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On Serre’s conjecture for mod \(\ell \) Galois representations over totally real fields. (English) Zbl 1227.11070
This fundamental paper proposes a generalisation of Serre’s Modularity Conjecture [J.-P. Serre, Duke Math. J. 54, 179–230 (1987; Zbl 0641.10026)] to totally real fields, which is a spectacular theorem of C. Khare, J.-P. Wintenberger and M. Kisin [“Serre’s modularity conjecture. I, II. Invent. Math. 178, No. 3, 485–504, 505–586 (2009; Zbl 1304.11041, Zbl 1304.11042)], M. Kisin [“Modularity of 2-adic Barsotti-Tate representations”, Invent. Math. 178, No. 3, 587–634 (2009; Zbl 1304.11043)]. Moreover, the generalised modularity conjecture is placed in a ‘mod-\(\ell\) Langlands’ framework.
We first explain the conjecture. Let \(\ell\) be a prime number and \(K\) a totally real field with integer ring \(\mathcal{O}\). Let \(f\) be a nonzero Hilbert modular cusp form over \(K\) of level \(\mathfrak{n}\) (for some weight) which is an eigenfunction for all Hecke operators \(T_{\mathfrak{p}}\), which are indexed by the prime ideals \(\mathfrak{p}\) of \(\mathcal{O}\) and commute with each other. Let \(a_{\mathfrak{p}}\) be the eigenvalue of \(T_{\mathfrak{p}}\); it is an algebraic integer, which we consider inside \(\overline{\mathbb{Q}}_\ell\) via a fixed embedding \(\overline{\mathbb{Q}} \hookrightarrow \overline{\mathbb{Q}}_\ell\). To \(f\) is attached a continuous Galois representation \[ \rho_f: \mathrm{Gal}(\overline{K}/K) \to \mathrm{GL}_2(\overline{\mathbb{Q}}_\ell), \] which is unramified outside \(\mathfrak{n}\ell\), totally odd, meaning that the determinant of the image of any complex conjugation is \(-1\), and which is characterised by \(\mathrm{Tr}(\mathrm{Frob}_{\mathfrak{p}}) = a_{\mathfrak{p}}\) for all prime ideals \(\mathfrak{p}\) coprime to \(\mathfrak{n}\ell\). By reduction and semisimplification one obtains a continuous Galois representation \(\overline{\rho}_f: \mathrm{Gal}(\overline{K}/K) \to \mathrm{GL}_2(\overline{\mathbb{F}}_\ell)\).
The weak version of the generalisation of Serre’s Modularity Conjecture, which is attributed to ‘folklore’ by the authors, states the following:
Conjecture. Any continuous, irreducible and totally odd Galois representation \(\overline{\rho}: \mathrm{Gal}(\overline{K}/K) \to \mathrm{GL}_2(\overline{\mathbb{F}}_\ell)\) is modular, i.e., it is isomorphic to \(\overline{\rho}_f\) for some \(f\) as above.
With \(K = \mathbb{Q}\) one recovers Serre’s original case. The strong form of Serre’s original conjecture states a recipe for a weight and a level (in most cases the minimal possible ones) in which an \(f\) with \(\overline{\rho} \cong\overline{\rho}_f\) can be found. The level is taken to be the prime-to-\(\ell\) Artin conductor of \(\overline{\rho}\), hence it only depends on the ramification away from \(\ell\), and the weight depends only on the ramification at \(\ell\).
A main point of the present article is to propose a weight recipe for the generalised conjecture in the case that \(\ell\) is unramified in \(K\). The recipe again only depends on the restriction of \(\overline{\rho}\) to the inertia groups at the prime ideals of \(\mathcal{O}\) above \(\ell\). Let \(G = \mathrm{GL}_2(\mathcal{O}/(\ell)) \cong \prod_{\Lambda\mid \ell} \mathrm{GL}_2(\mathcal{O}/\Lambda)\), where the product runs over the prime ideals \(\Lambda\) of \(\mathcal{O}\) lying over \(\ell\). The authors formulate their weight conjecture in a geometric way. Instead of with Hilbert modular forms they prefer to work with holomorphic automorphic representations \(\pi\) of \(\mathrm{GL}_2(\mathbb{A}_{K,f})\) with attached residual mod \(\ell\) Galois representation \(\overline{\rho}_\pi\). Via the Jacquet-Langlands correspondence (and level raising) all such \(\overline{\rho}_\pi\) are known to occur in the \(\ell\)-torsion of a Shimura curve for some quaternion algebra \(D\) over \(K\) that is split at precisely one infinite place and at all places above \(\ell\). More precisely, there is a compact open subgroup \(U\) of \((D \otimes_K \mathbb{A}_{K,f})^\times\) of level prime to \(\ell\) such that \(\overline{\rho}_\pi\) occurs as a subquotient of \((\mathrm{Pic}^0(X_{U'})[\ell](\overline{K}) \otimes V)^G\), where \(X_{U'}\) is the Shimura curve of level \(U' = \ker(U \to G)\) and \(V\) is an \(\overline{\mathbb{F}}_\ell[G]\)-module, which may be taken to be irreducible. This leads the authors to call isomorphism classes of irreducible \(\overline{\mathbb{F}}_\ell[G]\)-modules Serre weights and to say that a given \(\overline{\rho}\) is modular of weight \(V\) if \(\overline{\rho}\) occurs for \(V\) (and \(U\)) as above. Alternatively, the modularity can also be rephrased in terms of the étale cohomology of \(X_U\) for the locally constant étale sheaf associated with \(V\).
