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**On lattices in semi-stable representations: a proof of a conjecture of Breuil.**
*(English)*
Zbl 1133.14020

Let \(k\) be a perfect field of characteristic \(p\geq 3\), \(W(k)\) its ring of Witt vectors, \(K_0:= W(k)[1/p]\), \(K/K_0\) a finite totally ramified extension and \(e:= e(K/K_0)\) the absolute ramification index. In the general theory of \(p\)-adic Galois representations of local fields, the study of the so-called semi-stable \(p\)-adic Galois representations of the group \(G:= \text{Gal}(\overline K/K)\), with respect to this particular set-up, is of crucial interest, and it has undergone an increasing intensity in the course of the last ten years. Being closely related to integral \(p\)-adic Hodge theory and to integral crystalline cohomology in algebraic geometry, semi-stable \(p\)-adic representations of \(G\) were first both explicitely constructed and basically classified by P. Colmez and J.-M. Fontaine [Invent. Math. 140, No. 1, 1–43 (2000; Zbl 1010.14004)] a few years ago, essentially by means of J.-M. Fontaine’s theory of weakly admissible filtered \((\varphi,N)\)-modules [Astérisque 65, 3–80 (1979; Zbl 0429.14016)], which appear as certain finite-dimensional filtered Dieudonné modules.

Since the Galois group \(G= \text{Gal}(\overline K/K)\) is compact, any continuous representation \(\rho: G\to \text{GL}_n(\mathbb{Q}_p)\) admits a \(G\)-stable \(\mathbb{Z}_p\)-lattice, and it is therefore natural to ask the question of whether there also exists a (functorially) corresponding lattice structure within the framework of the associated filtered \((\varphi,N)\)-modules. After a first attack to this problem by J.-M. Fontaine and G. Laffaille [Ann. Sci. Éc. Norm. Supér. (4) 15, 547–608 (1982; Zbl 0579.14037)], which only worked for the special case of \(e= 1\) and \(N= 0\) a decisive break-through was obtained in the late 1990s by C. Breuil, who introduced an appropriately modified theory of filtered \((\varphi,N)\)-modules and, in this context, established certain integral structures, the so-called strongly divisible lattices inside his special filtered \((\varphi,N)\)-modules. For such a strongly divisible lattice \({\mathcal M}\), C. Breuil provided a functorial construction of a \(G\)-stable \(\mathbb{Z}_p\)-lattice \(T_{st}({\mathcal M})\) in a semi-stable Galois representation of \(G\) and he formulated a prediction (called Breuil’s conjecture) stating the following: For any nonnegative number \(r\leq p -2\) the functor \(T_{st}\) describes an anti-equivalence of categories between the category of strongly divisible lattices of weight \(r\) and the category of \(G\)-stable \(\mathbb{Z}_p\)-lattices in semi-stable representations of \(G\) with Hodge-Tate weights in \(\{0,\dots, r\}\).

This conjecture of C. Breuil’s [Integral \(p\)-adic Hodge theory. Algebraic geometry 2000, Azumino. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 36, 51–80 (2002; Zbl 1046.11085)] has been proved for the special cases \(r\leq 1\) (C. Breuil) and \(e\cdot r< p-1\) [X. Caruso, Ph.D. thesis, Université de Paris-Sud 11, Paris, 2005].

In the paper under review, the author completes the picture of Breuil’s conjecture by providing a general affirmative proof of its statement. His approach is heavily based on M. Kisin’s recently developed theory of crystalline representations in the framework of \((\varphi,N)\)-modules [cf.: M. Kisin, in: Crystalline representations and \(F\)-crystals, Algebraic geometry and number theory. Basel: Birkhäusser. Progress in Mathematics 253, 459–496 (2006; Zbl 1184.11052)], which is used to construct the category of so-called quasi-strongly divisible lattices and, in the sequel, to establish an anti-equivalence between this category and the category of \(G_\infty\)-stable \(\mathbb{Z}_p\)-lattices in semi-stable Galois representations of the group \(G\).

The full faithfulness of Breuil’s functor \(T_{st}\) is then derived from this result by showing that a quasi-strongly divisible lattice is already strongly divisible if and only if the corresponding \(G_\infty\)-stable \(\mathbb{Z}_p\)-lattice is \(G\)-stable. The crucial idea of this approach is to apply an extended version of a theorem of G. Faltings [J. Am. Math. Soc. 12, No. 1, 117–144 (1999; Zbl 0914.14009)], the proof of which uses the construction of the Cartier dual for quasi-strongly-divisible lattices due to X. Caruso [Ph.D. thesis, Université de Paris-Sud 11, Paris, 2005]. Finally, the essential surjectivity of Breuil’s functor \(T_{st}\) is proved in the last section of the present paper, mainly by skillfully combining the foregoing various preparations.

