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I've heard that elliptic curves have applications in strange places, due to their connection to elliptic functions and then elliptic integrals, which have nice algebraic addition formulas. I see papers named "The History of the Universe is an Elliptic Curve," and countless authors have successfully convinced me that they are a subject of great utility as well as theoretical interest. However, I'd love to see more concrete examples than just reference to "electrodynamic" or "string theory". In particular, I am asking more about applications of elliptic curves than than I am the theoretical places in which they appear.

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  • $\begingroup$ The cryptography and factorization both use elliptic curves over prime finite fields $\mathbb{F}_p$. But there are elliptic curves over any field, in particular : $\mathbb{Q}, \mathbb{C}, \overline{\mathbb{F}}_p,K/\mathbb{Q}$ all with their own specificity (that's why elliptic curves are such a vast subject) $\endgroup$
    – reuns
    Apr 30, 2017 at 1:02
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    $\begingroup$ The title deliberately excludes cryptography and integer factorization connections, but question doesn't; you may want to include that restriction explicitly so as to make your intention more precise. $\endgroup$ Apr 30, 2017 at 1:42

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