##
**Deforming Galois representations and the conjectures of Serre and Fontaine-Mazur.**
*(English)*
Zbl 1076.11035

The author achieves a major breakthrough in the theory of deformations of \(\bmod p\) representations of the absolute Galois group \(G_{\mathbb Q}\) of \(\mathbb Q\). He proves that a continuous representation \(\rho: G_{\mathbb Q} \rightarrow \text{GL}_2(\mathbb F)\), with \(\mathbb F\) a finite field of characteristic \(p>5\), and with some mild assumptions on the behaviour of \(\rho\) when restricted to a decomposition group at \(p\) (we refer to this as the local behaviour of \(\rho\) at \(p\)), lifts to a representation \(\tilde{\rho}: G_ {\mathbb Q} \rightarrow \text{GL}_2(W(\mathbb F))\), with \(W(F)\) the Witt vectors of \(\mathbb F\), that is ramified at finitely many primes. For the precise assumptions on the local behaviour of \(\rho\) at \(p\) we refer to Theorem 1 of the paper: for instance, when \(\rho\) is odd, if \(\rho\) is ramified at \(p\), then the assumptions on \(\rho\) in Theorem 1 of the paper are fulfilled. In this case the author in fact shows that there is a lift \(\tilde{\rho}\) as above that is ramified at finitely many primes and is potentially semistable at \(p\) in the sense of Fontaine. Such \(p\)-adic representations \(\tilde{\rho}\), with these 2 additional properties, are said to be geometric in the sense of J.-M. Fontaine and B. C. Mazur [in Coates, John (ed.) et al., Elliptic curves, modular forms, & Fermat’s last theorem. Proceedings ot the conference on elliptic curves and modular forms held at the Chinese University of Hong Kong, 1993. Cambridge, MA: International Press. Ser. Number Theory, 41–78 (1995; Zbl 0839.14011)]. The paper builds on earlier work of the author [Invent. Math. 138, No. 3, 537–562 (1999; Zbl 0968.11024)], but there are new ideas in this paper which are needed to get the results here that there exist geometric lifts for a wide class of 2-dimensional odd mod \(p\) Galois representations of \(G_{\mathbb Q}\).

To put this paper in context it is useful to recall a very important conjecture of J.-P. Serre [Duke Math. J. 54, No. 1, 179–230 (1987; Zbl 0641.10026)] which was perhaps the only reason why, before Theorem 1 of this paper was proven, one would have expected a result of this kind, as from the viewpoint of pure Galois theory, there is no compelling reason to expect that \(\rho\) should lift to a \(p\)-adic representation.

Consider a representation \(\rho\) as above, and assume further that \(\rho\) is absolutely irreducible and odd: we say that such a \(\rho\) is of Serre type. Serre conjectured that \(\rho\) arises from a newform on a congruence subgroup of \(\text{ SL}_2(\mathbb Z)\). This means that there is a newform \(f \in S_k(\Gamma_1(N))\) of weight \(k\) and level \(N\) such that the continuous representation \(\rho_f\colon G_{\mathbb Q} \rightarrow \text{GL}_2({\mathcal O})\) associated to \(f\) by the work of Eichler, Shimura and Deligne, with \({\mathcal O}\) the ring of integers of a finite extension of \(\mathbb Q_p\), when reduced modulo the maximal ideal of \({\mathcal O}\), gives a representation of \(G_{\mathbb Q}\) that is isomorphic to \(\rho\). It is known that such a \(\rho_f\) is ramified at finitely many primes, and is potentially semistable at \(p\), i.e., is geometric. Serre’s conjecture is still wide open, although it is widely believed to be true. The paper under review gives significant evidence in favor of the conjecture.

The fact that in the cases mentioned above the lifting \(\widetilde{\rho}\) that is constructed is geometric has led to several applications, of which we will mention a few.

