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Representations p-adiques et equations differentielles
- Source :
- Mathématiques [math]. Université Pierre et Marie Curie-Paris VI, 2001. Français, Inventiones Mathematicae, Inventiones Mathematicae, 2002, 148, pp.219--284. ⟨10.1007/s002220100202⟩
- Publication Year :
- 2001
- Publisher :
- HAL CCSD, 2001.
-
Abstract
- In this paper, we associate to every $p$-adic representation $V$ a $p$-adic differential equation $\mathbf{D}^{\dagger}_{\mathrm{rig}}(V)$, 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 $mathbf{D}_{mathrm{cris}}(V)$ and $\mathbf{D}_{\mathrm{st}}(V)$ from $\mathbf{D}^{\dagger}_{\mathrm{rig}}(V)$, which allows us to recognize semi-stable or crystalline representations; the connection is then either unipotent or trivial on $\mathbf{D}^{\dagger}_{\mathrm{rig}}(V)[1/t]$. In general, the connection has an infinite number of regular singularities, but we show that $V$ 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. A recent theorem of Y. Andr\'e gives a complete description of the structure of such an 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 ($H^1_g=H^1_{st}$), 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).<br />Comment: 71 pages. In French. Uses Xypic. 3rd Version: this revised version includes a proof of Fontaine's monodromy conjecture and some applications. Submitted for publication
- Subjects :
- Pure mathematics
{é}quations diff{é}rentielles $p$-adiques
Differential equation
General Mathematics
[MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC]
Mathematics::Number Theory
monodromie $p$-adique
[MATH] Mathematics [math]
Unipotent
01 natural sciences
cristallines
Mathematics - Algebraic Geometry
th{é}orie de Hodge $p$-adique
0103 physical sciences
isocristaux surconvergents
0101 mathematics
[MATH]Mathematics [math]
Mathematics
Conjecture
Functor
Mathematics - Number Theory
P{é}riodes $p$-adiques
Hodge theory
010102 general mathematics
de de Rham
11Gxx, 11Sxx, 12H25, 13K05, 14F30
semi-stables
Mathematics - Commutative Algebra
[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT]
Monodromy
Gravitational singularity
010307 mathematical physics
[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]
repr{é}sentations $p$-adiques ordinaires
Structured program theorem
Subjects
Details
- Language :
- French
- ISSN :
- 00209910 and 14321297
- Database :
- OpenAIRE
- Journal :
- Mathématiques [math]. Université Pierre et Marie Curie-Paris VI, 2001. Français, Inventiones Mathematicae, Inventiones Mathematicae, 2002, 148, pp.219--284. ⟨10.1007/s002220100202⟩
- Accession number :
- edsair.doi.dedup.....588f05d1c8fd6dcaab7d68b40b42363e