An introduction to the deformation theory of Galois representations.

*(English)*Zbl 0901.11015
Cornell, Gary (ed.) et al., Modular forms and Fermat’s last theorem. Papers from a conference, Boston, MA, USA, August 9–18, 1995. New York, NY: Springer. 243-311 (1997).

It is well known by now that one of the main ingredients of A. Wiles’s proof of Fermat’s Last Theorem (or better, of the Shimura-Taniyama-Weil Conjecture for semi-stable elliptic curves) is the deformation theory of Galois representations. In the underlying paper, the reader will find the very best introduction and overview by the founder of the theory. It consists of two parts: (I) Galois representations, group representations and the deformation theory of Galois representations, and (II) Functorial approach of deformation theory, tangent spaces and deformation conditions.

Let \(K\) be a number field, and let \(S\) be a finite set of non-archimedean places of \(K\). Denote by \(\overline{K}_S\) the maximal algebraic extension of \(K\) in the algebraic closure \(\overline{K}\) of \(K\). The corresponding Galois group is written \(G_{K,S}\). \(G_{K,S}\) is endowed with the Krull topology so that it becomes a profinite quotient of \(G_K:=\text{Gal}(\overline{K}/K)\). In general, \(G_{K,S}\) is not very well known, but one knows at least that for any prime \(p\), \(G_{K,S}\) satisfies the \(p\)-finiteness condition, i.e. for all open subgroups \(\Pi_0\subset\Pi=G_{K,S}\) of finite index, there are only finitely many continuous homomorphisms \(\Pi_0\rightarrow{\mathbb{Z}}/p{\mathbb{Z}}\). Each place \(v\) of \(K\) gives rise to a continuous homomorphism \(i_v:G_{K_v}=\text{Gal}(\overline{K}_v/K_v)\rightarrow G_{K,S}\), and one is particularly interested in understanding the “package” \(\{G_{K,S},i_v\}\), for all places \(v\) of \(K\), where the \(i_v\) are determined up to conjugation. For non-archimedean \(v\), \(v\not\in S\), one has Frobenius elements \(\phi_v=i_v(\text{Frob}_v)\in G_{K,S}\) unique up to conjugation. A fruitful approach to understanding the package \(\{G_{K,S},i_v\}\) is by studying the representations of \(G_{K,S}\). This leads to Galois representations \(\rho:G_{K,S}\rightarrow\text{GL}_N(A)\), where \(A\) denotes a coefficient ring, i.e. a complete noetherian local ring with finite residue field \(k\), such that \(\rho\) is a continuous homomorphism. Such group representations are in 1-1 correspondence with continuous \(A\)-algebra homomorphisms \(\tau:A[[\Pi]]\rightarrow M_N(A)\), where \(A[[\Pi]]\) is the completed group ring. Associated to \(\rho\) there is the residual representation \(\overline{\rho}:\Pi\rightarrow\text{GL}_N(k)\). Then \(\overline{\rho}\) is absolutely irreducible iff \(\tau\) is surjective. A homomorphism \(h:A_1\rightarrow A_0\) of coefficient rings \(A_0\), \(A_1\), and a representation \(\rho_0:\Pi\rightarrow\text{GL}_N(A_0)\) give rise to the notion of deformation of \(\rho_0\) to the coefficient ring \(A_1\), i.e. a strict equivalence class of liftings \[ \begin{matrix} \Pi & \buildrel\rho_1\over\longrightarrow & \text{GL}_N(A_1)\\ & \searrow&\downarrow\scriptstyle{h}\\ && \text{GL}_N(A_0) \end{matrix} \] where two liftings \(\rho_1\), \(\rho_1'\) are strictly equivalent if they differ by a conjugation by elements of \(\text{Ker}(h)\subset\text{GL}_N(A_1)\). Thus \(\rho\) is a deformation of its underlying \(\overline{\rho}\). Basic to the whole deformation theory of Galois representations is the following result: For a positive integer \(N\) and an absolutely irreducible representation \(\overline{\rho}:G_{K,S}\rightarrow\text{GL}_N(k)\), there exists a universal coefficient ring \(R=R(\overline{\rho})\) with residue field \(k\), and a universal deformation \(\rho^{\text{univ}}:G_{K,S}\rightarrow\text{GL}_N(R)\) of \(\overline{\rho}\) to \(R\). \(R\) is universal in the sense that any lifting \(\rho:G_{K,S}\rightarrow\text{GL}_N(A)\) for a coefficient ring \(A\) factors through \(\text{GL}_N(R)\). In other words, the functor \[ D_{\overline{\rho}}:\{\text{coefficient rings with residue field \(k\)}\}\rightarrow\text{Sets}, \] is representable by \(R\), i.e. \(D_{\overline{\rho}}(A)\simeq\operatorname{Hom}_{W(k)\text{-alg}}(R,A)\), where \(W(k)\) is the ring of Witt vectors of \(k\). \(\text{Spec}(R(\overline{\rho}))\) is called the universal deformation space. One may ask e.g. which points of \(\text{Spec}(R(\overline{\rho}))\) correspond to modular representations, or which points give irreducible representations on the étale cohomology of algebraic varieties (i.e. in Fontaine’s language, come from algebraic geometry). Several conjectures are related to \(\text{Spec}(R)\). On the other hand, there seems to be no relationship between the deformation space of a variety \(V\) defined over a number field and the deformation space of the various Galois representations ocurring in the étale cohomology of \(V\).

