arXiv subjects
Publications and source records attributed to Naoki Imai.
We construct the Langlands correspondence for connected reductive groups over finite fields, which we call the finite Langlands correspondence. We discuss also its relation with the categorical local Langlands correspondence.
We define Langlands parameters for connected reductive groups over finite fields and formulate the Langlands correspondence for finite fields using these parameters.
For a smooth formal scheme $\mathfrak{X}$ over the Witt vectors $W$ of a perfect field $k$, we construct a functor $\mathbb{D}_\mathrm{crys}$ from the category of prismatic $F$-crystals $(\mathcal{E},\varphi_\mathcal{E})$ (or prismatic $F$-gauges) on $\mathfrak{X}$ to the category of filtered $F$-crystals on $\mathfrak{X}$. We show that $\mathbb{D}_\mathrm{crys}(\mathcal{E},\varphi_\mathcal{E})$ enjoys strong properties when $(\mathcal{E},\varphi_\mathcal{E})$ is what we call locally filtered free (lff). Most significantly, we show that $\mathbb{D}_\mathrm{crys}$ actually induces an equivalence between the category of prismatic $F$-gauges on $\mathfrak{X}$ with Hodge--Tate weights in $[0,p-2]$ and the category of Fontaine--Laffaille modules on $\mathfrak{X}$. Finally, we use our functor $\mathbb{D}_\mathrm{crys}$ to enhance the study of prismatic Dieduonn\'e theory of $p$-divisible groups (as initiated by Ansch\"{u}tz--Le Bras) allowing one to recover the filtered crystalline Dieudonn\'e crystal from the prismatic Dieudonn\'e crystal. This in turn allows us to clarify the relationship between prismatic Dieudonn\'e theory and the work of Kim on classifying $p$-divisible groups using Breuil--Kisin modules.
This is a survey paper on the geometrization of the local Langlands correspondence by Fargues-Scholze.
We develop the Tannakian theory of (analytic) prismatic $F$-crystals on a smooth formal scheme $\mathfrak{X}$ over the ring of integers of a discretely valued field with perfect residue field. Our main result gives an equivalence between the $\mathcal{G}$-objects of prismatic $F$-crystals on $\mathfrak{X}$ and $\mathcal{G}$-objects on a newly-defined category of $\mathbb{Z}_p$-local systems on $\mathfrak{X}_\eta$: those of prismatically good reduction. Additionally, we develop a shtuka realization functor for (analytic) prismatic $F$-crystals on $p$-adic (formal) schemes and show it satisfies several compatibilities with previous work on the Tannakian theory of shtukas over such objects.
For a non-archimedean local field $F$ and a connected reductive group $G$ over $F$ equipped with a parabolic subgroup $P$, we show that the dualizing complex on $\mathrm{Bun}_P$, the moduli stack of $P$-bundles on the Fargues--Fontaine curve, can be described explicitly in terms of the modulus character of $P$. As applications, we identify various characters appearing in the theory of local and global Shimura varieties, show the Harris--Viehmann conjecture in the Hodge--Newton reducible case, and carry out some computations of the geometric Eisenstein functors for general parabolics.
For the integral canonical model $\mathscr{S}_{\mathsf{K}^p}$ of a Shimura variety $\mathrm{Sh}_{\mathsf{K}_0\mathsf{K}^p}(\mathbf{G},\mathbf{X})$ of abelian type at hyperspecial level $K_0=\mathcal{G}(\mathbb{Z}_p)$, we construct a prismatic $F$-gauge model for the `universal' $\mathcal{G}(\mathbb{Z}_p)$-local system on $\mathrm{Sh}_{\mathsf{K}_0\mathsf{K}^p}(\mathbf{G},\mathbf{X})$. We use this to obtain several new results about the $p$-adic geometry of Shimura varieties, notably an abelian-type analogue of the Serre--Tate deformation theorem (realizing an expectation of Drinfeld in the abelian-type case) and a prismatic characterization of these models at individual level.
We describe the structure of the supersingular Rapoport-Zink space associated to the group of unitary similitudes of signature (2,n-2) for an unramified quadratic extension of p-adic fields. In earlier work, two of the authors described the irreducible components in the category of schemes-up-to-perfection. The goal of this work is to remove the qualifier "up-to-perfection".
For a connected, reductive group $G$ over a finite field endowed with a cocharacter $\mu$, we define the zip cone of $(G,\mu)$ as the cone of all possible weights of mod $p$ automorphic forms on the stack of $G$-zips. This cone is conjectured to coincide with the cone of weights of characteristic $p$ automorphic forms for Hodge-type Shimura varieties of good reduction. We prove in full generality that the cone of weights of characteristic $0$ automorphic forms is contained in the zip cone, which gives further evidence to this conjecture. Furthermore, we determine exactly when the zip cone is generated by the weights of partial Hasse invariants, which is a group-theoretical generalization of a result of Diamond--Kassaei and Goldring--Koskivirta.
