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Given $E/K$ defined over a finite field extension $K$, can I find another curve $E'/K'$ and a group isomorphism $\phi: E/K \to E'/K'$, where $K'$ has an extension degree lower than that of $K$?

If $K = K'$ then we speak of isogenies defined over $K$, but I am specifically looking for an isomorphism defined over finite fields of different extension degree.

For instance given $E(\mathbb{F}_{p^2})$ for prime $p$, can I find a curve $E'(\mathbb{F}_r)$ for prime $r$ and an invertible map $\phi$ between the two that preserves addition of points?

(Motivation: implement addition in $E(\mathbb{F}_{p^2})$ using addition in curves defined over prime fields)

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  • $\begingroup$ If I were you, I would have asked at math.se where there are many people with good knowledge of this field. $\endgroup$
    – kelalaka
    Oct 19, 2022 at 15:09
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    $\begingroup$ Hence I didn't add it as an answer. Just a particular case, I am sure you could prove this doesn't work in an easier way, but one method would be to show it would be case by case. For instance, supersingular and ordinary would also not work due to the difference in the group structures. The remaining case would be ordinary to ordinary. $\endgroup$
    – Lev
    Oct 20, 2022 at 11:20
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    $\begingroup$ Thanks. I just want to point out that this argument holds when the extension degrees are 2 and 1. But it doesn't directly rule out the existence of such a map between say supersingular curves defined over $\mathbb{F}_{p^3}$ and $\mathbb{F}_{q^2}$ for primes $p, q$. (My intuition is that your argument can be generalized contradicting primality of $p, q$, perhaps with Weil conjectures?). So perhaps we could limit the scope (for now) to extension degrees 2 and 1, and non-supersingular elliptic curves. $\endgroup$
    – zugzwang
    Oct 20, 2022 at 11:28
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    $\begingroup$ I’m voting to close this question because already asked Math.se where it got more attention. $\endgroup$
    – kelalaka
    Oct 21, 2022 at 15:47
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    $\begingroup$ @kelalaka I think this question fits better here. Just have a look at the comments: The Math.SE question received nothing but clarification requests trying to resolve an evident linguistic divide, whereas here it already has a partial answer. $\endgroup$
    – yyyyyyy
    Oct 24, 2022 at 8:25

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Say you have $E_1/K_1 \rightarrow E_2/K_2$ How would you even define such isomorphism? Assuming it is a rational map then it can be expressed as a pair of polynomials. In what field are the coefficients in? This is expressed usually when you say "a map/curve is defined over $K$" i.e. the coefficients are from $K$.

Imagine in your example $K_1 = \mathbb{F}_p, K_2 = \mathbb{F}_t$ where $gcd(t,p)=1$ then clearly $K_1$ is not a subfield of $K_2$ and vice versa. Also their algebraic closures are different.

Now say you have $Q = (a,b) \in E_1(\mathbb{F}_p)$ so $Q=(a,b)$ is a pair of elements in $\mathbb{F}_p$. $(c,d) = \phi(Q) \in E_2(\mathbb{F}_t)$ is a pair of elements in $\mathbb{F}_t$.

In more detail, the value $c$ is calculated as $c = \frac{f(a,b)}{g(a,b)}$ where $f,g$ are polynomials in 2 variables with coefficients in $K_1$ or $K_2$ (for the sake of the argument).

If $f(x,y)\in K_1[x,y]$ then $f(a,b)$ is an element of $K_1$ and also has to be of $K_2$ since we assume $c$ is in $K_2$. Therefore $K_1$ is a subfield of $K_2$.

If $f(x,y)\in K_2[x,y]$ then when calculating $f(a,b)$ you get a problem. You need to do field operations between elements of $K_1$ ($a$ and $b$) and $K_2$ (coeffs in $f$). How do you do that? How do you multiply $v \in \mathbb{F}_5$ with $w \in \mathbb{F}_7$ for example? It does not make sense. It makes sense if there is some relation between $K_1$ and $K_2$ being subfields but not in general.

I think in general, you need $K_1$ to be a subfield of $K_2$ but I am not sure at the moment. But this is just meant to ilustrate why the question does not really make sense.

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  • $\begingroup$ Thanks for the image. Yes, of course such map cannot simply be a polynomial or rational mapping from $\mathbb F_p$ to $\mathbb F_t$, as operations do not make sense. But I can certainly define a map from $\mathbb F_p$ to $\mathbb F_t$ and vice versa by other means. For instance by fixing the representatives, embedding in $\mathbb Z$ and projecting. So I ask, is there a reason to rule out anything that is not a polynomial/rational map? $\endgroup$
    – zugzwang
    Nov 1, 2022 at 11:16

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