In the sequel, \(\Pi\) will be a profinite group (e.g. \(G_{K,S}\)), and \(\Lambda\) will denote a coefficient ring with residue field \(k\) of characteristic \(p\) (e.g. \(\Lambda=W(k)\)), and the coefficient ring \(A\) will be assumed to be a \(\Lambda\)-algebra. Denote by \(\widehat{\mathcal C}_{\Lambda}(A)\) the category of coefficient-\(\Lambda\)-algebras endowed with an \(A\)-augmentation. \(\mathcal C_{\Lambda}(A)\) will mean the full subcategory of \(\widehat{\mathcal C}_{\Lambda}(A)\) whose objects are artinian coefficient-\(\Lambda\)-algebras with \(A\)-augmentation. One writes \(\widehat{\mathcal C}_{\Lambda}\), resp. \(\mathcal C_{\Lambda}\) for \(\widehat{\mathcal C}_{\Lambda}(k)\), resp. \(\mathcal C_{\Lambda}(k)\). For a covariant functor \(D:{\mathcal C}_{\Lambda}\rightarrow\text{Sets}\), such that \(D(k)\) consists of a single element, one defines the Zariski tangent \(k\)-vector space \(t_D\) of \(D\) by \(t_D=D(k[\varepsilon])\), where \(k[\varepsilon]\) is the \(\Lambda\)-algebra defined by the relation \(\varepsilon^2=0\). Similarly in the relative case, for a covariant functor \(D:{\mathcal C}_{\Lambda}(A)\rightarrow\text{Sets}\) such that \(D(A)\) consists of a single point, one defines its Zariski tangent \(A\)-module \(t_{D,A}=D(A[\varepsilon])\), with \(A[\varepsilon]= A[T]/(T^2)=A\oplus A[\varepsilon]\), \(\varepsilon=T \pmod{T^2}\). One assumes the functor \(D\) satisfies hypothesis \(({\mathcal T}_k)\): \[ ({\mathcal T}_k):D(k[\varepsilon]\times_kk[\varepsilon])\rightarrow D(k[\varepsilon])\times D(k[\varepsilon])\quad \text{is a bijection.} \] Similarly in the relative case one has the hypothesis \(({\mathcal T}_A)\) for \(D(A[\varepsilon]\times_AA[\varepsilon])\). Functors on \(\mathcal C_{\Lambda}\) represented by objects of \(\widehat{\mathcal C}_{\Lambda}\) are called pro-representable. For a covariant functor \(D\) satisfying \(({\mathcal T}_A)\) and pro-representable by an \(A\)-augmented coefficient-\(\Lambda\)-algebra \(R\), one defines \(t_{D,\rho}\simeq\) subset of \(\operatorname{Hom}_{\Lambda\text{-alg}}(R,A[\varepsilon])\) consisting of those \(\Lambda\)-algebra homomorphisms whose composition with the projection \(A[\varepsilon]\rightarrow A\) is equal to the representation \(\rho:R\rightarrow A\). \(t_{D,\rho}\) is directly related to the \(R\)-module of continuous Kähler differentials \(\Omega_{R/\Lambda}\). Schlessinger’s theorem gives necessary and sufficient conditions for \(D\) to be (pro)-representable.

