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Weil groups and $F$-isocrystals

Richard Crew

TL;DR

The paper provides a Dieudonné-Manin–theoretic construction of the fundamental class $u_{L/K}$ for local Galois extensions via $F$-isocrystals, yielding a new, constructive route to local class field theory results. It represents the fundamental class by endomorphism algebras of $F$-isocrystals, builds explicit $L_K$-level realizations to realize the Weil group as a crossed product, and derives a concrete form of the norm residue symbol without relying on Tate–Nakayama. The work then develops a relative Weil group formalism and proves Shafarevich–Weil-type results, including the identification of the connecting homomorphism with the inverse norm residue and the functorial compatibility with transfers. Overall, it provides a cohomological, constructive alternative path to local reciprocity, with explicit cocycle descriptions and a robust framework for successive extensions. The results have implications for a streamlined, algebraic understanding of local class field theory via $F$-isocrystals and Weil groups, including constructive proofs of classical theorems.

Abstract

We show that much of local class theory can be deduced from the Dieudonné-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.

Weil groups and $F$-isocrystals

TL;DR

The paper provides a Dieudonné-Manin–theoretic construction of the fundamental class for local Galois extensions via -isocrystals, yielding a new, constructive route to local class field theory results. It represents the fundamental class by endomorphism algebras of -isocrystals, builds explicit -level realizations to realize the Weil group as a crossed product, and derives a concrete form of the norm residue symbol without relying on Tate–Nakayama. The work then develops a relative Weil group formalism and proves Shafarevich–Weil-type results, including the identification of the connecting homomorphism with the inverse norm residue and the functorial compatibility with transfers. Overall, it provides a cohomological, constructive alternative path to local reciprocity, with explicit cocycle descriptions and a robust framework for successive extensions. The results have implications for a streamlined, algebraic understanding of local class field theory via -isocrystals and Weil groups, including constructive proofs of classical theorems.

Abstract

We show that much of local class theory can be deduced from the Dieudonné-Manin structure theory for -isocrystals on an algebraically closed field of characteristic . 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.

Paper Structure

This paper contains 12 sections, 17 theorems, 104 equations.

Table of Contents

  1. The Fundamental Class
  2. $F$-isocrystals.
  3. The algebra $L_K$.
  4. Lubin-Tate $F$-isocrystals.
  5. The unramified case.
  6. The fundamental extension.
  7. The unramified case again.
  8. The norm residue symbol.
  9. Successive extensions.
  10. Weil Groups
  11. The formalism of Weil groups.
  12. The Shafarevich-Weil theorem.

Key Result

Theorem 1.1.1

The category of $F$-isocrystals on $K$ is semisimple, with one isomorphism class of simple objects for every element of $\mathbb{Q}$. The class of $\lambda\in\mathbb{Q}$ is the class of where $\lambda=r/d\in\mathbb{Q}$ in lowest terms (when $\lambda=0$ we take $r=0$ and $d=1$).

Theorems & Definitions (17)

  • Theorem 1.1.1: Dieudonné, Manin
  • Theorem 1.1.2: Dieudonné
  • Proposition 1.1.1
  • Lemma 1.2.1
  • Lemma 1.2.2
  • Proposition 1.3.1
  • Lemma 1.4.1
  • Theorem 1.4.1
  • Corollary 1.4.1
  • Theorem 1.4.2
  • ...and 7 more