[Paper Review] Serre weights and Breuil's lattice conjecture in dimension three

Daniel Le, Bao V. Le Hung+2 more|arXiv (Cornell University)|Aug 23, 2016

Algebraic Geometry and Number Theory27 references3 citations

TL;DR

This paper proves Breuil's lattice conjecture for $l_3$-representations over unramified $p$-adic fields, establishing that lattices in completed cohomology of $U(3)$-arithmetic manifolds are purely local—depending only on the Galois representation at places above $p$. It further proves the geometric Breuil–Mézard conjecture for potentially crystalline deformations with Hodge–Tate weights $(2,1,0)$ and verifies Serre weight conjectures in this setting, extending prior results to $l_3$ via modular representation theory and patching functors.

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, i.e., only depends on the Galois representation at places above $p$. This is a generalization to $\mathrm{GL}_3$ 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 $(0,1,2)$ as well as the Serre weight conjectures over an unramified field extending our previous results. We also prove results in modular representation theory about lattices in Deligne-Luzstig representations for the group $\mathrm{GL}_3(\mathbb{F}_q)$.

Motivation & Objective

• To generalize Breuil's lattice conjecture from $l_2$ to $l_3$-representations over unramified $p$-adic fields.
• To establish the geometric Breuil–Mézard conjecture for potentially crystalline deformation rings with Hodge–Tate weights $(2,1,0)$ in $l_3$.
• To verify the Serre weight conjectures of Herzig [Her09] over unramified fields, extending results from [LLHLM18].
• To prove structural results on lattices in Deligne–Lusztig representations for $l_3(\mathbb{F}_q)$ using injective envelopes and Weyl modules.
• To demonstrate that completed cohomology of $U(3)$-arithmetic manifolds realizes a purely local $p$-adic Langlands correspondence for $l_3$.

Proposed method

• Uses the patching functor method to construct $p$-adic Galois representations from completed cohomology of $U(3)$-arithmetic manifolds.
• Applies the theory of étale $l$-modules with descent data and semisimple Kisin modules to analyze deformation rings.
• Employs the extension graph and combinatorics of types and Serre weights to classify possible lattice structures.
• Constructs injective envelopes of irreducible representations for $l_3$ via Frobenius kernels and Weyl modules.
• Uses induction on defect and distance notions in the weight lattice to prove the structure theorem for lattices in Deligne–Lusztig representations.
• Applies the Taylor–Wiles method and solvable base change to show automorphy of Galois representations and non-vanishing of cohomology.

Experimental results

Research questions

• RQ1 Does the lattice in a tame type induced by completed cohomology of a $U(3)$-arithmetic manifold depend only on the local Galois representation at places above $p$?
• RQ2 Is the geometric Breuil–Mézard conjecture true for potentially crystalline deformation rings of $l_3$ with Hodge–Tate weights $(2,1,0)$?
• RQ3 Do the Serre weight conjectures of [Her09] hold over unramified extensions in the $l_3$ setting?
• RQ4 What is the structure of lattices in Deligne–Lusztig representations for $l_3(\mathbb{F}_q)$, particularly in terms of injective envelopes and Weyl modules?
• RQ5 Can the completed cohomology of $U(3)$-arithmetic manifolds realize a purely local $p$-adic Langlands correspondence for $l_3$?

Key findings

• The lattice in a tame type cut out by completed cohomology of a $U(3)$-arithmetic manifold is purely local, depending only on the Galois representation at places above $p$, under genericity and minimality assumptions.
• The geometric Breuil–Mézard conjecture is proven for $l_3$ with Hodge–Tate weights $(2,1,0)$, showing that components of deformation rings match with Serre weights via the Breuil–Mézard philosophy.
• The Serre weight conjectures of [Her09] are verified over unramified fields for $l_3$, extending the results of [LLHLM18] to higher rank.
• The structure of lattices in Deligne–Lusztig representations for $l_3(\mathbb{F}_q)$ is fully described using injective envelopes and Weyl modules, with a classification theorem established via distance-based induction.
• The universal deformation ring $R_{\tau}^{\mathrm{univ}}$ is finite over $\mathcal{O}_E$, implying the existence of a conjugate self-dual, minimally ramified, potentially crystalline lift of $\overline{r}$ of type $((0,1,2),\tau_v)$ at each $\widetilde{v} \in \Sigma_0^+$.
• The non-vanishing of $S(U,\sigma \otimes W)_{\mathfrak{m}}$ is established via local-global compatibility, confirming automorphy and multiplicity one in the mod $p$ cohomology.