On the importance of tilting Pasted image 20211106012609.png

\({X_\mathrm{FF}}\) is also roughly a moduli of untilts, whisch allow passing between \({ \mathbf{F} }_p\) and \({ {\mathbf{Q}}_p }\). A major goal is to go from characteristic zero to characteristic \(p\) (relatively easy) and then to go back to characteristic zero (relatively hard). The curve is useful because many linear algebraic objects of \(p{\hbox{-}}\)adic theory can be translated into vector bundles over \({X_\mathrm{FF}}\), and there is Grothendieck splitting type of theorem for those.

- Tags: - #MOC #arithmetic-geometry/p-adic-hodge-theory - Refs: - Seminar notes: https://www.math.columbia.edu/~phlee/SeminarNotes/FarguesFontaine.pdf#page=1 #resources/notes - attachments/Geometric Langlands, Perfectoid Spaces, Fargues-Fontaine Overview.pdf #resources/notes - https://arxiv.org/pdf/math/0607182.pdf#page=4 #resources/papers - http://people.math.harvard.edu/~lurie/ffcurve/Lecture1-Overview.pdf #resources/course-notes - Links: - Projects/2021 Fargues Fontaine/Fargues Fontaine Notes - Reading Group_Fargues Fontaine Curve - perfectoid MOC - tilting - p-adic Hodge theory - local Langlands