##
**On Serre’s reciprocity conjecture for 2-dimensional mod \(p\) representations of \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\).**
*(English)*
Zbl 1196.11076

The paper under review is an important step towards a proof of Serre’s conjecture [J.-P. Serre, Duke Math. J. 54, 179–230 (1987; Zbl 0641.10026)] on the modularity of certain Galois representations. In subsequent papers [Invent. Math. 178, No. 3, 485–504 (2009; Zbl 1304.11041)] and [Invent. Math. 178, No. 3, 505–586 (2009; Zbl 1304.11042)], the authors have carried out the strategy outlined in Section 6.2 to prove most cases of Serre’s Conjecture. Along with the results of M. Kisin [Invent. Math. 178, No. 3, 587–634 (2009; Zbl 1304.11043)], one can treat the missing cases and verify the full conjecture.

Let us first go over Serre’s modularity conjecture in detail before reviewing the contents of present article. A fundamental theorem due to Eichler, Shimura, Deligne and Deligne-Serre attaches to a newform a Galois representation. To be more precise, let \(f\) be a cuspidal eigenform of level \(N\), weight \(k\) and neben-character \(\chi\), with \(q\)-expansion \(q+\sum_{n=2}^{\infty}a_n q^n\). For any prime \(p\) and fixed embedding \(\iota_p:{\mathbb Q}(a_n: n \in \mathbb{N}) \hookrightarrow \overline{\mathbb{Q}}_p\), there is an irreducible Galois representation \(\rho_{f, \iota_p}: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{GL}_2(\overline{\mathbb{Q}}_p)\) which is odd (namely, \(\det(\rho_{f, \iota_p}(c))=-1\) for any complex conjugation \(c\)), and such that for any prime \(q \nmid Np\), the characteristic polynomial of \(\rho_{f, \iota_p}\) (evaluated at a Frobenius at \(q\)) is given by \(X^2-\iota_p(a_q)X + \iota_p(\chi(q)q^{k-1})\). By choosing an invariant lattice inside the representation space and reducing modulo the maximal ideal, then passing to semi-simplification, one obtains a continuous Galois representation \(\overline{\rho}_{f,\iota_p}: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{GL}_2(\overline{\mathbb{F}}_p)\).

A continuous Galois representation \(\overline{\rho}: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{GL}_2(\overline{\mathbb{F}}_p)\) is called \(S\)-type if it is odd and irreducible. Weak version of Serre’s Conjecture asserts that every \(\overline{\rho}\) of \(S\)-type comes from some modular form \(f_{\overline{\rho}}\) by the theorem recalled in the previous paragraph.

Strong form of Serre’s conjecture also specifies a weight \(k(\overline{\rho})\) and a level \(N(\overline{\rho})\) for the newform \(f_{\overline{\rho}}\). Away from \(p\), \(N(\overline{\rho})\) is the Artin conductor of \(\overline{\rho}\) and it encodes the ramification properties of \(\overline{\rho}\) outside \(p\), whereas \(k(\overline{\rho})\) contains the data on the ramification at \(p\). Strong version of Serre’s conjecture is known to follow from the weak version, thanks to the works of, among others, Buzzard, Carayol, Coleman, Diamond, Mazur and Ribet.

The current paper under review marks the following advances towards the proof of the conjecture described above:

(1) When \(2 \leq k(\overline{\rho}) < p \) or \(k(\overline{\rho})=p+1\), and \(\overline{\rho}|_{\mathbb{Q}_p(\mu_p)}\) is absolutely irreducible, the existence of minimally ramified lifts of \(\overline{\rho}\) (Theorem 3.3). This has the following meaning: Let \(\overline{\rho}: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{GL}_2(\mathbb{F})\) be an \(S\)-type representation, where \(\mathbb{F}\) is a finite field. A lift \(\rho\) of \(\overline{\rho}\) to the ring of integers of a finite extension of \(\text{Frac}(W(\mathbb{F}))\) is called minimal if it is minimally ramified at all primes \(\ell\). Away from \(p\), this means that \(N(\rho)=N(\overline{\rho})\). At \(\ell=p\) and when \(k(\overline{\rho}) \neq p+1\), we require that the Hodge-Tate weights of \(\rho\) are \((0, k(\overline{\rho})-1)\). When \(k(\overline{\rho})=p+1\), then we require that the Hodge-Tate weights are \((0,1)\) (semi-stable type) or Hodge-Tate weights are \((0,p)\) (crystalline type).

