$, the absolute ramification index $$\tilde{e} = [K:K_0]$$ and $$r$$ such that $$V$$ is semi-stable over $$K$$ if and only if there exist $$G_K$$-stable $$\mathbb{Z}_p$$-lattices $$L' \subset L$$ in a semi-stable representation $$W$$ of $$G_K$$ with Hodge-Tate weights in $$\{0,\dots,r\}$$ satisfying $$T/p^{\alpha}T \simeq L/L'$$.
Application 2: Let $$F$$ be a totally real field and $$\pi$$ a Hilbert eigenform of weight $$\underline{k} = (k_1, \dots, k_g)$$ with $$k_i \geq 2$$, integers all have the same parity. Let $$\rho_{\pi}$$ denote the two-dimensional $$p$$-adic Galois representation of $$G_F := \text{Gal}(\overline{F}/F)$$ attached to $$\pi$$. If $$\mathfrak{q} \; | \; p$$ is a prime of $$F$$ and $$G_{F_{\mathfrak{q}}}$$ denotes a decomposition group at $$\mathfrak{q}$$, then $$\rho_{\pi} \; | \; G_{F_{\mathfrak{q}}}$$ is potentially semi-stable with $$p$$-adic Hodge type corresponding to the weight $$\underline{k}$$. Moreover, the Weil-Deligne representation attached to $$\rho_{\pi} \; | \; G_{F_{\mathfrak{q}}}$$ via Fontaine’s construction corresponds to the local factor $$\pi_{\mathfrak{q}}$$ of $$\pi$$ via the local Langlands correspondence.
Reviewer: Xiao Xiao (Utica)

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14L05 Formal groups, $$p$$-divisible groups 11S15 Ramification and extension theory 11S20 Galois theory
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