A symplectic look at the Fargues–Fontaine curve
Forum of Mathematics, Sigma, Tome 10 (2022)
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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 $p^{th}$ 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 $\mathbf {F}_p(\!(z)\!)$.
@article{10_1017_fms_2021_83,
author = {Yank{\i} Lekili and David Treumann},
title = {A symplectic look at the {Fargues{\textendash}Fontaine} curve},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {10},
year = {2022},
doi = {10.1017/fms.2021.83},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2021.83/}
}
Yankı Lekili; David Treumann. A symplectic look at the Fargues–Fontaine curve. Forum of Mathematics, Sigma, Tome 10 (2022). doi: 10.1017/fms.2021.83
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