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We study the weight part of Serre’s conjecture for genericn-dimensional modpGalois representations. We first generalize Herzig’s conjecture to the case where the field is ramified atpand prove the weight elimination direction of the conjecture. We then introduce a new class of weights associated ton-dimensional local modprepresentations which we callextremal weights. Using a “Levi reduction” property of certain potentially crystalline Galois deformation spaces, we prove the modularity of these weights. As a consequence, we deduce the weight part of Serre’s conjecture for unit groups of some division algebras in generic situations. 

Abstract Let $K/{\mathbf{Q}}_p$K/\mathbf{Q}_{p}be unramified.Inside the Emerton–Gee stack ${\mathcal{X}}_2$\mathcal{X}_{2}, one can consider the locus of two-dimensional mod 𝑝 representations of $\mathrm{Gal}\left(\bar{K}/K\right)$\mathrm{Gal}(\overline{K}/K)having a crystalline lift with specified Hodge–Tate weights.We study the case where the Hodge–Tate weights are irregular, which is an analogue for Galois representations of the partial weight one condition for Hilbert modular forms.We prove that if the gap between each pair of weights is bounded by 𝑝 (the irregular analogue of a Serre weight), then this locus is irreducible.We also establish various inclusion relations between these loci. 

Let$$G$$be a split reductive group over the ring of integers in a$$p$$-adic field with residue field$$\mathbf {F}$$. Fix a representation$$\overline {\rho }$$of the absolute Galois group of an unramified extension of$$\mathbf {Q}_p$$, valued in$$G(\mathbf {F})$$. We study the crystalline deformation ring for$$\overline {\rho }$$with a fixed$$p$$-adic Hodge type that satisfies an analog of the Fontaine–Laffaille condition for$$G$$-valued representations. In particular, we give a root theoretic condition on the$$p$$-adic Hodge type which ensures that the crystalline deformation ring is formally smooth. Our result improves on all known results for classical groups not of type A and provides the first such results for exceptional groups. 

We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a $U(3)$ -arithmetic manifold is purely local, that is, only depends on the Galois representation at places above $$p$$ . This is a generalization to $$\text{GL}_{3}$$ of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil–Mézard conjecture for (tamely) potentially crystalline deformation rings with Hodge–Tate weights $(2,1,0)$ as well as the Serre weight conjectures of Herzig [‘The weight in a Serre-type conjecture for tame $$n$$ -dimensional Galois representations’, Duke Math. J.   149 (1) (2009), 37–116] over an unramified field extending the results of Le et al. [‘Potentially crystalline deformation 3985 rings and Serre weight conjectures: shapes and shadows’, Invent. Math.   212 (1) (2018), 1–107]. We also prove results in modular representation theory about lattices in Deligne–Lusztig representations for the group $$\text{GL}_{3}(\mathbb{F}_{q})$$ . 

We construct, over any CM field, compatible systems of $$l$$ -adic Galois representations that appear in the cohomology of algebraic varieties and have (for all $$l$$ ) algebraic monodromy groups equal to the exceptional group of type $$E_{6}$$ .