4 research outputs found
A geometric Jacquet-Langlands correspondence for paramodular Siegel threefolds
We 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
.Comment: Almost final version, to appear in Math.
THE LANGLANDS-RAPOPORT CONJECTURE (Automorphic forms, Automorphic representations, Galois representations, and its related topics)
We 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
On the ordinary Hecke orbit conjecture
We 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: 38 pages; v2 is a significantly revised version of v1; main results
unchange