My question is motivated by the following observations. Let $\mathrm{T}$ be a torus defined over a $p$-adic field $K$, then by theories of tori, we have it is uniquely determined by a free $\mathbb{Z}$-module with continuous $\mathrm{Gal}(K^{s}/K)$ where $K^s$ is the separable closure of $K$. And the $\mathbb{Z}$-module comes from taking the character lattice of $T_{K^s}$. In this case, if $\mathrm{T}$ splits over an unramified extension, then the $Gal(K^{s}/K)$ action factor through $\mathrm{Gal}(K^{ur}/K)$. On the other hand, I learned from Conrad's Luminy notes that the category of tori $\mathscr{T}$ defined over $\mathscr{O}_K$ is anti-equivalence to the category of $\pi_1(\mathscr{O}_K, \bar{\eta})$-modules. The functor is given by $\mathrm{Hom}(\mathscr{T},\mathbb{G}_m)_{\bar{\eta}}$, cf., Cor B.3.6 of http://math.stanford.edu/~conrad/papers/luminysga3smf.pdf. Here $\eta$ is the generic point of $\mathscr{O}_K$. If I understand this theorem correctly, this tells us {$\mathscr{T}$} are anti-equivalence to those free $\mathbb{Z}$-modules with $\mathrm{Gal}(K^{ur}/K)$ actions. So from the above discussions, we have every unramified torus $\mathrm{T}$ defined over a $p$-adic field has a natural smooth integral model $\mathscr{T}$ defined over $\mathscr{O}_K$. My question is:
1) If my arguments are correct, what the relation of $\mathscr{T}$ and the (connected) Néron-Raynaud module of $\mathrm{T}$.
2) Can we construct the integral smooth model of an unramified reductive group similarly? i.e., can we prove they are all equivalent to some combinatorial datum with Galois actions? Does this mean we can avoid using Bruhat-Tits theory in the unramified case? And if we can construct such an integral smooth model what's the relation with those constructed by Bruhat and Tits?
