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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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