Embedding elliptic curves into the general linear group

Question

Is it possible to embedd an elliptic curve $E:\;\; y^2=x^3+ax+b$, defined over an algebraically closed field $k$, into some $GL_n(k)$ ?

Answer 1

If $E\hookrightarrow\mathrm{GL}_n$ is a closed immersion of $k$-schemes ($k$ any field) inducing an isomorphism with the closed subscheme $Z$ of $\mathrm{GL}_n$, then $Z$ is necessarily affine, and, being isomorphic over $k$ to $E$, is also proper over $k$. It is therefore finite over $k$. But this implies that $E$ is finite over $k$, which is not the case.

Answer 2

By the Chevalley's theorem on algebraic groups (see for example this modern proof of B. Conrad), there exists a unique normal linear algebraic closed subgroup of your elliptic curve $E$ such that the quotient is an abelian variety (at least if you assume your base field is perfect, which is trivially the case in your situation). In particular, since the identity subgroup satisfies the requirement, we can conclude that an elliptic curve cannot embed into some $\mathrm{GL}_n$, since otherwise $E$ would be that unique normal subgroup.

Remark: This is probably overkill, but at least it answers the question.