Assume from now on that \(\Pi\) satisfies the \(p\)-finiteness condition. One may define \(\Lambda\)-deformation problems in a functorial way: \((i)\) The “absolute” \(\Lambda\)-deformation problem: this is given by the functor \(D_{\overline{\rho}}:\widehat{\mathcal C}_{\Lambda}\rightarrow\text{Sets}\), which associates the set \(D_{\overline{\rho}}(B)\) of deformations of \(\overline{\rho}:\Pi\rightarrow\text{GL}_N(k)\) to the coefficient-\(\Lambda\)-algebra \(B\). \((ii)\) For a lifting \(\rho:\Pi\rightarrow\text{GL}_N(A)\) of \(\overline{\rho}\) one has the “relative” \(\Lambda\)-deformation problem given by the functor \(D_{\rho}:\widehat{\mathcal C}_{\Lambda}(A)\rightarrow\text{Sets}\), with \(D_{\rho}(B):=\) the set of deformations of \(\rho\) to \(B\). For the absolute case one has the result (among others): If \(\overline{\rho}\) is absolutely irreducible, then \(D_{\overline{\rho}}\) is representable. For the relative case one must change the condition of representability to what is called “nearly representability”. For a deformation problem given by the functor \(D_{\rho}:\widehat{\mathcal C}_{\Lambda}(A)\rightarrow\text{Sets}\), one has a natural isomorphism of \(A\)-modules \(t_{\rho}=t_{D_{\rho},A}\simeq H^1(\Pi,\text{End}_A(V))\), where \(V=A^N\) is the representation module of \(\rho\), and where \(\text{End}_A(V)\) carries the adjoint action of \(\Pi\). As a matter of fact, under the usual conditions above, \(H^1(\Pi,\text{End}_A(V))\) is of finite type as an \(A\)-module. Furthermore, \[ t_{\rho}\simeq D_{\rho}(A[\varepsilon])\simeq H^1(\Pi,\text{End}_A(V))\simeq\text{Ext}^1_{A[\Pi]}(V,V). \] It often happens that one encounters a deformation problem with additional conditions. This can be made precise by the notion of a deformation condition, and then translates into the study of certain sub-\(A\)-modules of \(t_{\rho}\), i.e. of \(H^1(\Pi,\text{End}_A(V))\). A first example is provided by representations \(\rho\) of prescribed determinant. The corresponding sub-\(A\)-module turns out to be \(H^1(\Pi,\text{End}^0_A(V))\), where \(\text{End}^0_A(V)\subset\text{End}_A(V)\) consists of traceless endomophisms. A second example is given by Ramakrishna’s theory for \(\Pi=G_{{\mathbb{Q}},S}\) with \(\rho\) being finite flat at \(p\in S\). By a global Galois deformation problem is meant a deformation condition for the decomposition group \(\Pi=G_{K_v}\), \(\forall v\in S\). Cases of interest include those of minimal ramification when the residual characteristic \(\ell\) of \(K_v\) is not equal to \(p\), ordinaryness and ramification for \(\ell=p\), and finite flatness for \(K_v={\mathbb{Q}}_{\ell}\), \(\ell\neq 2\). These examples are discussed in detail.

For the entire collection see [Zbl 0878.11004].

