Universal norms and the Fargues-Fontaine curve

Abstract

We study the module of universal norms associated with a de Rham $p$-adic Galois representation in a perfectoid field extension. In particular, we compute precisely this module when the Hodge-Tate weights of a representation are greater than or equal to $0$. This generalises a result by Coates and Greenberg for Abelian varieties, and partially answers a question of theirs. Our method relies on the classification of vector bundles over the Fargues-Fontaine curve.