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Given two elliptic curves $E_1$ and $E_2$ over $\mathbb{C}$.

How to understand the statement "two (very) general elliptic curves are not isogenous"?

Question 1: Should it be "general" or "very general" here?

One way to think about this is to look at pairs $(E_1,E_2)\in \mathbb{A}_{j_1}\times \mathbb{A}_{j_2}$ and the set of pairs with $E_1$ isogenous to $E_2$ consits of finitely (countably) many hypersurfaces in the product. (Here $\mathbb{A}_j$ is the $j$-line as the coarse moduli of elliptic curves.)

Question 2: Is this the right way to think about this? Is there some reference in the literature about this?

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  • $\begingroup$ what do you mean by "very general"? $\endgroup$
    – ali
    Commented Apr 7, 2022 at 11:45
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    $\begingroup$ Maybe I'm totally off, but it seems to me that there should be countably many elliptic curves in a given isogeny class, however there are uncountably many isomorphism classes, and the statement should follow from that. $\endgroup$ Commented Apr 7, 2022 at 20:50
  • $\begingroup$ This may be helpful: in geometry, an arrangment is in general position if it is not part of a measure zero subset of the space of arrangements (e.g., no three planes intersect at a point). This may be the same notion, except applied to a relevant space. $\endgroup$
    – While I Am
    Commented Apr 8, 2022 at 0:34

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Question 1: Very general is safest and necessary outside of some specific situations as far as I know. There are some scenarios where it actually can be just general (over a number field, there are only finitely many elliptic curves isogeneous to a given one, but that's kind of hard, see here). I don't think this is the situation over $\Bbb C$, but I don't have an immediate reference for you off hand.

Question 2: Yeah, this is fine.

As Tabes Bridges mentions in the comments, there are only countably many isomorphism classes of curves in an isogeny class: an isogeny is determined by its kernel (this is a common exercise/lemma in an elliptic curves text) and there are only countably many finite subgroups of an elliptic curve and hence countably many curves isogenous to a given one. Because "very general" means "except for a countable union of proper closed subvarieties of the parameter space", this is enough to finish the problem.

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