Let \(K\) be a number field, and let \(S\) be a finite set of non-archimedean places of \(K\). Denote by \(\overline{K}_S\) the maximal algebraic extension of \(K\) in the algebraic closure \(\overline{K}\) of \(K\). The corresponding Galois group is written \(G_{K,S}\). \(G_{K,S}\) is endowed with the Krull topology so that it becomes a profinite quotient of \(G_K:=\text{Gal}(\overline{K}/K)\). In general, \(G_{K,S}\) is not very well known, but one knows at least that for any prime \(p\), \(G_{K,S}\) satisfies the \(p\)-finiteness condition, i.e. for all open subgroups \(\Pi_0\subset\Pi=G_{K,S}\) of finite index, there are only finitely many continuous homomorphisms \(\Pi_0\rightarrow{\mathbb{Z}}/p{\mathbb{Z}}\). Each place \(v\) of \(K\) gives rise to a continuous homomorphism \(i_v:G_{K_v}=\text{Gal}(\overline{K}_v/K_v)\rightarrow G_{K,S}\), and one is particularly interested in understanding the “package” \(\{G_{K,S},i_v\}\), for all places \(v\) of \(K\), where the \(i_v\) are determined up to conjugation. For non-archimedean \(v\), \(v\not\in S\), one has Frobenius elements \(\phi_v=i_v(\text{Frob}_v)\in G_{K,S}\) unique up to conjugation. A fruitful approach to understanding the package \(\{G_{K,S},i_v\}\) is by studying the representations of \(G_{K,S}\). This leads to Galois representations \(\rho:G_{K,S}\rightarrow\text{GL}_N(A)\), where \(A\) denotes a coefficient ring, i.e. a complete noetherian local ring with finite residue field \(k\), such that \(\rho\) is a continuous homomorphism. Such group representations are in 1-1 correspondence with continuous \(A\)-algebra homomorphisms \(\tau:A[[\Pi]]\rightarrow M_N(A)\), where \(A[[\Pi]]\) is the completed group ring. Associated to \(\rho\) there is the residual representation \(\overline{\rho}:\Pi\rightarrow\text{GL}_N(k)\). Then \(\overline{\rho}\) is absolutely irreducible iff \(\tau\) is surjective. A homomorphism \(h:A_1\rightarrow A_0\) of coefficient rings \(A_0\), \(A_1\), and a representation \(\rho_0:\Pi\rightarrow\text{GL}_N(A_0)\) give rise to the notion of deformation of \(\rho_0\) to the coefficient ring \(A_1\), i.e. a strict equivalence class of liftings \[ \begin{matrix} \Pi & \buildrel\rho_1\over\longrightarrow & \text{GL}_N(A_1)\\ & \searrow&\downarrow\scriptstyle{h}\\ && \text{GL}_N(A_0) \end{matrix} \] where two liftings \(\rho_1\), \(\rho_1'\) are strictly equivalent if they differ by a conjugation by elements of \(\text{Ker}(h)\subset\text{GL}_N(A_1)\). Thus \(\rho\) is a deformation of its underlying \(\overline{\rho}\). Basic to the whole deformation theory of Galois representations is the following result: For a positive integer \(N\) and an absolutely irreducible representation \(\overline{\rho}:G_{K,S}\rightarrow\text{GL}_N(k)\), there exists a universal coefficient ring \(R=R(\overline{\rho})\) with residue field \(k\), and a universal deformation \(\rho^{\text{univ}}:G_{K,S}\rightarrow\text{GL}_N(R)\) of \(\overline{\rho}\) to \(R\). \(R\) is universal in the sense that any lifting \(\rho:G_{K,S}\rightarrow\text{GL}_N(A)\) for a coefficient ring \(A\) factors through \(\text{GL}_N(R)\). In other words, the functor \[ D_{\overline{\rho}}:\{\text{coefficient rings with residue field \(k\)}\}\rightarrow\text{Sets}, \] is representable by \(R\), i.e. \(D_{\overline{\rho}}(A)\simeq\operatorname{Hom}_{W(k)\text{-alg}}(R,A)\), where \(W(k)\) is the ring of Witt vectors of \(k\). \(\text{Spec}(R(\overline{\rho}))\) is called the universal deformation space. One may ask e.g. which points of \(\text{Spec}(R(\overline{\rho}))\) correspond to modular representations, or which points give irreducible representations on the étale cohomology of algebraic varieties (i.e. in Fontaine’s language, come from algebraic geometry). Several conjectures are related to \(\text{Spec}(R)\). On the other hand, there seems to be no relationship between the deformation space of a variety \(V\) defined over a number field and the deformation space of the various Galois representations ocurring in the étale cohomology of \(V\).

In the sequel, \(\Pi\) will be a profinite group (e.g. \(G_{K,S}\)), and \(\Lambda\) will denote a coefficient ring with residue field \(k\) of characteristic \(p\) (e.g. \(\Lambda=W(k)\)), and the coefficient ring \(A\) will be assumed to be a \(\Lambda\)-algebra. Denote by \(\widehat{\mathcal C}_{\Lambda}(A)\) the category of coefficient-\(\Lambda\)-algebras endowed with an \(A\)-augmentation. \(\mathcal C_{\Lambda}(A)\) will mean the full subcategory of \(\widehat{\mathcal C}_{\Lambda}(A)\) whose objects are artinian coefficient-\(\Lambda\)-algebras with \(A\)-augmentation. One writes \(\widehat{\mathcal C}_{\Lambda}\), resp. \(\mathcal C_{\Lambda}\) for \(\widehat{\mathcal C}_{\Lambda}(k)\), resp. \(\mathcal C_{\Lambda}(k)\). For a covariant functor \(D:{\mathcal C}_{\Lambda}\rightarrow\text{Sets}\), such that \(D(k)\) consists of a single element, one defines the Zariski tangent \(k\)-vector space \(t_D\) of \(D\) by \(t_D=D(k[\varepsilon])\), where \(k[\varepsilon]\) is the \(\Lambda\)-algebra defined by the relation \(\varepsilon^2=0\). Similarly in the relative case, for a covariant functor \(D:{\mathcal C}_{\Lambda}(A)\rightarrow\text{Sets}\) such that \(D(A)\) consists of a single point, one defines its Zariski tangent \(A\)-module \(t_{D,A}=D(A[\varepsilon])\), with \(A[\varepsilon]= A[T]/(T^2)=A\oplus A[\varepsilon]\), \(\varepsilon=T \pmod{T^2}\). One assumes the functor \(D\) satisfies hypothesis \(({\mathcal T}_k)\): \[ ({\mathcal T}_k):D(k[\varepsilon]\times_kk[\varepsilon])\rightarrow D(k[\varepsilon])\times D(k[\varepsilon])\quad \text{is a bijection.} \] Similarly in the relative case one has the hypothesis \(({\mathcal T}_A)\) for \(D(A[\varepsilon]\times_AA[\varepsilon])\). Functors on \(\mathcal C_{\Lambda}\) represented by objects of \(\widehat{\mathcal C}_{\Lambda}\) are called pro-representable. For a covariant functor \(D\) satisfying \(({\mathcal T}_A)\) and pro-representable by an \(A\)-augmented coefficient-\(\Lambda\)-algebra \(R\), one defines \(t_{D,\rho}\simeq\) subset of \(\operatorname{Hom}_{\Lambda\text{-alg}}(R,A[\varepsilon])\) consisting of those \(\Lambda\)-algebra homomorphisms whose composition with the projection \(A[\varepsilon]\rightarrow A\) is equal to the representation \(\rho:R\rightarrow A\). \(t_{D,\rho}\) is directly related to the \(R\)-module of continuous Kähler differentials \(\Omega_{R/\Lambda}\). Schlessinger’s theorem gives necessary and sufficient conditions for \(D\) to be (pro)-representable.

