\(p\)-adic representations and differential equations. (Représentations \(p\)-adiques et équations différentielles.) (French) Zbl 1113.14016

Berger, Laurent

Invent. Math. 148, No. 2, 219-284 (2002).

Summary: We associate to every \(p\)-adic representation \(V\) a \(p\)-adic differential equation \({\mathbf D}^\dagger_{\text{rig}}(V)\), that is to say a module with a connection over the Robba ring. We do this via the theory of Fontaine’s \((\Phi, \Gamma_K)\)-modules.
This construction enables us to relate the theory of \((\Phi,\Gamma_K)\)-modules to \(p\)-adic Hodge theory. We explain how to construct \({\mathbf D}_{\text{cris}}(V)\) and \({\mathbf D}_{\text{st}}(V)\) from \({\mathbf D}^\dagger_{\text{rig}}(V)\), which allows us to recognize semi-stable or crystalline representations; the connection is then unipotent or trivial on \({\mathbf D}^\dagger_{\text{rig}}(V)[1/t]\).
In general, the connection has an infinite number of regular singularities, but \(V\) is de Rham if and only if those are apparent singularities. A structure theorem for modules over the Robba ring allows us to get rid of all singularities at once, and to obtain a “classical” differential equation, with a Frobenius structure.
Using this, we construct a functor from the category of de Rham representations to that of classical \(p\)-adic differential equations with Frobenius structure. A recent theorem of Y. André [Invent. Math. 148, No. 2, 285–317 (2002; Zbl 1081.12003)] gives a complete description of the structure of the latter object. This allows us to prove Fontaine’s \(p\)-adic monodromy conjecture: every de Rham representation is potentially semi-stable.
As an application, we can extend to the case of arbitrary perfect residue fields some results of Hyodo \((H^1_g=H^1_{\text{st}})\), of Perrin-Riou (the semi-stability of ordinary representations), of Colmez (absolutely crystalline representations are of finite height), and of Bloch and Kato (if the weights of \(V\) are \(\geq 2\), then Bloch-Kato’s exponential \(\exp_V\) is an isomorphism).

MSC:

14F30	\(p\)-adic cohomology, crystalline cohomology
11S20	Galois theory
12H25	\(p\)-adic differential equations
14F40	de Rham cohomology and algebraic geometry
14G20	Local ground fields in algebraic geometry

Citations:

Zbl 1081.12003