All together, the complete proof of Breuil’s conjecture involves the comparison of various categories and functors, which is carried out in an utmost lucid, rigorous, comprehensive and enlightening manner. The underlying theories previously developed by C. Breuil, M. Kisin, and X. Caruso are carefully reviewed and explained, which makes the important work under review largely self-contained and rounded off.

Since the Galois group \(G= \text{Gal}(\overline K/K)\) is compact, any continuous representation \(\rho: G\to \text{GL}_n(\mathbb{Q}_p)\) admits a \(G\)-stable \(\mathbb{Z}_p\)-lattice, and it is therefore natural to ask the question of whether there also exists a (functorially) corresponding lattice structure within the framework of the associated filtered \((\varphi,N)\)-modules. After a first attack to this problem by J.-M. Fontaine and G. Laffaille [Ann. Sci. Éc. Norm. Supér. (4) 15, 547–608 (1982; Zbl 0579.14037)], which only worked for the special case of \(e= 1\) and \(N= 0\) a decisive break-through was obtained in the late 1990s by C. Breuil, who introduced an appropriately modified theory of filtered \((\varphi,N)\)-modules and, in this context, established certain integral structures, the so-called strongly divisible lattices inside his special filtered \((\varphi,N)\)-modules. For such a strongly divisible lattice \({\mathcal M}\), C. Breuil provided a functorial construction of a \(G\)-stable \(\mathbb{Z}_p\)-lattice \(T_{st}({\mathcal M})\) in a semi-stable Galois representation of \(G\) and he formulated a prediction (called Breuil’s conjecture) stating the following: For any nonnegative number \(r\leq p -2\) the functor \(T_{st}\) describes an anti-equivalence of categories between the category of strongly divisible lattices of weight \(r\) and the category of \(G\)-stable \(\mathbb{Z}_p\)-lattices in semi-stable representations of \(G\) with Hodge-Tate weights in \(\{0,\dots, r\}\).

This conjecture of C. Breuil’s [Integral \(p\)-adic Hodge theory. Algebraic geometry 2000, Azumino. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 36, 51–80 (2002; Zbl 1046.11085)] has been proved for the special cases \(r\leq 1\) (C. Breuil) and \(e\cdot r< p-1\) [X. Caruso, Ph.D. thesis, Université de Paris-Sud 11, Paris, 2005].

In the paper under review, the author completes the picture of Breuil’s conjecture by providing a general affirmative proof of its statement. His approach is heavily based on M. Kisin’s recently developed theory of crystalline representations in the framework of \((\varphi,N)\)-modules [cf.: M. Kisin, in: Crystalline representations and \(F\)-crystals, Algebraic geometry and number theory. Basel: Birkhäusser. Progress in Mathematics 253, 459–496 (2006; Zbl 1184.11052)], which is used to construct the category of so-called quasi-strongly divisible lattices and, in the sequel, to establish an anti-equivalence between this category and the category of \(G_\infty\)-stable \(\mathbb{Z}_p\)-lattices in semi-stable Galois representations of the group \(G\).

The full faithfulness of Breuil’s functor \(T_{st}\) is then derived from this result by showing that a quasi-strongly divisible lattice is already strongly divisible if and only if the corresponding \(G_\infty\)-stable \(\mathbb{Z}_p\)-lattice is \(G\)-stable. The crucial idea of this approach is to apply an extended version of a theorem of G. Faltings [J. Am. Math. Soc. 12, No. 1, 117–144 (1999; Zbl 0914.14009)], the proof of which uses the construction of the Cartier dual for quasi-strongly-divisible lattices due to X. Caruso [Ph.D. thesis, Université de Paris-Sud 11, Paris, 2005]. Finally, the essential surjectivity of Breuil’s functor \(T_{st}\) is proved in the last section of the present paper, mainly by skillfully combining the foregoing various preparations.

All together, the complete proof of Breuil’s conjecture involves the comparison of various categories and functors, which is carried out in an utmost lucid, rigorous, comprehensive and enlightening manner. The underlying theories previously developed by C. Breuil, M. Kisin, and X. Caruso are carefully reviewed and explained, which makes the important work under review largely self-contained and rounded off.

Reviewer: Werner Kleinert (Berlin)

### MSC:

14F30 | \(p\)-adic cohomology, crystalline cohomology |

14L05 | Formal groups, \(p\)-divisible groups |

11S31 | Class field theory; \(p\)-adic formal groups |

14G20 | Local ground fields in algebraic geometry |

13K05 | Witt vectors and related rings (MSC2000) |

11S20 | Galois theory |