WEIGHT STRUCTURES AND MOTIVIC COMPARISONS OF P-ADIC COHOMOLOGY THEORIES FOR RIGID ANALYTIC SPACES

In this thesis, we study weight structures and p-adic cohomology theories for smooth rigid analytic spaces in the motivic context. The work consists of two independent components. The first part concerns about weight structures, a powerful tool for studying motives introduced by Bondarko. More precisely, we construct a bounded monoidal weight structure on compact rigid analytic motives over a complete non-archimedean field K, using Galois descent. This extends the weight structure on rigid analytic motives with good reduction studied by F. Binda, M. Gallauer and A. Vezzani. In particular, the full subcategory of compact rigid analytic motives over K admits a symmetric monoidal functor, called the weight complex functor, landing in a stable infinite-category of bounded chain complexes. The second part establishes a comparison between two p-adic cohomology theories for smooth rigid analytic spaces over C_p, the completed algebraic closure of Q_p. Specifically, we prove that, for adic étale motives over C_p, the vector bundles on the Fargues--Fontaine curve arising from their Hyodo--Kato cohomology coincide with their de Rham--Fargues--Fontaine cohomology. The latter provides an overconvergent refinement of crystalline vector bundles, albeit constructed on the generic fiber. This equivalence is formulated in the setting of symmetric monoidal infinite-categories and respects the full motivic structure. Moreover, we enrich both realizations with Galois actions, obtaining G_{\breve{Q}_p}-equivariant solid quasi-coherent sheaves on the Fargues--Fontaine curve; in this equivariant setting, the comparison equivalence becomes canonical. In addition, we show that the Fargues--Fontaine cohomology defined via the d\'{e}calage functor is also motivic and agrees with the de Rham-Fargues--Fontaine cohomology through their mutual comparisons with Hyodo--Kato cohomology. Combining these two aspects, we construct two spectral sequences converging to the Hyodo--Kato cohomology of smooth quasi-compact rigid analytic spaces over Q_p (without reduction assumptions) and to the de Rham--Fargues--Fontaine cohomology of such spaces over C_p. In particular, for a smooth quasi-compact rigid analytic space over Q_p (resp. over C_p) and each i≥0, its i-th Hyodo--Kato cohomology (resp. de Rham--Fargues--Fontaine cohomology) admits a finite increasing filtration. For the Hyodo--Kato case, this filtration is the weight filtration in the sense of Deligne; for the de Rham--Fargues--Fontaine case, this is a new type of filtration, distinct from the Harder--Narasimhan filtration.