Moduli of Fontaine–Laffaille modules and local–global compatibility mod p

Abstract: The mod $p$-local Langlands program generated from the observation that certain invariants on local Galois deformation rings can be predicted by the mod $p$ representation theory of $p$-adic $\mathbf{GL}_n$. A first attempt to give evidence for this program is in the expected local–global compatibility, namely that the correspondence will be realized in Hecke eigenspaces of the cohomology of locally symmetric spaces with infinite level at p. In this talk we prove one direction of this expectation, namely that the smooth $\mathbf{GL}_n(\mathbb{Q}_{p^f})$ action on Hecke eigenspaces in the mod $p$ cohomology of compact unitary groups with infinite level at $p$ determines the local Galois parameter at $p$-adic places, when the latter parameters are Fontaine–Laffaille. This is joint work in progress with D. Le, B. Le Hung, C. Park and Z. Qian.

commutative algebraalgebraic geometrynumber theory

Audience: researchers in the topic

Recent Advances in Modern p-Adic Geometry (RAMpAGe)

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