In a discussion today on the Shafarevich-Tate group of an elliptic curve, the following structure and question came up. I will abuse many notations and be very vague about some things, but am very open to suggestions for clarification. Not to mention that some, if not all, of the following is incorrect.

A key ingredient in the rich structure of number fields is Hilbert 90, asserting that the first galois cohomology of the multiplicative group is trivial. Elliptic curves, on the other hand, have no analogy (as far as I know). The discussion was about looking for a natural object to inject an elliptic curve into, that has trivial first galois cohomology.

Given an elliptic curve, an interesting looking (well, maybe not) object is the direct limit of Weil restriction of the curve, going over all the galois extensions of the base field and the morphisms are the natural inclusions. As a $G_K$ module the Weil restriction is the induced module $Ind_{G_L}^{G_K} E$, so by Shapiro's lemma the cohomology is isomorphic to $H^1(G_L,E)$. Applying the direct limit, we indeed see that the first cohomology of the direct limit of the Weil restriction is trivial (since $G_L$ keeps getting smaller).

And so many questions are raised regarding this direct limit. The first one is in the title, and is my question.

Is the direct limit of Weil restrictions, going over all galois extensions of the base field, of an elliptic curve a scheme?