With \(\overline{\rho}\) the authors associate a set \(W(\overline{\rho})\) of Serre weights. More precisely, with \(\overline{\rho}_\Lambda\), the restriction of \(\overline{\rho}\) to a decomposition group at \(\Lambda \mid \ell\), they associate a set \(W_\Lambda(\overline{\rho})\) of irreducible \(\overline{F}_\ell[\mathrm{GL}_2(\mathcal{O}/\Lambda)]\)-modules. These sets are defined very explicitly in terms of the classification of \(\overline{\rho}_\Lambda\). The set \(W(\overline{\rho})\) then consists precisely of the \(\overline{\mathbb{F}}_\ell[G]\)-modules \(\bigotimes_{\Lambda \mid \ell} V_\Lambda\) for \(V_\Lambda \in W_\Lambda(\overline{\rho})\).
The very important ‘weight conjecture’ (Conjecture 3.14) asserts the following.
Conjecture. Let \(\overline{\rho}\) be modular. Then the set \(W(\overline{\rho})\) is equal to the set of all Serre weights \(V\) such that \(\overline{\rho}\) is modular of weight \(V\).
In other words, if \(\overline{\rho}\) is known to be modular of some weight, then the conjecture specifies precisely all the Serre weights for which \(\overline{\rho}\) should be modular. The authors check that the conjecture is compatible with twisting and determinants. Several results have already been achieved towards the weight conjecture, notably by Gee (see, for instance, [T. Gee, Invent. Math. 184, No. 1, 1–46 (2011; Zbl 1280.11029)]) and Schein (see, for instance, [M. M. Schein, J. Reine Angew. Math. 622, 57–94 (2008; Zbl 1230.11070)]).
Another very important part of the paper concerns mod-\(\ell\) Langlands correspondences. In the spirit of a ‘global mod-\(\ell\) Langlands correspondence’, the authors associate with a representation \(\overline{\rho}\) as before a smooth representation \(\pi^D(\overline{\rho})\) of \((D \otimes \hat{\mathbb{Z}})^\times\) over \(\overline{\mathbb{F}}_\ell\), where \(D\) is a quaternion algebra which is either totally definite or has precisely one split infinite place (this distinction is useful for treating the cases \([K:\mathbb{Q}]\) even or odd separately).
In two very important conjectures (Conjecture 4.7 and 4.9) a description of \(\pi^D(\overline{\rho})\) as a restricted tensor product of smooth admissible representations \(\pi_{\mathfrak{p}}\) of \(D_{\mathfrak{p}}^\times\) is proposed. This ‘local-global compatibility conjecture’ is an analogue of a conjecture of M. Emerton [Local-global compatibility in the \(p\)-adic Langlands programme for \(\mathrm{GL}_{2,\mathbb{Q}}\), Preprint]. We give a little more detail.
For \(\mathfrak{p} \nmid \ell\), the authors define smooth admissible representations \(\pi^{D_{\mathfrak{p}}}(\overline{\rho}_{\mathfrak{p}})\) of \(D_{\mathfrak{p}}^\times\), depending only on \(\overline{\rho}_{\mathfrak{p}}\), the restriction of \(\overline{\rho}\) to a decomposition group at \(\mathfrak{p}\). For split \(D_{\mathfrak{p}}\) they use Emerton’s description, whereas the construction for quaternion algebras \(D_{\mathfrak{p}}\) is new and relies on work of M.-F. Vignéras [Number theory, Proc. 15th Journ. Arith., Ulm/FRG 1987, Lect. Notes Math. 1380, 254–266 (1989; Zbl 0694.12012)]. The local-global compatibility at \(\mathfrak{p} \nmid \ell\) asserts that \(\pi_{\mathfrak{p}}\) should be isomorphic to \(\pi^{D_{\mathfrak{p}}}(\overline{\rho}_{\mathfrak{p}})\). This statement is a strong form of level-lowering for Hilbert modular forms and results on level-lowering (e.g. by K. Fujiwara [“Deformation rings and Hecke algebras in the totally real case”, Preprint], F. Jarvis [Math. Ann. 313, No. 1, 141–160 (1999; Zbl 0978.11020)] and A. Rajaei [J. Reine Angew. Math. 537, 33–65 (2001; Zbl 0982.11023)]) imply parts of the conjecture. For \(\Lambda \mid \ell\) the authors do not completely specify the representation \(\pi_{\Lambda}\), but if \(K\) and \(D\) are unramified at \(\Lambda\), they conjecture that the Jordan-Hölder factors of the socle of \(\pi_{\Lambda}\) under a maximal compact subgroup of \(D_\Lambda^\times\) are precisely the elements of \(W_\Lambda(\overline{\rho}^\vee)\). They show that with this specification the weight conjecture is a consequence of the local-global compatibility conjecture.

MSC:
11F80 Galois representations
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F33 Congruences for modular and \(p\)-adic modular forms
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