(1) R. L. Taylor [J. Inst. Math. Jussieu 1, No. 1, 125–143 (2002; Zbl 1047.11051)] has used the results of the paper to provide further important evidence towards Serre’s conjecture. He proved that for many \(\rho\) of Serre type, the lifts \(\widetilde{\rho}\) that the paper under review constructs, and hence \(\rho\) itself, arise from the \(p\)-power torsion of an abelian variety \(A\) defined over \(\mathbb Q\) which has endomorphisms by a field \(K\) such that \(\dim(A)=[K: \mathbb Q]\), something which is implied by Serre’s conjecture.

(2) The reviewer [C. Khare, Math. Res. Lett. 7, No. 4, 455–462 (2000; Zbl 0983.11028)] has used the existence of the lifts \(\widetilde{\rho}\) to prove results that allow one to verify Serre’s conjecture for many \(\rho\) of Serre type assuming that there is a totally real solvable extension \(F\) such that \(\rho|_{G_F}\) arises from a Hilbert modular form for \(F\).

(3) In turn, Taylor [Am. J. Math. 125, No. 3, 549–566 (2003; Zbl 1031.11031)] has devised a strategy that uses the previous application as an ingredient to prove Serre’s conjecture for many \(\rho\) when the finite field \(F\) involved is small; for instance, \(|F|=5\) [R. Taylor, loc. cit.], \(|F|=7\) [J. Manoharmayum, Math. Res. Lett. 8, No. 5–6, 703–712 (2001; Zbl 1001.11023)], \(|F|=9\) [J. S. Ellenberg, “Serre’s conjecture over \(F_9\)”, preprint, arXiv.org/abs/math/0107147].

(4) The reviewer [Invent. Math. 154, No. 1, 199–222 (2003; Zbl 1042.11031)] has used the results of the paper to give new proofs of the results of A. J. Wiles [Ann. Math. (2) 141, No. 3, 443–551 (1995; Zbl 0823.11029)] and Taylor and Wiles [Ann. Math. (2) 141, No. 3, 553–572 (1995; Zbl 0823.11030)].

The method by which the author constructs the lifting \(\tilde{\rho}\) is ingenious and intricate. Let us focus on the case when \(\rho\) is odd. The author in fact constructs not just one geometric lifting \(\widetilde{\rho}\) but infinitely many such liftings. The liftings are indexed by very carefully chosen finite sets of auxiliary primes \(Q\). The lifting corresponding to such an auxiliary set \(Q\), let us denote it by \(\rho_Q\), is unramified outside the set of primes at which \(\rho\) ramifies, \(p\) and the primes in \(Q\). For a prime \(q \in Q\), \(\rho_Q|_{D_q}\) is “special” where \(D_q\) is a decomposition group at \(q\), i.e., up to twist is an extension of the trivial representation by the \(p\)-adic cyclotomic character. The author uses the technique of introducing auxiliary sets of primes \(Q\), a technique that has played a prominent role in number theory dating back at least to Chebotarev’s proof of his density theorem by “crossing with auxiliary cyclotomic extensions”, and subsequently used by Artin in the proof of his reciprocity law, and more recently by Wiles in his celebrated work [op. cit.]. We will not go into the very delicate considerations involved in the choice of the auxiliary sets \(Q\); suffice it to say that the \(Q\)’s are chosen to annihilate a certain dual Selmer group, using the Poitou-Tate duality theorems and the Chebotarev density theorem. Once the set \(Q\) is chosen, the method consists of lifting \(\rho\) in stages to mod \(p^2,p^3\),\(\ldots\) representations; by this we mean that for each \(n\) the author constructs a representation \(\rho_n\: G_Q \rightarrow \text{GL}_2(W(F)/(p^n))\) such that \(\rho_1\) is \(\rho\) and \(\rho_n\) mod \(p^{n-1}\) is \(\rho_{n-1}\). These representations are unramified outside \(S \cup Q\), where \(S\) is the set consisting of \(p\) and all the primes at which \(\rho\) is ramified, and have severe restrictions on the inertial behaviour at primes in \(S\) and on the local behaviour at primes in \(Q\).

