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Tame class field theory over local fields

Rahul Gupta, Amalendu Krishna, Jitendra Rathore

TL;DR

This work extends tame class field theory to higher-dimensional quasi-projective schemes X over a local field with smooth compactifications, constructing a continuous reciprocity map from the tame idele class group C^t(X) to the abelian tame etale fundamental group pi ab, t1(X) and describing its kernel and cokernel in terms of finite and divisible pieces. The authors establish a robust framework connecting tame class groups with motivic cohomology and etale realization, proving a finite/divisible structure for the kernels J^t(X) and kernel components in various dimensions, and a finiteness result for motivic and etale cohomology over local fields. Central to the approach is identifying C^t(X) with motivic cohomology with compact support and showing compatibility with realization maps, allowing the transfer of finiteness and duality results from motivic and etale cohomology to class field theory. The paper also treats both bad and good reduction cases, obtaining comprehensive results on p-part and prime-to-p parts, and derives applications to motivic cohomology and Chow groups, reflecting the deep symmetry between arithmetic ramification, K-theory, and motivic frameworks. Overall, the results yield a higher-dimensional, tame local-class-field theory that aligns with Grothendieck’s ideas on ramification, while providing new structural insights via motivic and etale realization techniques with significant consequences for arithmetic geometry over local fields.

Abstract

For a quasi-projective scheme $X$ admitting a smooth compactification over a local field of residue characteristic $p > 0$, we construct a continuous reciprocity homomorphism from a tame class group to the abelian tame etale fundamental group of $X$. We describe the prime-to-$p$ parts of its kernel and cokernel. This generalizes the higher dimensional unramified class field theory over local fields by Jannsen-Saito and Forre. We also prove a finiteness theorem for the geometric part of the abelian tame etale fundamental group, generalizing the results of Grothendieck and Yoshida for the unramified fundamental group.

Tame class field theory over local fields

TL;DR

This work extends tame class field theory to higher-dimensional quasi-projective schemes X over a local field with smooth compactifications, constructing a continuous reciprocity map from the tame idele class group C^t(X) to the abelian tame etale fundamental group pi ab, t1(X) and describing its kernel and cokernel in terms of finite and divisible pieces. The authors establish a robust framework connecting tame class groups with motivic cohomology and etale realization, proving a finite/divisible structure for the kernels J^t(X) and kernel components in various dimensions, and a finiteness result for motivic and etale cohomology over local fields. Central to the approach is identifying C^t(X) with motivic cohomology with compact support and showing compatibility with realization maps, allowing the transfer of finiteness and duality results from motivic and etale cohomology to class field theory. The paper also treats both bad and good reduction cases, obtaining comprehensive results on p-part and prime-to-p parts, and derives applications to motivic cohomology and Chow groups, reflecting the deep symmetry between arithmetic ramification, K-theory, and motivic frameworks. Overall, the results yield a higher-dimensional, tame local-class-field theory that aligns with Grothendieck’s ideas on ramification, while providing new structural insights via motivic and etale realization techniques with significant consequences for arithmetic geometry over local fields.

Abstract

For a quasi-projective scheme admitting a smooth compactification over a local field of residue characteristic , we construct a continuous reciprocity homomorphism from a tame class group to the abelian tame etale fundamental group of . We describe the prime-to- parts of its kernel and cokernel. This generalizes the higher dimensional unramified class field theory over local fields by Jannsen-Saito and Forre. We also prove a finiteness theorem for the geometric part of the abelian tame etale fundamental group, generalizing the results of Grothendieck and Yoshida for the unramified fundamental group.
Paper Structure (56 sections, 112 theorems, 159 equations)

This paper contains 56 sections, 112 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. An application to motivic cohomology
  5. Outline of proofs
  6. Notations
  7. The tame fundamental group
  8. Recollection of ramification filtration
  9. Definition of $\pi^{\rm ab, {\rm t}}_1(X)$
  10. Relation with $\pi^{\rm ab}_1(X)$ and $\pi^{\rm ab}_1(\overline{X})$
  11. Structure of $J^{\rm t}(X)$ when $\text{\rm dim}(X) = 1$
  12. The tame class group
  13. Lower Milnor $K$-theory of semilocal rings
  14. Tame class group
  15. Contravariance of tame class group
  16. ...and 41 more sections

Key Result

Theorem 1.1

If $X$ is geometrically connected over $k$, then where $F$ is a finite group and $r$ is the ${\mathfrak f}$-rank of the special fiber of the Néron model of ${\rm Alb}(\overline{X})$.

Theorems & Definitions (215)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Definition 2.1
  • Remark 2.2
  • ...and 205 more