On congruent isomorphisms for tori

Abstract

Let $F$ and $F^{\prime }$ be two $l$-close nonarchimedean local fields, where $l$ is a positive integer, and let $\mathrm{T}$ and ${\mathrm{T}}^{\prime }$ be two tori over $F$ and $F^{\prime }$, respectively, such that their cocharacter lattices can be identified as modules over the ''at most $l$-ramified'' absolute Galois group ${\Gamma }_F/I_F^l\cong {\Gamma }_{F^{\prime }}/I_{F^{\prime }}^l$. 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 $\mathrm{T}(F)/\mathrm{T}(F{)}_m\cong {\mathrm{T}}^{\prime }(F^{\prime })/{\mathrm{T}}^{\prime }(F^{\prime }{)}_m$, where $\mathrm{T}(F{)}_m$ and $\mathrm{T}(F^{\prime }{)}_m$ are the minimal congruent filtration subgroups of $\mathrm{T}(F)$ and $\mathrm{T}(F^{\prime })$, 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.

27 pages