On the weights of mod \(p\) Hilbert modular forms.

*(English)*Zbl 1280.11029The article under review is the first breakthrough on the weight conjecture in the generalisation of Serre’s modularity conjecture to Hilbert modular forms. Since the appearance of the present article, the results have been greatly generalised and the techniques have been applied in other contexts.

Let \(F\) be a totally real number field with integer ring \(\mathcal{O}\) and let \(\rho: \mathrm{Gal}(\overline{F}/F) \to \mathrm{GL}_2(\mathbb{F}_p)\) be a totally odd irreducible Galois representation. In generalisation of Serre’s modularity conjecture [J.-P. Serre, Duke Math. J. 54, 179–230 (1987; Zbl 0641.10026)] (proved by C. Khare and J.-P. Wintenberger [Invent. Math. 178, No. 3, 485–504 (2009; Zbl 1304.11041)]) one expects \(\rho\) to be modular, in the sense that it comes from some Hilbert modular form over \(F\). Serre’s original conjecture specifies a weight and a level (which in almost all cases are minimal) at which a corresponding modular form exists.

In an important work, K. Buzzard, F. Diamond and F. Jarvis [Duke Math. J. 155, No. 1, 105–161 (2010; Zbl 1227.11070)] propose a precise conjecture for the weights in the Hilbert modular forms case (if \(p\) is unramified in \(F\)). The authors use the Jacquet-Langlands correspondence and realise the Galois representations coming from Hilbert modular forms in the cohomology of Shimura curves with coefficient modules. The coefficient modules are irreducible \(\overline{\mathbb{F}}_p[G]\)-modules with \(G = \mathrm{GL}_2(\mathcal{O}/(p))\). They are called Serre weights. In the case \(F=\mathbb{Q}\) they are in natural bijection with the ‘usual’ weights for classical modular forms.

The Shimura curves used by Buzzard, Diamond and Jarvis are attached to indefinite quaternion algebras over \(F\) that are split at exactly one infinite place and all places above \(p\). In the present article, Gee prefers to work with definite quaternion algebras over \(F\) that are also split at all places above \(p\). The Jacquet-Langlands correspondence allows him to make a conjecture similar to the one of Buzzard, Diamond and Jarvis. More precisely, for a given \(\rho\) he specifies a set \(W(\rho)\) of ‘predicted weights’, which should be the set of weights for which \(\rho\) is modular. In fact, in the present article Gee introduces the notions of regular and weakly regular weights. It is for those that the methods in the article work are done. The regular weights predicted by Buzzard, Diamond and Jarvis agree with those in the article under review.

The main results in the article are the following. If \(\rho\) is modular of some regular weight, then \(\rho\) is in the set of predicted weights \(W(\rho)\). Conversely, under some additional assumptions, if \(\rho\) is modular of some weight and \(\sigma\) is a regular predicted weight in \(W(\rho)\), then \(\rho\) is modular of weight \(\sigma\).

The proof combines difficult calculations in \(p\)-adic Hodge theory, modularity lifting theorems and combinatorial arguments. More precisely, on the one hand, the definition of predicted weights involves crystalline lifts of \(\rho|G_{F_v}\) (for \(v\mid p\)). On the other hand, modularity and modularity lifting theorems naturally lead to potentially Barsotti-Tate lifts. The main technical part of the article uses the theory of Breuil modules and strongly divisible modules to relate potentially Barsotti-Tate lifts with crystalline lifts.

We give a very rough and imprecise sketch of the arguments for the main results. Assume one is given a \(\rho\) which is known to be modular of some Serre weight \(\sigma\). First a potentially Barsotti-Tate lift is constructed, which then by the results on \(p\)-adic Hodge theory is related to a crystalline lift of a certain type, showing \(\sigma \in W(\rho)\). Conversely, suppose one is given a modular \(\rho\) and a predicted weight \(\sigma \in W(\rho)\). One first has a crystalline lift of \(\rho\), which by the developed \(p\)-adic Hodge theory is related to a potentially Barsotti-Tate lift of a certain type related to \(\sigma\). Modularity lifting then assures that this potentially Barsotti-Tate lift is modular. This then implies that it can only be modular of certain weights. The regularity condition enters here to ensure that in fact it can only be modular of one weight, namely \(\sigma\), proving that \(\rho\) is indeed modular of weight \(\sigma\).

A number of papers have already taken up and extended the results and the techniques of the article under review. We only mention here that the original weight conjecture of Buzzard, Diamond and Jarvis has recently been settled under only very minor assumptions by T. Gee, T. Liu and D. Savitt [“The weight part in Serre’s conjecture for \(\mathrm{GL}(2)\)”, Preprint, arxiv:1309.0527].

