Is there a reference that shows, for a field $k$, that abelian $k$-varieties are $\Bbb A^1_k$-rigid?
A smooth variety $X$ over $k$ is $\Bbb A^1_k$-rigid if and only if the canonical map $$ {Hom}_{Var/k}(\mathop{\mathrm{Spec}} L,X)\rightarrow {Hom}_{Var/k}(\Bbb A_L^1,X) $$ is a bijection for every finitely generated separable extension $L/k$, where $\Bbb A_L^1=\Bbb A_k^1\times_{\mathop{\mathrm{Spec}}k} \mathop{\mathrm{Spec}}L$.
I do not have a reference, but isn't this easy to prove? Any morphism from $\mathbf A^1$ to an abelian variety $X$ is constant. Indeed, such a morphism extends to a morphism $f$ from $\mathbf P^1$ to $X$. For all global differential forms $\omega$on $X$, the pull-back $f^\star\omega$ is a global differential form on $\mathbf P^1$. Therefore, $f^\star\omega=0$ for all $\omega$. Since the cotangent bundle of $X$ is trivial, this implies that $f$ is constant.