Une conjecture $C_{\rm st}$ pour la cohomologie \`a support compact

TL;DR

The paper proves that certain p-adic functions eliminate higher-degree Galois cohomology on the Fargues-Fontaine curve, enabling new conjectures for p-adic analytic varieties' cohomology with compact support.

Contribution

It introduces p-adic analogs of logarithmic functions that trivialize Galois cohomology, facilitating the formulation of $C_{dR}$ and $C_{st}$ conjectures for compact support cohomology.

Findings

Adding p-adic logs kills Galois cohomology in degrees ≥ 1.

Results are analogous to folklore for $f B^+_{dR}$.

Enables new conjectures for p-adic analytic varieties' cohomology.

Abstract

Let $\mathbf{B}$ be the ring of analytic functions on the Fargues-Fontaine curve $Y_{\rm FF}$. We show that adding $p$-adic analogs of $\log p$ and $\log 2\pi i$ kills its Galois cohomology in degrees~$\geq 1$. The analogous result for $\mathbf{B}^+_{\rm dR}$ is folklore. This makes it possible to formulate $C_{\rm dR}$ and $C_{\rm st}$-type conjectures for compact support cohomology of $p$-adic analytic varieties.