Weil groups and $F$-isocrystals

TL;DR

This paper demonstrates how local class theory can be derived from the Dieudonné-Manin structure theory for F-isocrystals, providing new proofs of classical results and answering a longstanding question.

Contribution

It offers a novel approach linking F-isocrystals to local class theory, including new proofs of Dwork's formula and the local Shafarevich-Weil theorem.

Findings

New proof of Dwork's formula for the norm residue symbol

Constructive proof of the local Shafarevich-Weil theorem

Establishes a connection between F-isocrystals and local class theory

Abstract

We show that much of local class theory can be deduced from the Dieudonn\'e-Manin structure theory for $F$-isocrystals on an algebraically closed field of characteristic $p>0$. As a consequence we get a new proof of a formula of Dwork for the norm residue symbol, as well as a "constructive" proof of the local Shafarevich-Weil theorem. This last answers a question of Morava.