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On congruent isomorphisms for tori

Anne-Marie Aubert, Sandeep Varma

Abstract

Let $F$ and $F'$ be two $l$-close nonarchimedean local fields, where $l$ is a positive integer, and let $\mathrm{T}$ and $\mathrm{T}'$ be two tori over $F$ and $F'$, respectively, such that their cocharacter lattices can be identified as modules over the ''at most $l$-ramified'' absolute Galois group $Γ_F/I_F^l \congΓ_{F'}/I_{F'}^l$. In the spirit of the work of Kazhdan and Ganapathy, for every positive integer $m$ relative to which $l$ is large, we construct a congruent isomorphism $\mathrm{T}(F)/\mathrm{T}(F)_m\cong\mathrm{T}'(F')/\mathrm{T}'(F')_m$, where $\mathrm{T}(F)_m$ and $\mathrm{T}(F')_m$ are the minimal congruent filtration subgroups of $\mathrm{T}(F)$ and $\mathrm{T}(F')$, respectively, defined by J.-K.~Yu. We prove that this isomorphism is functorial and compatible with both the isomorphism constructed by Chai and Yu and the Kottwitz homomorphism for tori. We show that, when $l$ is even larger relative to $m$, it moreover respects the local Langlands correspondence for tori.

On congruent isomorphisms for tori

Abstract

Let and be two -close nonarchimedean local fields, where is a positive integer, and let and be two tori over and , respectively, such that their cocharacter lattices can be identified as modules over the ''at most -ramified'' absolute Galois group . In the spirit of the work of Kazhdan and Ganapathy, for every positive integer relative to which is large, we construct a congruent isomorphism , where and are the minimal congruent filtration subgroups of and , respectively, defined by J.-K.~Yu. We prove that this isomorphism is functorial and compatible with both the isomorphism constructed by Chai and Yu and the Kottwitz homomorphism for tori. We show that, when is even larger relative to , it moreover respects the local Langlands correspondence for tori.
Paper Structure (36 sections, 24 theorems, 20 equations)

This paper contains 36 sections, 24 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. A crude version of the main result
  3. Statement of the main result --- more precise version
  4. The case of split tori
  5. Deligne's triples.
  6. The case of $\mathrm{GL}_1$, from Del84
  7. General split tori
  8. Standard and congruent isomorphisms
  9. Using the work of Chai and Yu
  10. The organization of the paper
  11. Notation and review.
  12. Discretely valued Henselian fields
  13. Objects associated to a DVHF
  14. Passage to completion
  15. Ramification theory
  16. ...and 21 more sections

Key Result

Theorem 1.1.1

Suppose a local field $F$ is $l$-close to a local field $F'$, and a torus $\mathrm{T}'/F'$ is a transfer of a torus $\mathrm{T}/F$. Then:

Theorems & Definitions (58)

  • Theorem 1.1.1
  • Theorem 1.2.1
  • Proposition 1.4.1
  • Remark 2.1.1
  • Remark 2.2.1
  • Remark 2.2.3
  • Remark 2.3.2
  • Remark 2.3.3
  • Lemma 2.4.1
  • proof
  • ...and 48 more