(2) Proof of Serre’s conjecture in low levels and weights (Theorem 5.2, 5.4 and 5.6). In particular, the authors establish that there are no \(S\)-type \(\overline{\rho}\) with the following invariants \(N(\overline{\rho})\), \(k(\overline{\rho})\) (which goes hand in hand with Serre’s conjecture as in each of these cases, the space \(S_{k(\overline{\rho})}(\Gamma_1(N(\overline{\rho})))\) of cusp forms is trivial):

\(\bullet\) There is no \(S\)-type \(\overline{\rho}\) with \(N(\overline{\rho})=1\), \(k(\overline{\rho})=2\), for any \(p\),

\(\bullet\) There is no semi-stable \(S\)-type \(\overline{\rho}\) with \(N(\overline{\rho})=\{2,3,5,7,13\}\), \(k(\overline{\rho})=2\),

\(\bullet\) There is no \(S\)-type \(\overline{\rho}\) with \(N(\overline{\rho})=1\), \(2\leq k(\overline{\rho}) \leq 8\) or \(k(\overline{\rho})=14\) (this final case only when \(p \neq 11\)).

The authors also prove that if \(\overline{\rho}\) is of \(S\)-type with \(N(\overline{\rho})=1\) and \(k(\overline{\rho})=12\), then \(\overline{\rho}\) comes from the Ramanujan \(\Delta\)-function for any \(p\).

(3) A strategy for reducing the odd level case of Serre’s conjecture to a modularity lifting conjecture (MLC) which reads as follows:

MLC: Let \(\mathcal{O}\) be the ring of integers of a finite extension of \(\mathbb{Q}_p\) and \(\rho: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{GL}_2(\mathcal{O})\) be a continuous, absolutely irreducible odd representation which is ramified at only finitely many primes and de Rham at \(p\) with Hodge-Tate weights \((k-1,0)\) for \(k\geq 2\). Assume that the reduction \(\overline{\rho}\) is modular. Then \(\rho\) is isomorphic to an integral model of a \(p\)-adic representation \(\rho_f\) arising from a newform \(f\).

We now start reviewing the techniques involved in the proof of the results (1)-(3).

To produce minimal lifts, the authors rely on Taylor’s potential modularity theorem, which asserts (roughly) that for an \(S\)-type \(\overline{\rho}\) (for which \(k(\overline{\rho})\) satisfies the hypotheses of (1)), there exists a totally real field \(F\) such that the restriction of \(\overline{\rho}\) to \(\text{Gal}(\overline{F}/F)\) arises from some Hilbert modular form satisfying certain local properties. One then resorts to \(R=T\) type results of Fujiwara and others, relating the relating Hecke algebras and the universal deformation ring of \(\overline{\rho}\) along with a theorem of Böckle which imply that the universal minimally ramified deformation ring of \(\overline{\rho}\) is finite flat as a \(\mathbb{Z}_p\)-module and in turn that minimally ramified lifts exist.

In order to establish (2), the authors prove in Theorem 4.2 that there is a compatible family \((\rho_\ell)\) which is minimally ramified in an appropriate sense, such that a member of this family at \(p\) reduces mod \(p\) to given \(\overline{\rho}\). Along with Theorem 4.2, the authors use non-existence statements for certain abelian varieties (due to Fontaine, Brumer, Kramer and Schoof) as well as the known cases of modularity lifting (due to Skinner and Wiles) to conclude the assertions of (2).