We construct a moduli space $\mathsf{LP}_G$ of $\mathrm{SL}_2$-parameters over $\mathbb{Q}$, and show that it has good geometric properties (e.g. explicitly parametrized geometric connected components and smoothness). We construct a Jacobson--Morozov morphism $\mathsf{JM}\colon \mathsf{LP}_G\to\mathsf{WDP}_G$ (where $\mathsf{WDP}_G$ is the moduli space of Weil--Deligne parameters considered by several other authors). We show that $\mathsf{JM}$ is an isomorphism over a dense open of $\mathsf{WDP}_G$, that it induces an isomorphism between the discrete loci $\mathsf{LP}^{\mathrm{disc}}_G\to\mathsf{WDP}_G^{\mathrm{disc}}$, and that for any $\mathbb{Q}$-algebra $A$ it induces a bijection between Frobenius semi-simple equivalence classes in $\mathsf{LP}_G(A)$ and Frobenius semi-simple equivalence classes in $\mathsf{WDP}_G(A)$ with constant (up to conjugacy) monodromy operator.
For a connected reductive group $G$ over a finite field, we define partial Hasse invariants on the stack of $G$-zip flags. We obtain similar sections on the flag space of Shimura varieties of Hodge-type. They are mod $p$ automorphic forms which cut out a single codimension one stratum. We study their properties and show that such invariants admit a natural factorization through higher rank automorphic vector bundles. We define the socle of an automorphic vector bundle, and show that partial Hasse invariants lie in this socle.
We study the supersingular locus of a reduction at an inert prime of the Shimura variety attached to $\mathrm{GU}(2,n-2)$. More concretely, we realize irreducible components of the supersingular locus as closed subschemes of flag schemes over Deligne--Lusztig varieties defined by explicit conditions after taking perfections. Moreover we study the intersections of the irreducible components. Stratifications of Deligne--Lusztig varieties defined using powers of Frobenius action appear in the description of the intersections.
For a connected reductive group $G$ over a finite field, we study automorphic vector bundles on the stack of $G$-zips. In particular, we give a formula in the general case for the space of global sections of an automorphic vector bundle in terms of the Brylinski--Kostant filtration. Moreover, we give an equivalence of categories between the category of automorphic vector bundles on the stack of $G$-zips and a category of admissible modules with actions of a zero-dimensional algebraic subgroup a Levi subgroup and monodromy operators.
We construct a family of affinoids in the Lubin-Tate perfectoid space and their formal models such that the middle cohomology of their reductions realizes the local Langlands correspondence and the local Jacquet-Langlands correspondence for the simple supercuspidal representations. The reductions of the formal models are isomorphic to the perfections of some Artin-Schreier varieties, whose cohomology realizes primitive Galois representations. We show also the Tate conjecture for Artin-Schreier varieties associated to quadratic forms.
Let $\ell$ be a prime number different from the residue characteristic of a non-archimedean local field $F$. We give formulations of $\ell$-adic local Langlands correspondences for connected reductive algebraic groups over $F$, which we conjecture to be independent of a choice of an isomorphism between the $\ell$-adic coefficient field and the complex number field.
We study the mod $\ell$ Weil representation of a finite unitary group and related Deligne--Lusztig inductions. In particular, we study their decomposition as representations of a symplectic group, and give a construction of a mod $\ell$ Howe correspondence for $(\mathrm{Sp}_{2n},\mathrm{O}_2^-)$ including the case where $p=2$.
We define etale cohomology of the moduli spaces of mixed characteristic local shtukas so that it gives smooth representations including the case where the relevant elements of the Kottwitz set are both non-basic. Then we relate the etale cohomology of different moduli spaces of mixed characteristic local shtukas using convolution morphisms, duality morphisms and twist morphisms. As an application, we show the Kottwitz conjecture in some new cases including the cases for all inner forms of $\mathrm{GL}_3$ and minuscule cocharacters. We study also some non-minuscule cases and show that the Kottwitz conjecture is true for any inner form of $\mathrm{GL}_2$ and any cocharacter if the Langlands parameter is cuspidal. On the other hand, we show that the Kottwitz conjecture does not hold as it is in non-minuscule cases if the Langlands parameter is not cuspidal. Further, we show that a generalization of the Harris--Viehmann conjecture for the moduli spaces of mixed characteristic local shtukas does not hold in Hodge--Newton irreducible cases.
We construct extended Weil representations of unitary groups over finite fields geometrically, and show that they are Shintani lifts for Weil representations.