We formulate a few conjectures on some hypothetical coherent sheaves on the stacks of arithmetic local Langlands parameters, including their roles played in the local-global compatibility in the Langlands program. We survey some known results as evidences of these conjectures.

Coherent sheaves on the stack of Langlands parameters

Following the idea of [Far16], we develop the foundations for the geometric Langlands program on the Fargues-Fontaine curve. In particular, we define a category of-adic sheaves on the stack BunG of G-bundles on the Fargues-Fontaine curve, prove a geometric Satake equivalence over the Fargues-Fontaine curve, and study the stack of L-parameters. As applications, we prove finiteness results for the cohomology of local Shimura varieties and general moduli spaces of local shtukas, and define L-parameters associated with irreducible smooth representations of G(E), a map from the spectral Bernstein center to the Bernstein center, and the spectral action of the category of perfect complexes on the stack of L-parameters on the category of-adic sheaves on BunG.

Geometrization of the local Langlands correspondence

In this paper, we give a geometric construction of the Jacquet-Langlands transfer for automorphic forms of higher weights by studying the geometry of the mod $p$ fibres of different Hodge type Shimura varieties which satisfy a mild assumption and the cohomological correspondences between them.

A Geometric Jacquet-Langlands Transfer for automorphic forms of higher weights

Fargues and Scholze proved the geometric Satake equivalence over the Fargues– Fontaine curve. On the other hand, Zhu proved the geometric Satake equivalence using a Witt vector aﬃne Grassmannian. In this paper, we explain the relation between the two version of the geometric Satake equivalence via nearby cycle.

Relation between the two geometric Satake equivalence via nearby cycle

A new decomposition-integrated air quality index prediction model

Let $k$ be an algebraically closed field of characteristic $p$. Denote by $W(k)$ the ring of Witt vectors of $k$. Let $F$ denote a totally ramified finite extension of $W(k)[1/p]$ and $\mathcal{O}$ the its ring of integers. For a connected reductive group scheme $G$ over $\mathcal{O}$, we study the category $P_{L^+G}(Gr_G,\Lambda)$ of $L^+G$-equivariant perverse sheaves in $\Lambda$-coefficient on the affine Grassmannian $Gr_G$ where $\Lambda=\mathbb{Z}_{\ell}$ and $\mathbb{F}_{\ell}$ and prove it is equivalent as a tensor category to the category of finitely generated $\Lambda$-representations of the Langlands dual group of $G$.

Jize Yu

Papers

The integral geometric Satake equivalence in mixed characteristic.

Which component of air quality index drives stock price volatility in China: a decomposition-based forecasting method