Weights in Serre's conjecture for \(\text{GL}_n\) via the Bernstein-Gelfand-Gelfand complex
The nice article under review is concerned with the weights in an \(n\)-dimensional generalization of Serre's Modularity Conjecture [\textit{J.-P. Serre}, Duke Math. J. 54, 179--230 (1987; Zbl 0641.10026)]. More precisely, its aim is to determine for any \textit{modular} Galois representation \(\rho: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{GL}_n(\overline{\mathbb{F}}_p)\) of a \(p\)-minute weight (see below) which Fontaine-Lafaille numbers can at most occur for the restriction of \(\rho\) to a decomposition group at~\(p\). The notions of modularity and Serre weights adopted by the author need explanation. Differing from the notion of modularity in Serre type conjectures by Ash, Doud, Pollack, Sinnott [\textit{A. Ash, D. Doud} and \textit{D. Pollack}, Duke Math. J. 112, No. 3, 521--579 (2002; Zbl 1023.11025); \textit{A. Ash} and \textit{W. Sinnott}, Duke Math. J. 105, No. 1, 1--24 (2000; Zbl 1015.11018)] and \textit{F. Herzig} [``The weight in a Serre-type conjecture for tame \(n\)-dimensional Galois representations'', Duke Math. J. 149, No. 1, 37--116 (2009; Zbl 1232.11065)], the author calls a \(\rho\) as above modular if it occurs in the étale cohomology of a certain compact Shimura variety defined over an imaginary quadratic field in which \(p\) splits. A Serre weight is defined as an isomorphism class of an irreducible \(\overline{\mathbb{F}}_p[\mathrm{GL}_n(\mathbb{F}_p)]\)-module, and \(\rho\) is modular of Serre weight~\(V\) if it occurs in the étale cohomology with the locally constant étale sheaf associated with~\(V\). The Serre weights are parametrized by so-called \(p\)-restricted tuples \((a_1,\dots,a_n)\) (by definition: \(a_1 \geq a_2 \geq \dots \geq a_n\) and \(0 \leq a_i - a_{i+1} \leq p-1\) for \(1 \leq i \leq n-1\)) such that \(0 \leq a_n < p-1\). The weight is called \(p\)-minute if the attached \(p\)-restricted tuple satisfies in addition \(a_1-a_n < p-n\) and \(a_1+a_2+\dots+a_{n-1} - (n-1)a_n < p-1\). The main theorem of the article states that a modular Galois representation \(\rho\) (as above) of a \(p\)-minute weight is such that the restriction of \(\rho\) to a decomposition group at~\(p\) is in the essential image of a Fontaine-Lafaille functor (associated with \(a_n\)) and its Fontaine-Lafaille numbers are in the set \(\{a_i + (n-i) \mid 1 \leq i \leq n\}\). In particular, the theorem gives a strong restriction on the Serre weights for which a given \(\rho\) can be modular. The author includes a discussion on how his results are related to the above mentioned conjectures of Ash, Doud, Pollack, Sinnott and Herzig. The computations of the Fontaine-Lafaille numbers are performed using Faltings' comparison theorem between étale and crystalline cohomology and the integral Bernstein-Gelfand-Gelfand resolution of \textit{P. Polo} and \textit{J. Tilouine} [Astérisque 280, 97--135 (2002; Zbl 1035.17030)].
scientific article; zbMATH DE number 5880813
scientific article