##
**Automorphy for some \(l\)-adic lifts of automorphic mod \(l\) Galois representations. II.**
*(English)*
Zbl 1169.11021

Building on the methods developed by L. Clozel, M. Harris and the author [Publ. Math., Inst. Hautes Étud. Sci. 108, 1–181 (2008; Zbl 1169.11020)], along with the ideas from the work of M. Kisin [Moduli of finite flat group schemes and modularity, Ann. Math. (2) 170, No. 3, 1085–1180 (2009; Zbl 1201.14034)], the author proves the modularity of certain lifts of Galois representations which satisfy a list of hypotheses. Most notably, a certain “minimality” condition which is assumed in the paper cited above can be removed, bypassing the need of Ihara’s lemma for unitary groups. Thanks to the results of the paper under review, the Sato-Tate conjecture follows (combining with the results of M. Harris, N. Shepherd-Barron and the author [Ann. Math. (2) 171, No. 2, 779–813 (2010; Zbl 1263.11061)]) for elliptic curves over a totally real field with somewhere multiplicative reduction.

We copy-paste a sample modularity theorem that the author provides in his introduction (which is Corollary 5.5 in the main body of the article):

Theorem: Let \(n \in \mathbb{Z}^+\) be even and suppose \(\ell > \max\{3,n\}\) is a prime. Let \[ r: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\longrightarrow \mathrm{GSp}_n(\mathbb{Z}_\ell) \] be a continuous irreducible representation which satisfies the following hypotheses:

(\(H_R\)) \(r\) ramifies at only finitely many primes.

(\(H_{C+HT}\)) \(r\big{|}_{\text{Gal}(\overline{\mathbb{Q}}_\ell/\mathbb{Q}_\ell)}\) is crystalline and \(\dim_{\mathbb{Q}_\ell} (\text{gr}^i(r\otimes B_{dR})^{\text{Gal}(\overline{\mathbb{Q}}_\ell)/\mathbb{Q}_\ell)})=0\) unless \(i \in \{0,1,\dots, n-1\}\), in which case it has dimension 1.

(\(H_S\)) There is a prime \(q\neq \ell\) such that the semi-simplification \(r\big{|}_{G_{\mathbb{Q}_\ell}}^{\text{ss}}\) is unramified and \(r\big{|}_{G_{\mathbb{Q}_\ell}}^{\text{ss}}(\text{Frob}_\ell)\) has eigenvalues \(\{\alpha q^i: i=0,1, \dots, n-1\}\) for some \(\alpha\).

(\(H_I\)) The image of \(r \bmod {\ell}\) contains \(\mathrm{Sp}_n(\mathbb{F}_\ell)\).

(\(H_A\)) \(r \bmod {\ell}\) arises from a cuspidal automorphic representation \(\pi_0\) of \(\mathrm{GL}_n(\mathbb{A})\) for which \(\pi_{0,\infty}\) has trivial infinitesimal character and \(\pi_{0,q}\) is an unramified twist of the Steinberg representation.

Then, \(r\) arises from a cuspidal automorphic representation \(\pi\) of \(\mathrm{GL}_n(\mathbb{A})\) for which \(\pi_\infty\) has trivial infinitesimal character and \(\pi_q\) is an unramified twist of the Steinberg representation.

The content of the hypotheses are clearly explained by the author. As pointed out above, the “minimality” assumption (\(H_M\)) which was needed for the results of L. Clozel, M. Harris and the author (loc. cit.) is removed, in the expense of proving a “\(R^{\text{red}}=T\)” theorem, rather than a “\(R=T\)” theorem. To deal with a non-minimal prime \(p\), first reduce to the case \(p \cong 1 \mod \ell\) using a base change argument. The author then considers two deformation problems. Let \(\sigma\) be a generator of the tame inertia in \(\text{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)\). The first deformation problem is to consider all framed deformations (in the sense of Kisin) in which \(\sigma\) has characteristic polynomial \((x-1)^n\), and the second deformation is to consider those in which \(\sigma\) has characteristic polynomial

\((x-\zeta_1)\cdots (x-\zeta_n)\), where \(\zeta_i\) are distinct \(\ell\)th roots of unity. Let \(R^{(1)}_p\) and \(R^{(2)}_p\) denote the respective framed (local) deformation spaces. Via a variant of the Taylor-Wiles patching argument, the author obtains two framed (global) deformation rings \(R^{(1)}=R^{(1)}_p[[Y_1,\dots,Y_r]]\) and \(R^{(2)}=R^{(2)}_p[[Y_1,\dots,Y_r]]\). The main point of this is that the second deformation problem may be treated via Kisin’s variant of Taylor-Wiles method. The author then uses this along with the fact that first and second deformation problems become equal \(\mod \lambda\) (where \(\lambda\) is the prime above \(\ell\) in the coefficient ring) to obtain the “\(R^{\text{red}}=T\) theorem” mentioned above.

