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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 $\phi^*: \hat{E_1} \to \hat{E_2}$.

It is clear that $\phi^*$ captures local properties of $E_1$ and $E_2$ near the identities, whereas $\phi$ is a global homomorphism between $E_1$ and $E_2$. According to this post, 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 (an isogeny) between $E_1$ and $E_2$.

Suppose we don't know if $E_1$ and $E_2$ are isogenous or not, and suppose we are given a homomorphism $f: \hat{E_1} \to \hat{E_2}$. My question:

  • Is it possible to extend $f$ to a homomorphism 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 $f: E_1 \to E_2$ that does not necessarily need to be isogenous?

  • What happens when $\hat{E_1} =\hat{E_2}$?

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

I hope I will get some help here. Thanks

Learner
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1 Answers1

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It seems that you’re asking whether the inclusion $$ \mathop{\rm Hom}(E_1,E_2)\to\mathop{\rm Hom}(\hat E_1,\hat E_2) $$ is onto. Far from it! In characteristic zero, elliptic curves’ homomorphism modules are $\Bbb Z$-modules of rank one or two; the corresponding modules for their formal groups are $\Bbb Z_p$-modules, of similarly bounded rank.

Lubin
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