Torus gerbes over curves Setup: Let $k$ be an algebraically closed field. Let $C$ be a smooth connected curve over $k$. Let $K(C)$ be the function field of $C$.
Tsen's Theorem implies that every $\mathbb{G}_m$-gerbe over $K(C)$ splits, i.e., the etale cohomology $H^2_{et}(K(C),\mathbb{G}_m)=0.$
Now, let $T$ be a not-necessarily split torus over $K(C).$
Question: Do we always have that $H^2_{et}(K(C),T)=0$?
Tsen's theorem implies that we do have the desired vanishing if $T$ is split or quasi-split, i.e., an induced torus.
A paper that I am reading seems to suggest that the desired vanishing is a piece of common sense. However, I cannot find an explicit reference for it, nor can I prove it. Any help is greatly appreciated.
 A: I am just posting my comment as one answer.  Let $K$ be a field, and let $T$ be a $K$-group scheme such that there exists a field extension $K'/K$ that is finite and separable (i.e., étale), and there exists an isomorphism $i$ of $k'$-group schemes from $T_{K'}:=\text{Spec}\ K'\times_{\text{Spec}\ K}T$ to the split torus $\mathbb{G}_{m,K'}^d$, i.e., $T$ is a torus.
By adjointness, there is a natural morphism of $K$-group schemes to the Weil restriction, $T\to R_{K'/K}T_{k'}.$  The composition with $R_{K'/K}i$ gives a morphism of $K$-schemes, $$T\to R_{K'/K}\mathbb{G}_{m,K'}^d.$$  Since it is true after basechange to $K'$, also this morphism of $K$-group schemes is a closed immersion whose quotient, $Q$, is also a torus.  Therefore we have a short exact sequence of $K$-group schemes, $$1 \to T \to R_{K'/K}\mathbb{G}_{m,K'}^d \to Q \to 1.$$
The associated long exact sequence of étale cohomology gives the following, $$H^1_{\text{et}}(\text{Spec}\ K',\mathbb{G}_{m,K'})^d \to H^1_{\text{et}}(\text{Spec}\ K,Q) \to H^2_{\text{et}}(\text{Spec}\ K,T)\to H^2_{\text{et}}(\text{Spec}\ K',\mathbb{G}_{m,K'})^d.$$
By Hilbert's Theorem 90, the first group in this sequence is trivial.  Standard results about Brauer groups prove that the last group is trivial if and only if the Brauer group of $K'$ is trivial if and only if every $\textbf{PGL}_\ell$-torsor over $K'$ has a $K'$-point (for all positive integers $\ell$).
Thus, the Galois cohomology group $H^2_{\text{et}}(\text{Spec}\ K, T)$ is trivial if every $\textbf{PGL}_\ell$-torsor over $K'$ has a $K'$-point and every $Q$-torsor over $K$-has a $K$-point.  For $K$ and $K'$ equal to function fields of curves over an algebraically closed field $k$, this can be proved directly, cf., Tsen's original proof.
In fact, since all such torsors are dense Zariski open subschemes of smooth projective varieties that are separably rationally connected, this also follows from the "rationally connected fibration theorem", i.e., the Kollár-Miyaoka-Mori conjecture.