In Section 6 the author gives the following application of his modularity lifting theorem:

Theorem: Suppose \(K\) is a totally real field and \(E/K\) is an elliptic curve with multiplicative reduction at some prime. Then

(1) \(\text{symm}^m H^1(E)\) is potentially automorphic for any odd integer \(m\),

(2) For any positive integer \(m\), the \(L\)-function \(L(\text{symm}^m H^1(E))\) has meromorphic continuation to the whole complex plane and satisfies the expected functional equation.

(3) The Sato-Tate conjecture is true for \(E\).

The condition that \(E\) has multiplicative reduction somewhere is because of the hypothesis \((H_S)\) in the modularity lifting theorem stated above.

In a recent preprint, T. Barnet-Lamb, D. Geraghty, M. Harris and the author [A family of Calabi-Yau varieties and potential automorphy. II, see Publ. Res. Inst. Math. Sci. 47, No. 1, 29–98 (2011; Zbl 1264.11044)] have significantly improved the results of this paper and generalized them in various aspects. The main inputs for the improvements are as follows:

(A) Based on the work of G. Laumon and B.-C. Ngô [Ann. Math. (2) 168, No. 2, 477–573 (2008; Zbl 1179.22019)] and of J.-L.Waldspurger [J. Inst. Math. Jussieu 5, No. 3, 423–525 (2006; Zbl 1102.22010)]; S.-W. Shin [Galois representations arising from some compact Shimura varieties, Ann. Math. (2) 173, No. 3, 1645–1741 (2011; Zbl 1269.11053)] and G. Chenevier, L. Clozel, J.-P. Labesse and M. Harris were able to construct Galois representations for all “regular algebraic, essentially conjugate self-dual” cuspidal automorphic representations of \(\mathrm{GL}_n\) of the adeles of a CM field. Thanks to this advance, the hypothesis (\(H_S\)) may be dropped.

(B) The generalization by D. Geraghty of the modularity lifting theorems of L. Clozel, M. Harris and the author (loc. cit.) and of the paper under review to the ordinary case, allowing one to change weight using Hida families.

We copy-paste a sample modularity theorem that the author provides in his introduction (which is Corollary 5.5 in the main body of the article):

Theorem: Let \(n \in \mathbb{Z}^+\) be even and suppose \(\ell > \max\{3,n\}\) is a prime. Let \[ r: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\longrightarrow \mathrm{GSp}_n(\mathbb{Z}_\ell) \] be a continuous irreducible representation which satisfies the following hypotheses:

(\(H_R\)) \(r\) ramifies at only finitely many primes.

(\(H_{C+HT}\)) \(r\big{|}_{\text{Gal}(\overline{\mathbb{Q}}_\ell/\mathbb{Q}_\ell)}\) is crystalline and \(\dim_{\mathbb{Q}_\ell} (\text{gr}^i(r\otimes B_{dR})^{\text{Gal}(\overline{\mathbb{Q}}_\ell)/\mathbb{Q}_\ell)})=0\) unless \(i \in \{0,1,\dots, n-1\}\), in which case it has dimension 1.

(\(H_S\)) There is a prime \(q\neq \ell\) such that the semi-simplification \(r\big{|}_{G_{\mathbb{Q}_\ell}}^{\text{ss}}\) is unramified and \(r\big{|}_{G_{\mathbb{Q}_\ell}}^{\text{ss}}(\text{Frob}_\ell)\) has eigenvalues \(\{\alpha q^i: i=0,1, \dots, n-1\}\) for some \(\alpha\).

(\(H_I\)) The image of \(r \bmod {\ell}\) contains \(\mathrm{Sp}_n(\mathbb{F}_\ell)\).

(\(H_A\)) \(r \bmod {\ell}\) arises from a cuspidal automorphic representation \(\pi_0\) of \(\mathrm{GL}_n(\mathbb{A})\) for which \(\pi_{0,\infty}\) has trivial infinitesimal character and \(\pi_{0,q}\) is an unramified twist of the Steinberg representation.

Then, \(r\) arises from a cuspidal automorphic representation \(\pi\) of \(\mathrm{GL}_n(\mathbb{A})\) for which \(\pi_\infty\) has trivial infinitesimal character and \(\pi_q\) is an unramified twist of the Steinberg representation.

The content of the hypotheses are clearly explained by the author. As pointed out above, the “minimality” assumption (\(H_M\)) which was needed for the results of L. Clozel, M. Harris and the author (loc. cit.) is removed, in the expense of proving a “\(R^{\text{red}}=T\)” theorem, rather than a “\(R=T\)” theorem. To deal with a non-minimal prime \(p\), first reduce to the case \(p \cong 1 \mod \ell\) using a base change argument. The author then considers two deformation problems. Let \(\sigma\) be a generator of the tame inertia in \(\text{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)\). The first deformation problem is to consider all framed deformations (in the sense of Kisin) in which \(\sigma\) has characteristic polynomial \((x-1)^n\), and the second deformation is to consider those in which \(\sigma\) has characteristic polynomial

\((x-\zeta_1)\cdots (x-\zeta_n)\), where \(\zeta_i\) are distinct \(\ell\)th roots of unity. Let \(R^{(1)}_p\) and \(R^{(2)}_p\) denote the respective framed (local) deformation spaces. Via a variant of the Taylor-Wiles patching argument, the author obtains two framed (global) deformation rings \(R^{(1)}=R^{(1)}_p[[Y_1,\dots,Y_r]]\) and \(R^{(2)}=R^{(2)}_p[[Y_1,\dots,Y_r]]\). The main point of this is that the second deformation problem may be treated via Kisin’s variant of Taylor-Wiles method. The author then uses this along with the fact that first and second deformation problems become equal \(\mod \lambda\) (where \(\lambda\) is the prime above \(\ell\) in the coefficient ring) to obtain the “\(R^{\text{red}}=T\) theorem” mentioned above.