Let \(F\) be a totally real number field with integer ring \(\mathcal{O}\) and let \(\rho: \mathrm{Gal}(\overline{F}/F) \to \mathrm{GL}_2(\mathbb{F}_p)\) be a totally odd irreducible Galois representation. In generalisation of Serre’s modularity conjecture [J.-P. Serre, Duke Math. J. 54, 179–230 (1987; Zbl 0641.10026)] (proved by C. Khare and J.-P. Wintenberger [Invent. Math. 178, No. 3, 485–504 (2009; Zbl 1304.11041)]) one expects \(\rho\) to be modular, in the sense that it comes from some Hilbert modular form over \(F\). Serre’s original conjecture specifies a weight and a level (which in almost all cases are minimal) at which a corresponding modular form exists.

In an important work, K. Buzzard, F. Diamond and F. Jarvis [Duke Math. J. 155, No. 1, 105–161 (2010; Zbl 1227.11070)] propose a precise conjecture for the weights in the Hilbert modular forms case (if \(p\) is unramified in \(F\)). The authors use the Jacquet-Langlands correspondence and realise the Galois representations coming from Hilbert modular forms in the cohomology of Shimura curves with coefficient modules. The coefficient modules are irreducible \(\overline{\mathbb{F}}_p[G]\)-modules with \(G = \mathrm{GL}_2(\mathcal{O}/(p))\). They are called Serre weights. In the case \(F=\mathbb{Q}\) they are in natural bijection with the ‘usual’ weights for classical modular forms.

The Shimura curves used by Buzzard, Diamond and Jarvis are attached to indefinite quaternion algebras over \(F\) that are split at exactly one infinite place and all places above \(p\). In the present article, Gee prefers to work with definite quaternion algebras over \(F\) that are also split at all places above \(p\). The Jacquet-Langlands correspondence allows him to make a conjecture similar to the one of Buzzard, Diamond and Jarvis. More precisely, for a given \(\rho\) he specifies a set \(W(\rho)\) of ‘predicted weights’, which should be the set of weights for which \(\rho\) is modular. In fact, in the present article Gee introduces the notions of regular and weakly regular weights. It is for those that the methods in the article work are done. The regular weights predicted by Buzzard, Diamond and Jarvis agree with those in the article under review.

The main results in the article are the following. If \(\rho\) is modular of some regular weight, then \(\rho\) is in the set of predicted weights \(W(\rho)\). Conversely, under some additional assumptions, if \(\rho\) is modular of some weight and \(\sigma\) is a regular predicted weight in \(W(\rho)\), then \(\rho\) is modular of weight \(\sigma\).

The proof combines difficult calculations in \(p\)-adic Hodge theory, modularity lifting theorems and combinatorial arguments. More precisely, on the one hand, the definition of predicted weights involves crystalline lifts of \(\rho|G_{F_v}\) (for \(v\mid p\)). On the other hand, modularity and modularity lifting theorems naturally lead to potentially Barsotti-Tate lifts. The main technical part of the article uses the theory of Breuil modules and strongly divisible modules to relate potentially Barsotti-Tate lifts with crystalline lifts.

We give a very rough and imprecise sketch of the arguments for the main results. Assume one is given a \(\rho\) which is known to be modular of some Serre weight \(\sigma\). First a potentially Barsotti-Tate lift is constructed, which then by the results on \(p\)-adic Hodge theory is related to a crystalline lift of a certain type, showing \(\sigma \in W(\rho)\). Conversely, suppose one is given a modular \(\rho\) and a predicted weight \(\sigma \in W(\rho)\). One first has a crystalline lift of \(\rho\), which by the developed \(p\)-adic Hodge theory is related to a potentially Barsotti-Tate lift of a certain type related to \(\sigma\). Modularity lifting then assures that this potentially Barsotti-Tate lift is modular. This then implies that it can only be modular of certain weights. The regularity condition enters here to ensure that in fact it can only be modular of one weight, namely \(\sigma\), proving that \(\rho\) is indeed modular of weight \(\sigma\).

A number of papers have already taken up and extended the results and the techniques of the article under review. We only mention here that the original weight conjecture of Buzzard, Diamond and Jarvis has recently been settled under only very minor assumptions by T. Gee, T. Liu and D. Savitt [“The weight part in Serre’s conjecture for \(\mathrm{GL}(2)\)”, Preprint, arxiv:1309.0527].

Reviewer: Gabor Wiese (Luxembourg)

##### 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 |

11S37 | Langlands-Weil conjectures, nonabelian class field theory |

14L15 | Group schemes |

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