Rigid cohomology and isocrystals

Padova, June 10-16, 2001

During this week, we would like to discuss some recent developements of the cohomological theories which have been inspired by Dwork’s work. The style of the conference will be a workshop in which the participants will give some lectures to be chosen among the following. In order to better indicate the ”cut” we would like to have, we suggest for every conference a ”line of developing” and references. We expect for each subject a time of one hour (perhaps more in some cases).

Depending upon the participants’ demands, we might also organize during the week some informal conferences of lighter-introductive level.

Lecture 1. – Convergent and overconvergent isocrystals. The full-faithfulness problem.

We would like to have an account of the relation between the two categories, with a view to the question of the full-faithfullness of the natural functor between them.

Ref. : [Be86], [Be96], [Ch00], [Cr87], [Et99a], [Et99b], [Et00], [Ka73], [Ke00], [LT99], [Ro75], [Ts96], [Ts97].

Lecture 2. – Frobenius action, F -isocrystals and slope filtration.

We would like here to have a view of the role played by another actor featuring the rigid world: the Frobenius functor.

Ref. : [Ch98], [CL99a], [CL99b], [CL00], [Cr87], [Cr92], [dJ98a], [dJ98b], [DO00], [EL93], [Ka79], [Ke00], [Ma97], [Ts98a].

Lecture 3. – Local and global monodromy groups. The monodromy conjecture.

We would like to give an account of the p-adic analog of the quasi-unipotency of the local monodromy in the l-adic case.

Ref. : [An00], [Cr87], [Cr92], [Cr98], [dJ98a], [dJ98b], [Ke00], [Ts97], [Ts98a].

Lecture 4. – Local structure of overconvergent differential equations.

We would like to investigate the applications arising from the work of G. Christol and Z. Mebkhout on the structure of p-adic differential equations, namely: index theorem, local monodromy, slope filtrations.

Ref. : [An00], [CM00], [Gac99], [Lo98], [Po00].

Lecture 5. – Comparison theorems between rigid cohomology and other cohomology theories.

We would like to link rigid cohomology (and all the constructions we can make out of it) to ”classical” cohomology theories, and, in some cases, to log-cohomology.

Ref. : [BC94], [Be97a], [Be97b], [Ch99], [CL00], [EL97].

Lecture 6. – Dagger analytic spaces and their de Rham cohomology.

We would like to discuss ”the category of rigid spaces with overconvergent structure sheaf” as a good setting to develop de Rham cohomology for rigid spaces.

Ref. : [Gr00a], [Gr00b], [Mr71].

Lecture 7. – Applications to ramification theory and to the Swan conductor.

In this part, we would like to study the analogies between the usual ramification and singularities theories (in the classical and étale cases) and the rigid setting: unipotent F -isocrystals, p-adic representations, Swan conductor, link with Fontaine’s theory. This part, perhaps, should be given in several lectures.

Ref. : [CM00], [Cr00], [Ga95], [Gac99], [Ma95], [Ma97], [Me85], [Ts96], [Ts98c].

We now give some hints of possible introductory talks:

Introductory talk A. – Some examples of Dwork cohomologies : trigonometric sums, hypergeometric equation. Isocrystals.

Ref. : [Be84], [CR94], [Dw74], [Dw82], [Ro86], [Ka73].

Introductory talk B. – Construction of rigid cohomology and rigid cohomology with compact supports.

Ref. : [Be86], [Be96], [Be97a], [vdP86].

Introductory talk C. – Report on cohomology with constant coefficients as a Weil cohomology.

Ref. : [Be97a], [Be97b], [Ch98], [CL98], [CL00], [CT00], [Me97], [Pe00], [Ts99].

Introductory talk D. – L-functions of F -isocrystals.

Ref. : [EK99], [EL93], [EL97], [Et99b], [Et00], [Ka73], [Wa99], [Wa00a], [Wa00b].

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