Previous Special Year Seminar

We give a new local characterization of the Local Langlands Correspondence, using deformation spaces of p-divisible groups, and show its existence by a comparison with the cohomology of some Shimura varieties. This reproves results of Harris-Taylor...

Picard moduli spaces parametrize principally polarized abelian varieties with complex multiplication by the ring of integers in an imaginary-quadratic field. The loci where the abelian varieties split off an elliptic curve in a controlled way are...

Some automorphic forms, despite the fact they are algebraic, do not have any interpretation as cohomology classes on a Shimura variety: therefore nothing is known at present on their expected arithmetic properties. I shall explain how such forms...

Let F be a locally compact non-Archimedean field, p its residue characteristic and G a connected reductive algebraic group over F . The classical Satake isomorphism describes the Hecke algebra (over the field of complex numbers) of double classes in...

I will outline the proof of various cases of the local-global compatibility statement alluded to in the title, and also explain its applications to the Fontaine--Mazur conjecture, and to a conjecture of Kisin.

I will outline the proof of various cases of the local-global compatibility statement alluded to in the title, and also explain its applications to the Fontaine—Mazur conjecture, and to a conjecture of Kisin.

We will discuss a generalisation of Serre's conjecture on the possible weights of modular mod p Galois representations for a broad class of reductive groups. In good cases (essentially when the Galois representation is tamely ramified at p) the...