On congruent isomorphisms for tori

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.