Can an algebraic group only have trivial elements over $k$ Let $G$ be an algebraic group over $k$ such that $G(k) = \{e\}$ is the trivial group. Does this imply that $G_{\overline{k}}$ is trivial?
I think the answer is no. I think you can just take an elliptic curve over $\mathbf{Q}$ without a non-trivial rational point. That works right?
 A: Very good question, and the answer is No ( as you said ) . ( But is $k$ a finite field ? )
Because the algebraic closure by notion specifies that we take an algebraic extension which is algebraically closed. So even though the group defined over the base field may be trivial, but once you lift it to some extension, you will find some points ( and that is why this 'closure' has such a significance in Galois theory where one considers $\rm{Gal}(\mathbb{\bar{Q}}/\mathbb{Q})$ ) .
Even your remark was quite beautiful, yes. One can consider an elliptic curve over some base field, (and as  you can always get a solution when you go over some extension in general case ), hence  the $E(\mathbb{\bar Q})$ is not trivial. 
But to remark something which is off-topic the group you have mentioned has something to do with the base field $\mathbb{Q}$, the  semisimple linear algebraic group $G$ over $\mathbb Q$, the Tamagawa number of $G$ (which is the standard terminology for the volume of $G(\mathbb Q)\setminus G(\mathbb A)$ with respect to Tamagawa measure) should be equal to $1$, and when $G$ is an elliptic curve rather than a linear group, there are many interesting things that happened, and which gave rise to a series of seminal works, and the Tamagawa numbers are widely used for tori ( by T.Ono ) , and also later the Birch and Swinnerton-Dyer found an analogue of the tamagawa number of elliptic curve ( anlogous to the work of T.Ono in defining the tamagawa numbers of Tori ) that played a central part in defining the Birch and Swinnerton-Dyer conjectures.
