On congruent isomorphisms for tori

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.