I've been trying to understand some of the idiosyncrasies associated to algebraic groups over non-algebraically closed fields $K$ of characteristic $p > 0$.


Let $G$ is a connected almost absolutely simple adjoint algebraic group over $K$ and $\widetilde{G}$ is the simply connected cover of $G$ also defined over $K$ with central isogeny $\pi : \widetilde{G} \rightarrow G$. Let $\widetilde{T} \subseteq \widetilde{G}$ be a maximal $K$-torus and let $T= \pi(\widetilde{T}) \subseteq G$.


Is it possible for the $\mathbb{Z}$-rank of the $K$-defined character groups $X\_K(\widetilde{T})$ and $X\_K(T)$ to be different? In other words, is the $K$-split part of $\widetilde{T}$ always the same rank as the $K$-split part of $T$?



Rank of the character group of a maximal $K$-torus for semisimple and adjoint algebraic groups
The map $\widetilde{T} \rightarrow T$ is an isogeny, and an isogeny between tori over any field $K$ induces a finite-index inclusion between the associated geometric character groups as ${\rm{Gal}}(K\_s/K)$-lattices, hence an isomorphism on associated rational vector space, so same dimension for spaces of Galois-invariants; i.e., same rank for $K$-rational character groups. However, this is a central isogeny, and that is really the key to the good behavior.


For isogenies $f:G \rightarrow G'$ between connected semisimple $K$-groups such that $\ker f$ is *not* central in $G$ (which can only occur in positive characteristic $p$, such as Frobenius isogenies $F\_{G/K}:G \rightarrow G^{(p)}$) it can happen that $G'$ has larger $K$-rank than $G$. This might sound paradoxical if you are not familiar with it, since you might reason that if $T' \subset G'$ is a maximal $K$-torus (say containing a maximal split $K$-torus of $G'$) then the identity component of the underlying reduced scheme of $f^{-1}(T')$ seems to be a smooth $K$-subgroup scheme of $G$, hence a torus mapping onto $T'$ via an isogeny, so it has the same $K$-rank as $T'$ by the first paragraph above. That reasoning is valid *provided* that $f^{-1}(T')\_{\rm{red}}$ really is a smooth $K$-subgroup scheme of $G$. For perfect $K$ such conditions hold (since the underlying reduced scheme of a finite type $K$-group scheme is a smooth $K$-subgroup scheme for such $K$). But it can *fail* when $K$ is imperfect. 


For example, if $K$ is a local function field of characteristic $p$ and $A$ is a central division algebra of dimension $p^2$ over $K$ then $G := {\rm{SL}}\_1(A)$ is a $K$-anisotropic form of ${\rm{SL}}\_p$ but $G^{(p)}$ is $K$-split since $A^{(p)}$ is $K$-split (by local class field theory). So $F\_{G/K}:G \rightarrow G^{(p)}$ is an isogeny from a $K$-anisotropic absolutely simple semisimple $K$-group onto a $K$-split one. And of course this is a non-central isogeny. I suspect that this is the kind of phenomenon you were trying to find when formulating the OP.


[EDIT: I should probably have explained why in the case of central isogenies there are *no* surprises. That is, if $f:G \rightarrow G'$ is a central isogeny between connected reductive $K$-groups for a field $K$ (i.e., the scheme-theoretic kernel $\ker f$ centralizes $G$ in the functorial sense) and if $T' \subset G'$ is a maximal $K$-torus then the scheme-theoretic preimage $T := f^{-1}(T')$ is a maximal $K$-torus of $G$ (in particular, smooth and connected, even if $f$ is inseparable or has disconnected kernel). The reason is that to prove the $K$-group scheme $T$ is a maximal $K$-torus it suffices to do so after a ground field extension, since by Grothendieck's theorem on the "geometric maximality" of maximal tori over the ground field in a smooth connected affine $K$-group we do not lose the maximality hypothesis on $T'$ after a ground field extension. Hence, we may assume $K$ is algebraically closed. 


With $K = \overline{K}$, all choices of $T'$ are $G'(K)$-conjugate and the map $G(K) \rightarrow G'(K)$ is surjective, so it suffices to treat the case of a *single* $T'$. Ah, but then we simply choose a maximal $K$-torus $S$ in $G$, so $T' := f(S)$ is a maximal $K$-torus in $G' = f(G)$, and thus the problem is to show that the inclusion $S \subset f^{-1}(f(S))$ of $K$-group schemes is an equality. Since $f$ is necessarily faithfully flat, so $G' = G/(\ker f)$ as fppf group sheaves, it suffices to show that $\ker f \subset S$ as subfunctors of $G$. Since $\ker f$ is central in $G$ by hypothesis, so it centralizes $S$, and hence it suffices to show that the scheme-theoretic centralizer $Z\_G(S)$ of $S$ is equal to $S$. We know equality on $K$-points by the classical theory, so one just has to show that the group scheme $Z\_G(S)$ is smooth, which is to say that it has the "expected" tangent space (i.e., that of $S$). This is a problem on dual numbers, and is proved in section 9 of Chapter III of Borel's textbook in a more classical language.]


