A symplectic look at the Fargues-Fontaine curve

Abstract

We study a version of the Fukaya category of a symplectic 2-torus with coefficients in a locally constant sheaf of rings. The sheaf of rings includes a globally defined Novikov parameter that plays its usual role in organising polygon counts by area. It also includes a ring of constants whose variation around the the torus can be encoded by a pair of commuting ring automorphisms. When these constants are perfectoid of characteristic p, one of the holonomies is trivial and the other is the power map, it is possible in a limited way to specialise the Novikov parameter to 1. We prove that the Dehn twist ring defined there is isomorphic to the homogeneous coordinate ring of a scheme introduced by Fargues and Fontaine: their 'curve of p-adic Hodge theory' for the local field.