The author’s method can be summarised by saying that a well-chosen auxiliary set \(Q\) entails that if there is a mod \(p^n\) lift of \(\rho\) that satisfies certain specified local conditions at primes in \(S \cup Q\), then the mod \(p^n\) lift in fact has a mod \(p^{n+1}\) lift. Now this \(\operatorname{mod}p^{n+1}\) representation may not satisfy the specified conditions at the primes in \(S \cup Q\): the author adjusts this mod \(p^{n+1}\) lift so that it satisfies the local conditions at primes in \(S \cup Q\), and at the same time mod \(p^n\) it remains the same. The artistry in the choice of \(Q\) is in ensuring that one can carry out these two steps. There have been technical improvements to this paper made by Taylor [see op. cit., 2003].

It is quite certain that the beautiful principle for lifting mod \(p\) Galois representations of this paper can be applied to many situations, and will continue to have a wealth of arithmetic applications in the future.

To put this paper in context it is useful to recall a very important conjecture of J.-P. Serre [Duke Math. J. 54, No. 1, 179–230 (1987; Zbl 0641.10026)] which was perhaps the only reason why, before Theorem 1 of this paper was proven, one would have expected a result of this kind, as from the viewpoint of pure Galois theory, there is no compelling reason to expect that \(\rho\) should lift to a \(p\)-adic representation.

Consider a representation \(\rho\) as above, and assume further that \(\rho\) is absolutely irreducible and odd: we say that such a \(\rho\) is of Serre type. Serre conjectured that \(\rho\) arises from a newform on a congruence subgroup of \(\text{ SL}_2(\mathbb Z)\). This means that there is a newform \(f \in S_k(\Gamma_1(N))\) of weight \(k\) and level \(N\) such that the continuous representation \(\rho_f\colon G_{\mathbb Q} \rightarrow \text{GL}_2({\mathcal O})\) associated to \(f\) by the work of Eichler, Shimura and Deligne, with \({\mathcal O}\) the ring of integers of a finite extension of \(\mathbb Q_p\), when reduced modulo the maximal ideal of \({\mathcal O}\), gives a representation of \(G_{\mathbb Q}\) that is isomorphic to \(\rho\). It is known that such a \(\rho_f\) is ramified at finitely many primes, and is potentially semistable at \(p\), i.e., is geometric. Serre’s conjecture is still wide open, although it is widely believed to be true. The paper under review gives significant evidence in favor of the conjecture.

The fact that in the cases mentioned above the lifting \(\widetilde{\rho}\) that is constructed is geometric has led to several applications, of which we will mention a few.

(1) R. L. Taylor [J. Inst. Math. Jussieu 1, No. 1, 125–143 (2002; Zbl 1047.11051)] has used the results of the paper to provide further important evidence towards Serre’s conjecture. He proved that for many \(\rho\) of Serre type, the lifts \(\widetilde{\rho}\) that the paper under review constructs, and hence \(\rho\) itself, arise from the \(p\)-power torsion of an abelian variety \(A\) defined over \(\mathbb Q\) which has endomorphisms by a field \(K\) such that \(\dim(A)=[K: \mathbb Q]\), something which is implied by Serre’s conjecture.

(2) The reviewer [C. Khare, Math. Res. Lett. 7, No. 4, 455–462 (2000; Zbl 0983.11028)] has used the existence of the lifts \(\widetilde{\rho}\) to prove results that allow one to verify Serre’s conjecture for many \(\rho\) of Serre type assuming that there is a totally real solvable extension \(F\) such that \(\rho|_{G_F}\) arises from a Hilbert modular form for \(F\).

