Let $K$ be a $p$-adic field (a finite extension of the field of $p$-adic numbers ${\mathbb Q}\_p$).
Let $T$ be a $K$-torus with character group $X={\sf X}^\*(T)$ and cocharacter group $Y={\sf X}\_\*(T)=X^\vee$.
I would like to have *explicit cocycles* for all Galois cohomology classes in $H^1(K,T)$.


We can compute $H^1(K,T)$ via double duality.


The cup product pairing defines an isomorphism
$$ H^1(K,T)\overset\sim\longrightarrow {\rm Hom}\big(H^1(K,X),{\Bbb Q}/{\Bbb Z}\big),$$
where I write $H^1(K,X)$ for $H^1({\rm Gal}(\overline K/K),X)$. See Milne, Arithmetic Duality Theorems, Corollary I.2.3.


Let $L/K$ be a finite Galois extension of degree $d$ splitting $T$.
We have
$$H^1(K,T)=H^1(\Gamma\_{L/K}, Y\otimes\_{\Bbb Z} L^\times),$$
where $\Gamma\_{L/K}={\rm Gal}(L/K)$.
The finite group $\Gamma\_{L/K}$ acts on $X$ and $Y$, and we have
$$H^1(K,X)=H^1(\Gamma\_{L/K}, X).$$
The canonical pairing of $\Gamma\_{L/K}$-modules
$$ Y\times X\to {\Bbb Z}$$
induces an isomorphism
\begin{align\*} 
H^1(\Gamma\_{L/K}, X)&\overset\sim\longrightarrow 
 {\rm Hom}\bigg(H^{-2}\big(\Gamma\_{L/K},\, Y\otimes\_{\Bbb Z} ({\Bbb Q}/{\Bbb Z})\,\big),
\ {\Bbb Q}/{\Bbb Z}\bigg)\\
 &\overset\sim\longrightarrow {\rm Hom}\big(H^{-1}(\Gamma\_{L/K}, Y),{\Bbb Q}/{\Bbb Z}\big);
\end{align\*}
see Brown, Cohomology of Groups, Corollary VI.7.3.
Thus we obtain an isomorphism
\begin{equation\*}
\lambda\colon\,(Y\_\Gamma)\_{\rm tors}= H^{-1}(\Gamma\_{L/K}, Y)\overset\sim\longrightarrow H^1(K,T)=H^1(\Gamma\_{L/K}, Y\otimes\_{\Bbb Z} L^\times),
\end{equation\*}
where $\Gamma={\rm Gal}(\overline K/K)$, $\ Y\_\Gamma$ denotes the group of coinvariants of $\Gamma$ in $Y$,
and $(\ )\_{\rm tors}$ denotes the torsion subgroup of the group in the parentheses.



> 
> **Question.** Let $y\in Y$ be a cocharacter whose image in $Y\_\Gamma$ is of finite order (say, of order dividing $d$).
> Write explicitly a cocycle
> in $Z^1(\Gamma\_{L/K}, Y\otimes\_{\Bbb Z} L^\times)$ representing $\lambda[y]\in H^1(\Gamma\_{L/K}, Y\otimes\_{\Bbb Z} L^\times)$.
> 
> 
> 


I think that this is possible, because I know the corresponding formula for $K={\Bbb R}$.



Explicit cocycles for the first Galois cohomology of a $p$-adic torus
Answer of James S. Milne: Most probably, this homomorphism
$$\lambda\colon\, H^{-1}(\Gamma\_{L/K}, Y)\overset\sim\longrightarrow H^1(\Gamma\_{L/K}, Y\otimes\_{\Bbb Z} L^\times)$$
is just the cup-product with the fundamental class in $H^2(\Gamma\_{L/K}, L^\times)$.


