Can we get a homomorphism between two elliptic curves if we know a homomorphism between the respective formal groups?

Question

Let $E_1$ and $E_2$ be two elliptic curves over a field of characteristic $0$, and let $\hat{E_1}$ and $\hat{E_2}$ be two formal group laws associated with $E_1$ and $E_2$, respectively. It is known that an isogeny $\phi: E_1 \to E_2$ induces a homomorphism $\phi^*: \hat{E_1} \to \hat{E_2}$.

$\phi^*$ captures local properties of $E_1$ and $E_2$ near the identities, whereas $\phi$ is a global homomorphism between $E_1$ and $E_2$.

My question is:

• Can $\phi^*$ be lifted back to a homomorphism (not necessarily to $\phi$) between $E_1$ and $E_2$?
• In general, can we find at least one homomorphism $f^*: \hat{E_1} \to \hat{E_2}$ that can be lifted to a homomorphism $f: E_1 \to E_2$?

Assuming $\hat{E_1} =\hat{E_2}$, does the previous question give a positive answer?

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For the first question, I saw this post. According to that, we have \begin{align} & \phi(\hat{E_1}(x,y))=\hat{E_2}(\phi(x), \phi(y)) \\ & \phi^*(\hat{E_1}(x,y))=\hat{E_2}(\phi^*(x), \phi^*(y)) \end{align} and so I think, we may lift $\phi^*$ to a homomorphism between $E_1$ and $E_2$.

The second question, more general case, I have no idea.

For the third question with the assumption that $\hat{E_1} =\hat{E_2}$. We can consider the $p$-endomorphism $[p]_{\hat{E_1}}=[p]_{\hat{E_2}}$ (when $\hat{E_1}=\hat{E_2}$), then perhaps, we can do a sort of interpolation to extend it to globally as a homomorphism between $E_1$ and $E_2$. But, I do not have a clear mathematical model on this.

I hope I will get some help here. Thanks

Edit: I am editing my question following the comment of Keerthi Madapusi below: since I am considering elliptic curve, the respective formal group laws are 1-dimensional commutative formal group laws which are isomorphic to the additive formal group law, there is hardly non-commutative formal group laws in dimension 1.

Answer 1

This is a temporary answer, please let me know if any argument is incorrect:

SI am answering the third question only. With the assumption $\hat{E_1} =\hat{E_2}$, my answer is yes, provided the other comments are correct.

Answer 1: According to the comments of @Antoine Labelle and this post, if the elliptic curves are isogenous then we can ascend from a homomorphism between the respective formal group laws to that of a homomorphism between the elliptic curve.

On the other hand, according to the comment of @Chris Wurthrich from this post, if two elliptic curve have the same formal group law, then the curves are itself equal and their Weierstras equation is also equal to \begin{align*} f(X,Y)&=X+Y-a_1XY-a_2X^2Y-a_2XY^2-2a_3X^3Y+\cdots+(-2a_1a_3-2a_4)X^4Y+\cdots\\ &+(-2a_1^3a_3-2a_1^2a_4-4a_1a_2a_3-2a_2a_4-2a_3^2-3a_6)X^4Y+\cdots \end{align*} over any ring of characteristics $0$ or coprime to $6$. So if the formal group laws are equal, the parental elliptic curves are by default isogenous, and therefore, by the argument of the previous paragraph, we can ascend from a homomorphism between two formal group laws to a homomorphism between the parental elliptic curves.

Answer 2: This second answer is due to the comment of @Piotr Achinger.

Conider the elliptic curves over $\mathbb{Q}$. By the result of Honda, the invariant differential of a formal group law coincides with the $L$-function of the parental elliptic curve. Since the formal group laws are equal, i.e., $\hat{E_1}=\hat{E_2}$, their parental elliptic curves $E_1$ and $E_2$ will have the same $L$-function. Since the $L$-function is an invariant of an isogeny class of elliptic curves, the upshot is that, $E_1$ and $E_2$ are isogenous. Therefore, we can lift a homomorphism $\hat{E_1} \to \hat{E_2}$ into a homomorphism $E_1 \to E_2$.

I think it is also true for elliptic curves over the $p$-adic field $\mathbb{Q}_p$.

Edit: $p$-adic $L$-functions of elliptic curve over $\mathbb{Q}$ also determines isogeny class of elliptic curves as discussed in this post citing the paper

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