(1) I am currently trying to find out how to classify all elliptic curves over an arbitrary field. Several questions have accumulated while I was trying to answer this question. If you don't want to read the whole question, only read part (3) or (4). A satisfying answer to one of these would solve my problem.
(2) Can elliptic curve over imperfect field be reduced to Weierstrass form? This old question has never been answered, so I'm including it here. The easy definition of the j-invariant always uses the weierstrass-form. If the curve cannot be converted into weierstrass-form, how is the j-invariant defined in the first place? Is there a general way to do it that's independent of field and form?
(3) It seems like twists are the way to classify curves that happen to be isomorphic over the closure (or the seperable closure?) of a given (arbitrary) field. In this thread Mr. Silverman does a great way to explain how this works and even how to define an invariant from that method: J-invariant and isomorphism of elliptic curves over $\mathbb{Q}$
Is it possible to extend/ modify Mr. Silvermans answer, so that the classification works in arbitrary fields?
(4) I have tried to find an answer in the literature. The classification from Mr. Silvermans answer above is done in both his and in Mr. Husemöllers (Chapter 7) books, but, as far as I understand it, both only classify curves for perfect fields. It seems like something similar is possible for arbitrary fields, though. In the book "Rational Points on Varieties" by Poonen, Chapter 4, a theorem is included for twists of quasi-projective k-varieties. This classifies curves that become isomorphic over an arbitrary Galois extension and thus also over the separable closure. This is only useful, though, if we can classify the curves over the separable closure of a given field. As far as I understand it, to classify all the different curves that are isomorphic over the separable closure of a field would require some different method because the j-invariant will only work if the separable closure is equal to the algebraic closure (case of an perfect field). How do you classify all curves over the separable closure of a non perfect field?
Secondly, this book is a little bit hard. To be honest, at the moment, I don't even understand enough scheme theory do verify with certainty that elliptic curves are quasi-projective varieties in the scheme-theoretic sense. Are they? Furthermore, the book is pretty technical. If this is the way to do the classification in the case of (truly) arbitrary fields, then my last question is, wether there is a less technical and maybe less general (as I only want to do it for elliptic curves) way to do it? Do you know of any good reference for this?
Best Answer
This is too long for a comment. If it doesn't answer your question, please let me know. I also wrote this in a rush, so please double check my claims.
I don't know if you are focused on characteristic 2 or 3, but I think if $k$ is of characteristic $p>3$ then for elliptic curves $E_i$ for $i=1,2$ over $k$ one has that $(E_1)_{\overline{k}}\cong (E_2)_{\overline{k}}$ iff $(E_1)_{k^\mathrm{sep}}\cong (E_2)_{k^\mathrm{sep}}$. The reason is 'simple' (but uses heavy machinery). Consider the functor
$$\mathrm{Isom}(E_1,E_2):\mathbf{Sch}_{/k}\to\mathbf{Set},\qquad T\mapsto \mathrm{Isom}_{\mathbf{Sch}_{/T}}((E_1)_T,(E_2)_T)$$
If $E_1$ and $E_2$ are isomorphic over $\overline{k}$ then, in particular, it's not hard to see that $\mathrm{Isom}(E_1,E_2)$ is a fpqc torsor for $\mathrm{Aut}(E_1)$. This is a finite group scheme over $\mathrm{Spec}(k)$, and since affine morphisms satisfy fpqc descent (see Tag 0245) we know that the $\mathrm{Aut}(E_1)$-torsor $\mathrm{Isom}(E_1,E_2)$ is representable by some $k$-scheme $I$. But, since $\mathrm{char}(k)>3$ we know that $|\mathrm{Aut}(E_1)|$ is invertible in $k$ (cf. [Silv, Theorem 10.1]) and so we know that $\mathrm{Aut}(E_1)$ is smooth over $\mathrm{Spec}(k)$ (e.g. see [Mil, Corollary 11.31]), and so $I$ must also be smooth over $\mathrm{Spec}(k)$ (e.g. see Tag 02VL). But, then $I(k^\mathrm{sep})\ne\varnothing$ (see [Poon, Proposition 3.5.70]). Thus, $(E_1)_{k^\mathrm{sep}}\cong (E_2)_{k^\mathrm{sep}}$.
So, asuming $\mathrm{char}(k)>3$ one can use usual Galois descent arguments to classify elliptic curves over $k$ (e.g. see Theorem 27 of my blog post here).
For characteristic $p=2,3$, the same thing works for classifying forms of $E$ assuming that $p\nmid |\mathrm{Aut}(E)|$. If $p\mid |\mathrm{Aut}(E)|$ I'm not sure what happens.
References
[Mil] Milne, J.S., 2017. Algebraic groups: the theory of group schemes of finite type over a field (Vol. 170). Cambridge University Press.
[Poon] Poonen, B., 2017. Rational points on varieties (Vol. 186). American Mathematical Soc..
[Silv] Silverman, J.H., 2009. The arithmetic of elliptic curves (Vol. 106). Springer Science & Business Media.