Assume from now on that \(\Pi\) satisfies the \(p\)-finiteness condition. One may define \(\Lambda\)-deformation problems in a functorial way: \((i)\) The “absolute” \(\Lambda\)-deformation problem: this is given by the functor \(D_{\overline{\rho}}:\widehat{\mathcal C}_{\Lambda}\rightarrow\text{Sets}\), which associates the set \(D_{\overline{\rho}}(B)\) of deformations of \(\overline{\rho}:\Pi\rightarrow\text{GL}_N(k)\) to the coefficient-\(\Lambda\)-algebra \(B\). \((ii)\) For a lifting \(\rho:\Pi\rightarrow\text{GL}_N(A)\) of \(\overline{\rho}\) one has the “relative” \(\Lambda\)-deformation problem given by the functor \(D_{\rho}:\widehat{\mathcal C}_{\Lambda}(A)\rightarrow\text{Sets}\), with \(D_{\rho}(B):=\) the set of deformations of \(\rho\) to \(B\). For the absolute case one has the result (among others): If \(\overline{\rho}\) is absolutely irreducible, then \(D_{\overline{\rho}}\) is representable. For the relative case one must change the condition of representability to what is called “nearly representability”. For a deformation problem given by the functor \(D_{\rho}:\widehat{\mathcal C}_{\Lambda}(A)\rightarrow\text{Sets}\), one has a natural isomorphism of \(A\)-modules \(t_{\rho}=t_{D_{\rho},A}\simeq H^1(\Pi,\text{End}_A(V))\), where \(V=A^N\) is the representation module of \(\rho\), and where \(\text{End}_A(V)\) carries the adjoint action of \(\Pi\). As a matter of fact, under the usual conditions above, \(H^1(\Pi,\text{End}_A(V))\) is of finite type as an \(A\)-module. Furthermore, \[ t_{\rho}\simeq D_{\rho}(A[\varepsilon])\simeq H^1(\Pi,\text{End}_A(V))\simeq\text{Ext}^1_{A[\Pi]}(V,V). \] It often happens that one encounters a deformation problem with additional conditions. This can be made precise by the notion of a deformation condition, and then translates into the study of certain sub-\(A\)-modules of \(t_{\rho}\), i.e. of \(H^1(\Pi,\text{End}_A(V))\). A first example is provided by representations \(\rho\) of prescribed determinant. The corresponding sub-\(A\)-module turns out to be \(H^1(\Pi,\text{End}^0_A(V))\), where \(\text{End}^0_A(V)\subset\text{End}_A(V)\) consists of traceless endomophisms. A second example is given by Ramakrishna’s theory for \(\Pi=G_{{\mathbb{Q}},S}\) with \(\rho\) being finite flat at \(p\in S\). By a global Galois deformation problem is meant a deformation condition for the decomposition group \(\Pi=G_{K_v}\), \(\forall v\in S\). Cases of interest include those of minimal ramification when the residual characteristic \(\ell\) of \(K_v\) is not equal to \(p\), ordinaryness and ramification for \(\ell=p\), and finite flatness for \(K_v={\mathbb{Q}}_{\ell}\), \(\ell\neq 2\). These examples are discussed in detail.

For the entire collection see [Zbl 0878.11004].

Reviewer: W.W.J.Hulsbergen (Haarlem)