Equivariant $\mathcal D$-modules on rigid analytic spaces

Notes on Equivariant D-modules

Modular Curves and Rigid-analytic Spaces

1.1. Motivation. In the original work of Katz on p-adic modular forms [Kz], a key insight is the use of Lubin’s work on canonical subgroups in 1-parameter formal groups to define a relative theory of a “canonical subgroup” in p-adic families of elliptic curves whose reduction types are good but not too supersingular. The theory initiated by Katz has been refined in various directions (as in [AG...

Etale Cohomology of Rigid Analytic Spaces

The paper serves as an introduction to etale cohomology of rigid analytic spaces. A number of basic results are proved, e.g. concerning cohomological dimension, base change, invariance for change of base elds, the homotopy axiom and comparison for etale cohomology of algebraic varieties. The methods are those of classical rigid analytic geometry and along the way a number of known results on ri...

Descent for Coherent Sheaves on Rigid-analytic Spaces

If one assumes k to have a discrete valuation, then descent theory for coherent sheaves is an old result of Gabber. Gabber’s method was extended to the general case by Bosch and Görtz [BG]. Our method is rather different from theirs (though both approaches do use Raynaud’s theory of formal models [BL1], [BL2], we use less of this theory). We think that our approach may be of independent interes...

Quantum D-modules and Equivariant Floer Theory for Free Loop Spaces

The objective of this paper is to clarify the relationships between quantum Dmodule and equivariant Floer theory. Equivariant Floer theory was introduced by Givental in his paper “Homological Geometry”. He conjectured that the quantum D-module of a symplectic manifold is isomorphic to the equivariant Floer cohomology for the universal cover of the free loop space. First, motivated by the work o...