$p$-adic geometry

TL;DR

This paper reviews recent advances in $p$-adic geometry, including foundational results, applications to Langlands conjectures, and the development of universal $p$-adic cohomology theories, with speculations on a unified prime theory.

Contribution

It summarizes recent progress in $p$-adic geometry, highlighting new period maps, cohomology theories, and connections to Langlands conjectures, proposing a comprehensive prime-inclusive framework.

Findings

Degeneration of Hodge-to-de Rham spectral sequence for compact $p$-adic manifolds

Construction of new period maps on moduli spaces of abelian varieties

Development of universal $p$-adic cohomology theories

Abstract

We discuss recent developments in $p$-adic geometry, ranging from foundational results such as the degeneration of the Hodge-to-de Rham spectral sequence for "compact $p$-adic manifolds" over new period maps on moduli spaces of abelian varieties to applications to the local and global Langlands conjectures, and the construction of "universal" $p$-adic cohomology theories. We finish with some speculations on how a theory that combines all primes $p$, including the archimedean prime, might look like.