The MLC proposed by the authors is a generalization of the modularity lifting results used in the article under review to establish (2) above. In Section 6, the authors outline a strategy to prove Serre’s conjecture. First in Section 6.1, the authors reduce the level one case to MLC. The argument is “changing the prime” and it was used by the first named author [Duke Math. J. 134, No. 3, 557-589 (2006; Zbl 1105.11013)] in a similar manner (modified so that the known cases of MLC sufficed) in order to prove the level one case. The proof of Theorem 6.1 proceeds as follows. Let \(\overline{\rho}\) be of \(S\)-type. Using Theorem 4.2, it fits into a minimally ramified compatible family \((\rho_\ell)\). The reduction of any member \(\rho_\ell\) mod \(3\) is known to be reducible thence modular. Hence MLC implies that the whole family \((\rho_\ell)\) is modular, hence \(\overline{\rho}\) as well.

In Section 6.2, the authors develop an inductive strategy to “kill ramification” by changing the prime. The induction is on the number of prime factors dividing \(N(\overline{\rho})\), and the first step is settled in [Duke Math. J. 134, No. 3, 557–589 (2006; Zbl 1105.11013)]. For the induction step, one again chooses a minimally ramified compatible family \((\rho_\ell)\) using Theorem 4.2. If \(q\) is a prime dividing \(N(\overline{\rho})\), then one switches to \(\rho_q\), set \(\overline{\rho}_q= \rho_q \mod q\). If \(\overline{\rho}_q\) is reducible then it is already modular, otherwise it is of \(S\)-type and \(N(\overline{\rho}_q)\) is divisible by fewer primes then \(N(\overline{\rho})\), hence modular by the induction hypothesis. In either case, MLC implies that the whole family \((\rho_\ell)\) is modular, thus \(\overline{\rho}\) too.

In subsequent papers [Invent. Math. 178, No. 3, 485–504 (2009; Zbl 1304.11041) and Invent. Math. 178, No. 3, 505–586 (2009; Zbl 1304.11042)], the authors managed to prove Serre’s conjecture when

\(\bullet\) \(p>2\), \(N(\overline{\rho})\) odd,

\(\bullet\) \(p=2\), \(k(\overline{\rho})=2\).

The general case was reduced to proving the MLC when \(p=2\), \(k=2\) and \(\rho\) has non-solvable image. This has been proved by M. Kisin [Invent. Math. 178, No. 3, 587–634 (2009; Zbl 1304.11043)] extending the results of [Ann. Math. (2) 170, No. 3, 1085–1180 (2009; Zbl 1201.14034)] so as to include the case \(p=2\). Thus, the proof of Serre’s conjecture is now complete.

Let us first go over Serre’s modularity conjecture in detail before reviewing the contents of present article. A fundamental theorem due to Eichler, Shimura, Deligne and Deligne-Serre attaches to a newform a Galois representation. To be more precise, let \(f\) be a cuspidal eigenform of level \(N\), weight \(k\) and neben-character \(\chi\), with \(q\)-expansion \(q+\sum_{n=2}^{\infty}a_n q^n\). For any prime \(p\) and fixed embedding \(\iota_p:{\mathbb Q}(a_n: n \in \mathbb{N}) \hookrightarrow \overline{\mathbb{Q}}_p\), there is an irreducible Galois representation \(\rho_{f, \iota_p}: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{GL}_2(\overline{\mathbb{Q}}_p)\) which is odd (namely, \(\det(\rho_{f, \iota_p}(c))=-1\) for any complex conjugation \(c\)), and such that for any prime \(q \nmid Np\), the characteristic polynomial of \(\rho_{f, \iota_p}\) (evaluated at a Frobenius at \(q\)) is given by \(X^2-\iota_p(a_q)X + \iota_p(\chi(q)q^{k-1})\). By choosing an invariant lattice inside the representation space and reducing modulo the maximal ideal, then passing to semi-simplification, one obtains a continuous Galois representation \(\overline{\rho}_{f,\iota_p}: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{GL}_2(\overline{\mathbb{F}}_p)\).

A continuous Galois representation \(\overline{\rho}: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{GL}_2(\overline{\mathbb{F}}_p)\) is called \(S\)-type if it is odd and irreducible. Weak version of Serre’s Conjecture asserts that every \(\overline{\rho}\) of \(S\)-type comes from some modular form \(f_{\overline{\rho}}\) by the theorem recalled in the previous paragraph.

