9 research outputs found
A geometric Jacquet-Langlands correspondence for paramodular Siegel threefolds
Full text linkWe study the Picard-Lefschetz formula for the Siegel modular threefold of paramodular level and prove the weight-monodromy conjecture for its middle degree inner cohomology with arbitrary automorphic coefficients. We give some applications to the Langlands programme: Using Rapoport-Zink uniformisation of the supersingular locus of the special fiber, we construct a geometric Jacquet-Langlands correspondence between and a definite inner form, proving a conjecture of Ibukiyama. We also prove an integral version of the weight-monodromy conjecture and use it to deduce a level lowering result for cohomological cuspidal automorphic representations of .Almost final version, to appear in Math.
On the ordinary Hecke orbit conjecture
Full text linkWe prove the ordinary Hecke orbit conjecture for Shimura varieties of Hodge
type at primes of good reduction. We make use of the global Serre-Tate
coordinates of Chai as well as recent results of D'Addezio about the -adic
monodromy of isocrystals. The new ingredients in this paper are a general
monodromy theorem for Hecke-stable subvarieties for Shimura varieties of Hodge
type, and a rigidity result for the formal completions of ordinary Hecke
orbits. Along the way we show that classical Serre--Tate coordinates can be
described using unipotent formal groups, generalising results of Howe.Comment: 44 pages; v3 is a minorly revised version of v1; main results
unchange
Monodromy and Irreducibility of Igusa Varieties
Full text linkWe determine the irreducible components of Igusa varieties for Shimura varieties of Hodge type under a mild condition and use that to compute the irreducible components of central leaves. In particular, we show that a strong version of the discrete Hecke orbit conjecture is false in general. Our method combines recent work of D’Addezio on monodromy groups of compatible local systems with a generalisation of a method of Hida, using the Honda–Tate theory for Shimura varieties of Hodge type developed by Kisin–Madapusi Pera–Shin. We also determine the irreducible components of Newton strata in Shimura varieties of Hodge type by combining our methods with recent work of Zhou–Zhu.</p
THE LANGLANDS-RAPOPORT CONJECTURE (Automorphic forms, Automorphic representations, Galois representations, and its related topics)
Full text linkWe give a brief introduction to the Langlands-Rapoport conjecture, which describes the mod p points of Shimura varieties. We overview known results and explain what is missing to deal with the general case
Igusa Stacks and the Cohomology of Shimura Varieties
Full text linkWe 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 on that controls the cohomology of the corresponding Shimura variety. We use this sheaf and the spectral action of Fargues-Scholze to prove a compatibility between the cohomology of Shimura varieties of Hodge type and the semisimple local Langlands correspondence of Fargues-Scholze, generalizing the Eichler-Shimura relation of Blasius-Rogawski to arbitrary level at . When the given Shimura variety is proper, we show moreover that the sheaf is perverse, which allows us to prove new torsion vanishing results for the cohomology of Shimura varieties
Igusa Stacks and the Cohomology of Shimura Varieties
Full text linkWe 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 on that controls the cohomology of the corresponding Shimura variety. We use this sheaf and the spectral action of Fargues-Scholze to prove a compatibility between the cohomology of Shimura varieties of Hodge type and the semisimple local Langlands correspondence of Fargues-Scholze, generalizing the Eichler-Shimura relation of Blasius-Rogawski to arbitrary level at . When the given Shimura variety is proper, we show moreover that the sheaf is perverse, which allows us to prove new torsion vanishing results for the cohomology of Shimura varieties.121 pages, comments welcome
On the Piatetski-Shapiro construction for integral models of Shimura varieties
Full text linkWe study the Piatetski-Shapiro construction, which takes a totally real field F and a Shimura datum (G,X) and produces a new Shimura datum (H,Y). If F is Galois, then the Galois group Gamma of F acts on (H,Y), and we show that the Gamma-fixed points of the Shimura varieties for (H,Y) recover the Shimura varieties for (G,X) under some hypotheses. For Shimura varieties of Hodge type with parahoric level, we show that the same is true for the p-adic integral models constructed by Pappas--Rapoport, if p is unramified in F. We also study the Gamma-fixed points of the Igusa stacks of Zhang for (H,Y) and prove optimal results
On a conjecture of Pappas and Rapoport
Full text linkWe prove a conjecture of Pappas and Rapoport about the existence of ''canonical'' integral models of Shimura varieties of Hodge type with quasi-parahoric level structure at a prime . For these integral models, we moreover show uniformization of isogeny classes by integral local Shimura varieties, and prove a conjecture of Kisin and Pappas on local model diagrams
