This paper begins a series studying D-modules on the Feigin-Frenkel semi-infinite flag variety from the perspective of the Beilinson-Drinfeld factorization (or chiral) theory. Here we calculate Whittaker-twisted cohomology groups of Zastava spaces, which are certain finite-dimensional subvarieties of the affine Grassmannian. We show that such cohomology groups realize the nilradical of a Borel ...

We generalize a theorem of Tate and show that the second cohomology of the Weil group of a global or local field with coefficients in C∗ (or, more generally, with coefficients in the complex points of an algebraic torus over C) vanish, where the cohomology groups are defined using measurable cochains in the sense of Moore. We recover a theorem of Labesse stating that the admissible homomorphism...

We use functoriality of tropicalization and the geometry of projections of subvarieties of tori to show that the fibers of the tropicalization map are dense in the Zariski topology. For subvarieties of tori over fields of generalized power series, points in each tropical fiber are obtained “constructively” using Kedlaya’s transfinite version of Newton’s method.

1 Hodge structures 3 1.1 The Hodge decomposition . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 The Frölicher spectral sequence . . . . . . . . . . . . . . . . . 3 1.1.2 Hodge filtration and Hodge decomposition . . . . . . . . . . . 6 1.1.3 Hodge structures . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Morphisms of Hodge structures . . . . . . . . . . . . . . . . . . . . . 10 1...

Abstract Kottwitz’s conjecture describes the contribution of a supercuspidal representation to cohomology local Shimura variety in terms Langlands correspondence. A natural extension this concerns Scholze’s more general spaces shtukas. Using new Lefschetz–Verdier trace formula for v-stacks, we prove extended conjecture, disregarding action Weil group, and modulo virtual whose character vanishes...

This is an introduction to the work of Beilinson and Drinfeld [6] on the Langlands program. Mathematics Subject Classification (2000). Primary 11R39; Secondary 14D20.