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##### LFANT - 2019

Overall Objectives
New Software and Platforms
Partnerships and Cooperations
Bibliography

## Section: New Results

### $p$-adic rings and geometry

Participant : Xavier Caruso.

In [19], Xavier Caruso, Tristan Vaccon and Thibaut Verron laid the foundations of an algorithmic treatment of rigid $p$-adic geometry by introducting and studing Gröbner bases over Tate algebras. In addition, they designed a Buchberger-like and a F4-like algorithm for computing such Gröbner bases.

In [22], Xavier Caruso presents a survey on Fontaine's theory of $p$-adic period rings. These notes are based on a course given jointly by Laurent Berger and Xavier Caruso in Rennes in 2014; their aim is to detail the construction of the rings ${B}_{\text{crys}}$ and ${B}_{\text{dR}}$ (and some of their variants) and state several comparison theorems between étale and crystalline or de Rham cohomologies for $p$-adic algebraic varieties.