Deforming semistable Galois representations. (English) Zbl 0901.11016

Fontaine, Jean-Marc

Proc. Natl. Acad. Sci. USA 94, No. 21, 11138-11141 (1997).

Let \(G=\text{Gal}(\overline{\mathbb{Q}}/{\mathbb{Q}})\) and \(G_{\ell}=\text{Gal}(\overline{\mathbb{Q}}_{\ell}/{\mathbb{Q}}_{\ell})\subset G\) for chosen algebraic closures \(\overline{\mathbb{Q}}\) of \({\mathbb{Q}}\) and \(\overline{\mathbb{Q}}_{\ell}\) of \({\mathbb{Q}}_{\ell}\). Let \(S\) be a finite set of primes containing \(p\). \(E\) will denote a finite extension of \({\mathbb{Q}}_p\). By an \(E\)-representation of a profinite group \(J\) one means a finite dimensional \(E\)-vector space equipped with a linear and continuous \(J\)-action. An \(E\)-representation \(V\) of \(G\) is called geometric if \((i)\) it is unramified outside of a finite set of primes, \((ii)\) it is potentially semistable at \(p\). A geometric irreducible \(E\)-representation \(V\) of \(G\) is called a Hecke representation if there is a finite \({\mathbb{Z}}_p\)-algebra \(\mathcal H\), generated by Hecke operators acting on some automorphic representation space equipped with a continuous homomorphism \(\rho:G\rightarrow\text{GL}_d(\mathcal H)\), compatible with the action of the Hecke operators, such that \(V\) comes from \(\mathcal H\). For a two-dimensional representation of \(G\), geometric Hecke means that the representation is a Tate twist of one associated to a modular form. Such a representation comes from algebraic geometry, i.e. it is isomorphic to a subquotient of \(E\otimes_{{\mathbb{Q}}_p}H^m_{\acute{e}t}(\overline{X},{\mathbb{Q}}_p(j))\) for some variety \(X/{\mathbb{Q}}\) and suitable integers \(m\) and \(j\). It is conjectured that any geometric irreducible \(E\)-representation comes from algebraic geometry. One may even conjecture that any geometric irreducible representation is Hecke and that any geometric Hecke representation comes from algebraic geometry.
Inspired by Wiles’s work on the Taniyama-Weil conjecture (or FLT) one is interested in the deformation theory of Hecke representations. To this end, consider the category of noetherian complete \(\mathcal O_E\)-algebras with residue field \(k\), where \(\mathcal O_E\) is the ring of integers of \(E\) and \(k={\mathcal O}_E/\pi{\mathcal O}_E\) for a uniformizing parameter \(\pi\). Fix a geometric \(E\)-representation \(V\) of \(G\) and a full subcategory \(\mathcal D_p\) of \(\text{Rep}_{{\mathbb{Z}}_p}^f(G_p)\) (the category of \({\mathbb{Z}}_p\)-modules of finite length with continuous \(G_p\)-action), stable under subobjects, quotients and direct sums. Choose a \(G\)-stable \({\mathcal O}_E\)-lattice \(U\) of \(V\) and assume \(u=U/\pi U\) is absolutely irreducible. One says that \(V\) is of type \((S,{\mathcal D}_p)\) if it is unramified outside of \(S\) and lies in \(\mathcal D_p\). An \(E\)-representation \(V'\) of \(G\) is said to be \((S,{\mathcal D}_p)\)-close to \(V\) if \(V'\) is of type \((S,{\mathcal D}_p)\) and, for a given \(G\)-stable lattice \(U'\) of \(V'\), one has \(U'/\pi U'\simeq u\). This gives a local condition at \(p\), and one may try to extend Wiles’s methods.
For the deformation theory to work correctly, one needs that the category \(\mathcal D_p\) be semistable, i.e. any \(E\)-representation of \(G_p\) is potentially semistable. Three examples of semistable \(\mathcal D_p\) are worked out in some detail:
(i) \(\mathcal D_p^{cr}\) (crystalline): An \(E\)-representation \(V\) of \(G_p\) lies in \(\mathcal D_p^{cr}\) iff \(V\) is crystalline, the Hodge numbers \(h^r(V)=0\) for \(r>0\) or \(r<-p+1\), and \(V\) has no nonzero subobject \(V'\) with \(V'(-p+1)\) unramified. Moreover, if \(X\) is proper and smooth over \({\mathbb{Q}}_p\), and having good reduction, then \(H^r_{\acute{e}t}(X_{\overline{\mathbb{Q}}_p},{\mathbb{Z}}/p^n{\mathbb{Z}})\) is an object of \(\mathcal D_p^{cr}({\mathbb{Z}}_p)\) for \(n\in{\mathbb{N}}\) and \(0\leq r\leq p-2\). One may calculate deformation spaces, i.e. the cohomology \(H^1_{\mathcal D_p^{cr}}({\mathbb{Q}}_p,\mathfrak{gl}(U_n))\). In particular, if \(H^0({\mathbb{Q}}_p,\mathfrak{gl}(u))=k\), the deformation problem is smooth. The Hodge type does not change under deformation.
(ii) \(\mathcal D_p^{na}\) (the naive generalization of \(\mathcal D_p^{cr}\) to the semistable case): This category can be characterized as a suitable full subcategory of the category of semistable representations with crystalline semisimplification. Again, one may calculate \(H^1_{\mathcal D_p^{na}}({\mathbb{Q}}_p,\mathfrak{gl}(U_n))\), but the deformation problem is not always smooth. The Hodge type does not change under deformations.
(iii) \(\mathcal D_p^{st}\) (the good generalization of \(\mathcal D_p^{cr}\) to the semistable case, due to C. Breuil): In fact, for \(r\leq p-2\), one has categories \(\mathcal D_p^{st,r}\) such that if an \(E\)-representation \(V\) of \(G_p\) lies in \(\mathcal D_p^{st,r}\), then \(V\) is semistable and \(h^m(V)=0\) if \(m>0\) or \(m<-r\). The converse seems likely. Breuil proved the result: Let \(X\) be proper and smooth over \({\mathbb{Q}}_p\) having semistable reduction. Then \(H^r_{\acute{e}t}(X_{\overline{\mathbb{Q}}_p},{\mathbb{Z}}/p^n{\mathbb{Z}})\) is an object of \(\mathcal D_p^{st,r}({\mathbb{Z}}_p)\) for \(n\in{\mathbb{N}}\) and \(0\leq r\leq p-2\). The Hodge type may change under deformations.

Reviewer: W.W.J.Hulsbergen (Haarlem)

MSC:

11F80	Galois representations
11R39	Langlands-Weil conjectures, nonabelian class field theory
14F30	\(p\)-adic cohomology, crystalline cohomology

Keywords:

Galois representation; deformations; Hecke representations; deformation spaces; semistable categories; crystalline representations; Hodge type

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