Représentations p-adiques et équations différentielles

By Laurent Berger. 2001

309 citations recorded by the source corpus.

Abstract.In this paper, we associate to every p-adic representation V a p-adic differential equation D†rig(V), that is to say a module with a connection over the Robba ring. We do this via the theory of Fontaine’s (ϕ,ΓK)-modules.¶This construction enables us to relate the theory of (ϕ,ΓK)-modules to p-adic Hodge theory. We explain how to construct Dcris(V) and Dst(V) from D†rig(V), which allows us to recognize semi-stable or crystalline representations; the connection is then unipotent or trivial on D†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 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 (H1g=H1st), 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 ≥2, then Bloch-Kato’s exponential expV is an isomorphism).

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d27d98abc6d6728060fd0770807f0d1fc0392de5

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Citations in the corpus

Analytic families of finite-slope Selmer groups (2013, background): “(As pointed out in [Berger 2002], the étale case was first proved by O.”
Slope filtrations for relative Frobenius (2006, background): “This point of view has been put forth chiefly by Berger with striking consequences: he has proved Fontaine’s conjecture that de Rham representations are potentially semistable [3], and given an alternate proof of the Colmez-Fontaine theorem on admissibility of filtered (φ,N)-modules [4].”
Slope filtrations revisited (2005, methodology): “The pLMT also gives rise to a proof of Fontaine’s conjecture that every de Rham representation (of the absolute Galois group of a mixed characteristic local field) is potentially semistable, via a construction of Berger [3] linking the theory of (φ,Γ)-modules to the theory of”
Construction de (φ,Γ)-modules: représentations p-adiques et B-paires (2008, background): “Réciproquement, si x ∈ (B̃†rig[1/t]) ϕ=1, alors il existe n >0 tel que x= tnxn avec xn ∈ (B̃†rig) ϕ=pn et le lemme suit alors de la proposition 3.2 de [Berger 2002] qui nous dit que (B̃†rig) ϕ=pn = (B̃+rig) ϕ=pn .”
Relative P-adic Hodge Theory: Foundations (2013, background): “In case R = L, the morphism Spec(R1 ⊕ R2)→ Proj(P ) is faithfully flat, so it can be used to define quasicoherent sheaves and compute their cohomology; this is related to Berger’s construction of B-pairs [10].”
Bloch and Kato's exponential map: three explicit formulas (2002, background): “p-adic representations and differential equations We shall now recall some of the results of [2], which allow us to recover Dcris(V ) from the (φ,Γ)-module of V .”