In Section 6 the author gives the following application of his modularity lifting theorem:

Theorem: Suppose \(K\) is a totally real field and \(E/K\) is an elliptic curve with multiplicative reduction at some prime. Then

(1) \(\text{symm}^m H^1(E)\) is potentially automorphic for any odd integer \(m\),

(2) For any positive integer \(m\), the \(L\)-function \(L(\text{symm}^m H^1(E))\) has meromorphic continuation to the whole complex plane and satisfies the expected functional equation.

(3) The Sato-Tate conjecture is true for \(E\).

The condition that \(E\) has multiplicative reduction somewhere is because of the hypothesis \((H_S)\) in the modularity lifting theorem stated above.

In a recent preprint, T. Barnet-Lamb, D. Geraghty, M. Harris and the author [A family of Calabi-Yau varieties and potential automorphy. II, see Publ. Res. Inst. Math. Sci. 47, No. 1, 29–98 (2011; Zbl 1264.11044)] have significantly improved the results of this paper and generalized them in various aspects. The main inputs for the improvements are as follows:

(A) Based on the work of G. Laumon and B.-C. Ngô [Ann. Math. (2) 168, No. 2, 477–573 (2008; Zbl 1179.22019)] and of J.-L.Waldspurger [J. Inst. Math. Jussieu 5, No. 3, 423–525 (2006; Zbl 1102.22010)]; S.-W. Shin [Galois representations arising from some compact Shimura varieties, Ann. Math. (2) 173, No. 3, 1645–1741 (2011; Zbl 1269.11053)] and G. Chenevier, L. Clozel, J.-P. Labesse and M. Harris were able to construct Galois representations for all “regular algebraic, essentially conjugate self-dual” cuspidal automorphic representations of \(\mathrm{GL}_n\) of the adeles of a CM field. Thanks to this advance, the hypothesis (\(H_S\)) may be dropped.

(B) The generalization by D. Geraghty of the modularity lifting theorems of L. Clozel, M. Harris and the author (loc. cit.) and of the paper under review to the ordinary case, allowing one to change weight using Hida families.

Reviewer: Kâzım Büyükboduk (Bonn)

### MSC:

11F80 | Galois representations |

11G18 | Arithmetic aspects of modular and Shimura varieties |

11R34 | Galois cohomology |

### Citations:

Zbl 1102.22010; Zbl 1169.11020; Zbl 1201.14034; Zbl 1263.11061; Zbl 1264.11044; Zbl 1179.22019; Zbl 1269.11053
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\textit{R. Taylor}, Publ. Math., Inst. Hautes Étud. Sci. 108, 183--239 (2008; Zbl 1169.11021)

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### References:

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[2] | C. Breuil and A. Mezard, Multiplicités modulaires et représentations de GL2(Z p ) et de \(\textup{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)\) en =p, Duke Math. J., 115 (2002), 205–310. · Zbl 1042.11030 |

[3] | L. Clozel, M. Harris, and R. Taylor, Automorphy for some -adic lifts of automorphic mod Galois representations, this volume. · Zbl 1169.11020 |

[4] | D. Eisenbud, Commutative Algebra with a View Towards Algebraic Geometry, Springer, 1994. |

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[6] | M. Harris, N. Shepherd-Barron, and R. Taylor, Ihara’s lemma and potential automorphy, Ann. Math., to appear. · Zbl 1263.11061 |

[7] | M. Harris and R. Taylor, The Geometry and Cohomology of some Simple Shimura Varieties, Ann. Math. Stud., vol. 151, Princeton University Press, 2001. · Zbl 1036.11027 |

[8] | M. Kisin, Moduli of finite flat groups schemes and modularity, Ann. Math., to appear. · Zbl 1201.14034 |

[9] | H. Matsumura, Commutative Ring Theory, Cambridge University Press, 1986. · Zbl 0603.13001 |

[10] | C. Skinner and A. Wiles, Base change and a problem of Serre, Duke Math. J., 107 (2001), 15–25. · Zbl 1016.11017 |

[11] | R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. Math., 141 (1995), 553–572. · Zbl 0823.11030 |

[12] | R. Taylor and T. Yoshida, Compatibility of local and global Langlands correspondences, J. Amer. Math. Soc., 20 (2007), 467–493. · Zbl 1210.11118 |

[13] | A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. Math., 141 (1995), 443–551. · Zbl 0823.11029 |

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