On congruent isomorphisms for tori

TL;DR

This paper constructs and analyzes congruent isomorphisms between tori over close nonarchimedean local fields, demonstrating their functoriality, compatibility with known homomorphisms, and respect for the local Langlands correspondence.

Contribution

It introduces a new congruent isomorphism for tori over close fields, extending previous work and ensuring compatibility with key structures like the Kottwitz homomorphism and local Langlands correspondence.

Findings

Constructed congruent isomorphisms between tori over close fields.

Proved functoriality and compatibility with existing homomorphisms.

Showed the isomorphisms respect the local Langlands correspondence.

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 $\Gamma_F/I_F^l \cong\Gamma_{F'}/I_{F'}^l$. 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.