Does there exist an algorithm or something of the sort to reverse-engineer a curve from its modular form (weight two eigenform with complex coefficients)? I am aware that sometimes there isn’t a unique elliptic curve paired with a given modular form, but is there way to get just one?

  • 6
    $\begingroup$ Yes, this is possible. I think this can be done by computing the period lattice, see chapter 2 here homepages.warwick.ac.uk/~masgaj/book/fulltext/index.html $\endgroup$
    – Will Sawin
    Nov 11, 2022 at 1:33
  • 5
    $\begingroup$ I wouldn't call it "reverse engineering" as this is much older than the way to associate a modular form to the elliptic curve. For instance the Antwerp IV tables did this systematically, then Cremona's tables (Will's link above) now on lmfdb extended it. $\endgroup$ Nov 11, 2022 at 14:33


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.