(3) In turn, Taylor [Am. J. Math. 125, No. 3, 549–566 (2003; Zbl 1031.11031)] has devised a strategy that uses the previous application as an ingredient to prove Serre’s conjecture for many \(\rho\) when the finite field \(F\) involved is small; for instance, \(|F|=5\) [R. Taylor, loc. cit.], \(|F|=7\) [J. Manoharmayum, Math. Res. Lett. 8, No. 5–6, 703–712 (2001; Zbl 1001.11023)], \(|F|=9\) [J. S. Ellenberg, “Serre’s conjecture over \(F_9\)”, preprint, arXiv.org/abs/math/0107147].

(4) The reviewer [Invent. Math. 154, No. 1, 199–222 (2003; Zbl 1042.11031)] has used the results of the paper to give new proofs of the results of A. J. Wiles [Ann. Math. (2) 141, No. 3, 443–551 (1995; Zbl 0823.11029)] and Taylor and Wiles [Ann. Math. (2) 141, No. 3, 553–572 (1995; Zbl 0823.11030)].

The method by which the author constructs the lifting \(\tilde{\rho}\) is ingenious and intricate. Let us focus on the case when \(\rho\) is odd. The author in fact constructs not just one geometric lifting \(\widetilde{\rho}\) but infinitely many such liftings. The liftings are indexed by very carefully chosen finite sets of auxiliary primes \(Q\). The lifting corresponding to such an auxiliary set \(Q\), let us denote it by \(\rho_Q\), is unramified outside the set of primes at which \(\rho\) ramifies, \(p\) and the primes in \(Q\). For a prime \(q \in Q\), \(\rho_Q|_{D_q}\) is “special” where \(D_q\) is a decomposition group at \(q\), i.e., up to twist is an extension of the trivial representation by the \(p\)-adic cyclotomic character. The author uses the technique of introducing auxiliary sets of primes \(Q\), a technique that has played a prominent role in number theory dating back at least to Chebotarev’s proof of his density theorem by “crossing with auxiliary cyclotomic extensions”, and subsequently used by Artin in the proof of his reciprocity law, and more recently by Wiles in his celebrated work [op. cit.]. We will not go into the very delicate considerations involved in the choice of the auxiliary sets \(Q\); suffice it to say that the \(Q\)’s are chosen to annihilate a certain dual Selmer group, using the Poitou-Tate duality theorems and the Chebotarev density theorem. Once the set \(Q\) is chosen, the method consists of lifting \(\rho\) in stages to mod \(p^2,p^3\),\(\ldots\) representations; by this we mean that for each \(n\) the author constructs a representation \(\rho_n\: G_Q \rightarrow \text{GL}_2(W(F)/(p^n))\) such that \(\rho_1\) is \(\rho\) and \(\rho_n\) mod \(p^{n-1}\) is \(\rho_{n-1}\). These representations are unramified outside \(S \cup Q\), where \(S\) is the set consisting of \(p\) and all the primes at which \(\rho\) is ramified, and have severe restrictions on the inertial behaviour at primes in \(S\) and on the local behaviour at primes in \(Q\).

The author’s method can be summarised by saying that a well-chosen auxiliary set \(Q\) entails that if there is a mod \(p^n\) lift of \(\rho\) that satisfies certain specified local conditions at primes in \(S \cup Q\), then the mod \(p^n\) lift in fact has a mod \(p^{n+1}\) lift. Now this \(\operatorname{mod}p^{n+1}\) representation may not satisfy the specified conditions at the primes in \(S \cup Q\): the author adjusts this mod \(p^{n+1}\) lift so that it satisfies the local conditions at primes in \(S \cup Q\), and at the same time mod \(p^n\) it remains the same. The artistry in the choice of \(Q\) is in ensuring that one can carry out these two steps. There have been technical improvements to this paper made by Taylor [see op. cit., 2003].

It is quite certain that the beautiful principle for lifting mod \(p\) Galois representations of this paper can be applied to many situations, and will continue to have a wealth of arithmetic applications in the future.

Reviewer: Chandrashekhar B. Khare (Mumbai)

### MSC:

11F80 | Galois representations |