Strong form of Serre’s conjecture also specifies a weight \(k(\overline{\rho})\) and a level \(N(\overline{\rho})\) for the newform \(f_{\overline{\rho}}\). Away from \(p\), \(N(\overline{\rho})\) is the Artin conductor of \(\overline{\rho}\) and it encodes the ramification properties of \(\overline{\rho}\) outside \(p\), whereas \(k(\overline{\rho})\) contains the data on the ramification at \(p\). Strong version of Serre’s conjecture is known to follow from the weak version, thanks to the works of, among others, Buzzard, Carayol, Coleman, Diamond, Mazur and Ribet.

The current paper under review marks the following advances towards the proof of the conjecture described above:

(1) When \(2 \leq k(\overline{\rho}) < p \) or \(k(\overline{\rho})=p+1\), and \(\overline{\rho}|_{\mathbb{Q}_p(\mu_p)}\) is absolutely irreducible, the existence of minimally ramified lifts of \(\overline{\rho}\) (Theorem 3.3). This has the following meaning: Let \(\overline{\rho}: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{GL}_2(\mathbb{F})\) be an \(S\)-type representation, where \(\mathbb{F}\) is a finite field. A lift \(\rho\) of \(\overline{\rho}\) to the ring of integers of a finite extension of \(\text{Frac}(W(\mathbb{F}))\) is called minimal if it is minimally ramified at all primes \(\ell\). Away from \(p\), this means that \(N(\rho)=N(\overline{\rho})\). At \(\ell=p\) and when \(k(\overline{\rho}) \neq p+1\), we require that the Hodge-Tate weights of \(\rho\) are \((0, k(\overline{\rho})-1)\). When \(k(\overline{\rho})=p+1\), then we require that the Hodge-Tate weights are \((0,1)\) (semi-stable type) or Hodge-Tate weights are \((0,p)\) (crystalline type).

(2) Proof of Serre’s conjecture in low levels and weights (Theorem 5.2, 5.4 and 5.6). In particular, the authors establish that there are no \(S\)-type \(\overline{\rho}\) with the following invariants \(N(\overline{\rho})\), \(k(\overline{\rho})\) (which goes hand in hand with Serre’s conjecture as in each of these cases, the space \(S_{k(\overline{\rho})}(\Gamma_1(N(\overline{\rho})))\) of cusp forms is trivial):

\(\bullet\) There is no \(S\)-type \(\overline{\rho}\) with \(N(\overline{\rho})=1\), \(k(\overline{\rho})=2\), for any \(p\),

\(\bullet\) There is no semi-stable \(S\)-type \(\overline{\rho}\) with \(N(\overline{\rho})=\{2,3,5,7,13\}\), \(k(\overline{\rho})=2\),

\(\bullet\) There is no \(S\)-type \(\overline{\rho}\) with \(N(\overline{\rho})=1\), \(2\leq k(\overline{\rho}) \leq 8\) or \(k(\overline{\rho})=14\) (this final case only when \(p \neq 11\)).

The authors also prove that if \(\overline{\rho}\) is of \(S\)-type with \(N(\overline{\rho})=1\) and \(k(\overline{\rho})=12\), then \(\overline{\rho}\) comes from the Ramanujan \(\Delta\)-function for any \(p\).

(3) A strategy for reducing the odd level case of Serre’s conjecture to a modularity lifting conjecture (MLC) which reads as follows:

MLC: Let \(\mathcal{O}\) be the ring of integers of a finite extension of \(\mathbb{Q}_p\) and \(\rho: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{GL}_2(\mathcal{O})\) be a continuous, absolutely irreducible odd representation which is ramified at only finitely many primes and de Rham at \(p\) with Hodge-Tate weights \((k-1,0)\) for \(k\geq 2\). Assume that the reduction \(\overline{\rho}\) is modular. Then \(\rho\) is isomorphic to an integral model of a \(p\)-adic representation \(\rho_f\) arising from a newform \(f\).

We now start reviewing the techniques involved in the proof of the results (1)-(3).

