Vector bundles and \(p\)-adic Galois representations (Q2900278)

The authors have defined and studied the so-called ``Fargues-Fontaine curve in \(p\)-adic Hodge theory'' (work in progress). The paper under review is a survey of some of their constructions and results. The curve is attached to the data of two fields: \(F\) is a perfect field of characteristic \(p>0\) complete for a non-trivial absolute value (for example the fraction field of \(\projlim_{x \mapsto x^p} \mathcal{O}_{\mathbb{C}_p}/p\)), and \(E\) is a non-Archimedean locally compact field whose residue field is contained in \(F\) (for example \(E=\mathbb{Q}_p\)). The authors study some rings of analytic functions on subsets of the punctured unit disk \(\{ z \in F\), \(0<|z|<1\}\), and use these rings to construct a complete regular curve \(X=X_{F,E}\) defined over \(E\). The vector bundles on \(X_{F,E}\) that are semistable of slope \(0\) can then be interpreted in terms of certain Galois representations. More generally, Galois equivariant bundles correspond to the reviewer's \(B\)-pairs. The two main theorems about de Rham representations (``de Rham implies potentially semistable'' and ``weakly admissible implies admissible'') can be proved for vector bundles and then imply their classical counterparts.NEWLINENEWLINEThe contents of the article are (1) Curves and vector bundles (2) Bounded analytic functions (3) The rings of rigid analytic functions (4) The curve \(X\) in the case where \(F\) is algebraically closed (5) Galois descent (6) de Rham \(G_K\)-equivariant vector bundles (7) de Rham \(=\) potentially log-crystalline.NEWLINENEWLINEFor the entire collection see [Zbl 1235.00045].