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Représentations de hauteur finie et complexe syntomique

Abstract : The aim of this thesis is to study finite height crystalline representations in relative p-adic Hodge theory, and apply the results thus obtained towards the computation of continuous Galois cohomology of these representations via syntomic methods. In 1980’s, Fontaine initiated a program for classifying p-adic representations of the absolute Galois group of a p-adic local field by means of certain linear-algebraic objects functorially attached to the representations. One of the aspects of his program was to classify all p-adic representations of the Galois group in terms of étale (phi, Gamma)-modules. On the other hand, Fontaine showed that crystalline representations can be classified in terms of filtered phi-modules. Therefore, it is a natural question to ask for crystalline representations: Does there exist some direct relation between the filtered phi-module and the étale (phi, Gamma)-module? Fontaine explored this question himself, where he considered finite height represenations (defined in terms of (phi, Gamma)-modules) and examined their relationship with crystalline representations. This line of thought was further explored by Wach, Colmez, and Berger. In particular, Wach gave a description of finite height crystalline representations in terms of (phi, Gamma)-modules. In the relative case, the theory of (phi, Gamma)-modules has been developed by the works of Andreatta, Brinon and Iovita. Further, the analogous notion of crystalline representations was studied by Brinon. The first main contribution of our work is the notion of relative Wach modules. Motivated by the theory of Fontaine, Wach and Berger, we define and study some properties of relative Wach modules. Further, we explore their relation with Brinon’s theory of relative crystalline representations and associated F-isocrystals. The second result is concerned with the computation of Galois cohomology using syntomic complex with coefficients. This idea was utilized in a recent work of Colmez and Niziol, where they carry out the computation for cyclotomic twists of the trivial representation. Under certain technical assumptions, we show that for finite height crystalline representations, one can essentially generalize the local result of Colmez and Niziol.
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Submitted on : Friday, December 17, 2021 - 10:10:11 AM
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Abhinandan Abhinandan. Représentations de hauteur finie et complexe syntomique. Algebraic Geometry [math.AG]. Université de Bordeaux, 2021. English. ⟨NNT : 2021BORD0267⟩. ⟨tel-03485160⟩

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