To produce minimal lifts, the authors rely on Taylor’s potential modularity theorem, which asserts (roughly) that for an \(S\)-type \(\overline{\rho}\) (for which \(k(\overline{\rho})\) satisfies the hypotheses of (1)), there exists a totally real field \(F\) such that the restriction of \(\overline{\rho}\) to \(\text{Gal}(\overline{F}/F)\) arises from some Hilbert modular form satisfying certain local properties. One then resorts to \(R=T\) type results of Fujiwara and others, relating the relating Hecke algebras and the universal deformation ring of \(\overline{\rho}\) along with a theorem of Böckle which imply that the universal minimally ramified deformation ring of \(\overline{\rho}\) is finite flat as a \(\mathbb{Z}_p\)-module and in turn that minimally ramified lifts exist.

In order to establish (2), the authors prove in Theorem 4.2 that there is a compatible family \((\rho_\ell)\) which is minimally ramified in an appropriate sense, such that a member of this family at \(p\) reduces mod \(p\) to given \(\overline{\rho}\). Along with Theorem 4.2, the authors use non-existence statements for certain abelian varieties (due to Fontaine, Brumer, Kramer and Schoof) as well as the known cases of modularity lifting (due to Skinner and Wiles) to conclude the assertions of (2).

The MLC proposed by the authors is a generalization of the modularity lifting results used in the article under review to establish (2) above. In Section 6, the authors outline a strategy to prove Serre’s conjecture. First in Section 6.1, the authors reduce the level one case to MLC. The argument is “changing the prime” and it was used by the first named author [Duke Math. J. 134, No. 3, 557-589 (2006; Zbl 1105.11013)] in a similar manner (modified so that the known cases of MLC sufficed) in order to prove the level one case. The proof of Theorem 6.1 proceeds as follows. Let \(\overline{\rho}\) be of \(S\)-type. Using Theorem 4.2, it fits into a minimally ramified compatible family \((\rho_\ell)\). The reduction of any member \(\rho_\ell\) mod \(3\) is known to be reducible thence modular. Hence MLC implies that the whole family \((\rho_\ell)\) is modular, hence \(\overline{\rho}\) as well.

In Section 6.2, the authors develop an inductive strategy to “kill ramification” by changing the prime. The induction is on the number of prime factors dividing \(N(\overline{\rho})\), and the first step is settled in [Duke Math. J. 134, No. 3, 557–589 (2006; Zbl 1105.11013)]. For the induction step, one again chooses a minimally ramified compatible family \((\rho_\ell)\) using Theorem 4.2. If \(q\) is a prime dividing \(N(\overline{\rho})\), then one switches to \(\rho_q\), set \(\overline{\rho}_q= \rho_q \mod q\). If \(\overline{\rho}_q\) is reducible then it is already modular, otherwise it is of \(S\)-type and \(N(\overline{\rho}_q)\) is divisible by fewer primes then \(N(\overline{\rho})\), hence modular by the induction hypothesis. In either case, MLC implies that the whole family \((\rho_\ell)\) is modular, thus \(\overline{\rho}\) too.

In subsequent papers [Invent. Math. 178, No. 3, 485–504 (2009; Zbl 1304.11041) and Invent. Math. 178, No. 3, 505–586 (2009; Zbl 1304.11042)], the authors managed to prove Serre’s conjecture when

\(\bullet\) \(p>2\), \(N(\overline{\rho})\) odd,

\(\bullet\) \(p=2\), \(k(\overline{\rho})=2\).

The general case was reduced to proving the MLC when \(p=2\), \(k=2\) and \(\rho\) has non-solvable image. This has been proved by M. Kisin [Invent. Math. 178, No. 3, 587–634 (2009; Zbl 1304.11043)] extending the results of [Ann. Math. (2) 170, No. 3, 1085–1180 (2009; Zbl 1201.14034)] so as to include the case \(p=2\). Thus, the proof of Serre’s conjecture is now complete.

Reviewer: Kâzım Büyükboduk (Istanbul)

### MSC:

11F80 | Galois representations |

11R39 | Langlands-Weil conjectures, nonabelian class field theory |

11F11 | Holomorphic modular forms of integral weight |