Related papers: On categorical local langlands program for $GL_n$

Moduli spaces of global $\mathbb G$-shtukas play a crucial role in the Langlands program for function fields. We analyze their functoriality properties following a change of the curve and a change of the group scheme $\mathbb G$ under…

We study unramified unitary and unitary similitude groups in an odd number of variables. Using work of the first and third named authors on the Kottwitz Conjecture for the similitude groups, we show that the Fargues--Scholze local Langlands…

We give a new proof of the base change fundamental lemma for GL_n for certain Hecke functions by comparing the cohomology of two different moduli space of $D$-shtukas. The original proof, due to Clozel and Labesse, make use of the trace…

Let $F$ be a totally real number field. We prove that a character of the spherical Hecke algebra appearing in the completed cohomology of Hilbert modular varieties is modular if the associated Galois representation is absolutely…

We prove that V. Lafforgue's global Langlands correspondence is compatible with Fargues-Scholze's semisimplified local Langlands correspondence. As a consequence, we canonically lift Fargues-Scholze's construction to a non-semisimplified…

We construct Igusa stacks for all Shimura varieties of abelian type and derive consequences for the cohomology of these Shimura varieties. As an application, we prove that the Fargues--Scholze local Langlands correspondence agrees with the…

In this note we intend to look at the moduli stacks for global $G$-shtukas from a new perspective. We discuss a unifying interpretation of several moduli spaces (stacks) including moduli of global $G$-shtukas and (a variant of the) moduli…

This article provides a ``local'' complementary to the previous results concerning the local models for the moduli stacks of ``global'' $G$-shtukas. Here we study the geometry of Rapoport-Zink spaces for local $P$-shtukas by constructing…

Mixed-parity module emerges for instance when a de Rham Galois representation is being tensored with a square root of cyclotomic character, which produces half odd integers as the corresponding Hodge-Tate weights. We build the whole…

In this article we develop the theory of local models for the moduli stacks of global $G$-shtukas, the function field analogs for Shimura varieties. Here $G$ is a smooth affine group scheme over a smooth projective curve. As the first…

We study the cohomology of certain local systems on moduli spaces of principally polarized abelian surfaces with a level 2 structure. The trace of Frobenius on the alternating sum of the \'etale cohomology groups of these local systems can…

For arbitrary reductive groups $G$ defined over a finite field, we decompose Newton strata in the special fiber of moduli spaces of global $G$-shtukas into a product of Rapoport-Zink spaces and Igusa varieties. This allows us to compare the…

We give a group theoretic definition of "local models" as sought after in the theory of Shimura varieties. These are projective schemes over the integers of a $p$-adic local field that are expected to model the singularities of integral…

In this article we study motives corresponding to the moduli stacks of G-shtukas and their local models. In particular we deal with the question of describing their motivic fundamental invariants. As an application, we provide a criterion…

We give a categorical formulation of the $p$-adic local Langlands correspondence for $\mathrm{GL}_2(\mathbb{Q}_p)$,as an embedding of the derived category of locally admissible representations into the category of Ind-coherent sheaves on…

For a set of maps of based spaces $S$ we construct a version of Weiss' orthogonal calculus which depends only on the $S$-local homotopy type of the functor involved. We show that $S$-local homogeneous functors of degree $n$ are equivalent…

We study the integral models of meta-unitary Shimura varieties through the lens of Scholze's fiber product conjecture. Reformulating Bultel's original construction in terms of moduli stacks of Shtukas and Igusa stacks, we prove the validity…

We review the analog of Fontaine's theory of crystalline $p$-adic Galois representations and their classification by weakly admissible filtered isocrystals in the arithmetic of function fields over a finite field. There crystalline Galois…

We construct functorial Igusa stacks for all Hodge-type Shimura varieties, proving a conjecture of Scholze and extending earlier results of the fourth-named author for PEL-type Shimura varieties. Using the Igusa stack, we construct a sheaf…

Let $G$ and $\tilde G$ be reductive groups over a local field $F$. Let $\eta : \tilde G \to G$ be a $F$-homomorphism with commutative kernel and commutative cokernel. We investigate the pullbacks of irreducible admissible…