Abstract
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 $\text{GL}_{3}(\mathbb{F}_{q})$ .
Highlights
One of the most important developments in the Langlands program in recent years has been the p-adic local Langlands correspondence for GL2(Qp)
There has been some progress on several avatars of the p-adic local Langlands correspondence, namely, Serre weight conjectures, geometric Breuil–Mezard conjecture, and Breuil’s lattice conjecture
In [LLHLM18], we develop a technique for computing these Galois deformation rings when the descent data is tame and sufficiently generic
Summary
One of the most important developments in the Langlands program in recent years has been the p-adic local Langlands correspondence for GL2(Qp). The main ingredients used in [EGS15] are the Taylor–Wiles patching method, the geometric Breuil–Mezard conjecture for potentially Barsotti–Tate Galois deformation rings (building on the work of [Bre14]), and a classification of lattices in tame types (extending [Bre, BP13]). The key local argument is a careful study of the restriction of algebraic representations to rational points (Proposition 4.2.10), which lets us constrain the submodule structure of (part of) the GL3(Fq)-restriction of an algebraic Weyl module in terms of the extension graph This method does not work for all weights σ ∈ JH(R), as the corresponding lattices will not always have simple socle and cannot be embedded into a Weyl module. If R is a ring, we let Irr(Spec (R)) denote the set of minimal primes of R
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