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What is the eponymous relationship between ellipses and elliptic curves?

benjimin
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    I've no problem with this question, since it naturally arises from one's initial study of the topic, but I have to admit, I'm perplexed by the up-voting of this question, when compared with the reflexive and silent down-voting of other questions with far more context. And note that this isn't a question from several years ago. (ETA: Just to be clear, I'd far rather have this question be up-voted than those other ones be silently down-voted.) – Brian Tung Jan 07 '24 at 22:35
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    @BrianTung I would guess many people upvoted this question because it resonates with them, since they've asked themselves the same thing when they first heard about elliptic curves. It's not about "quality of the question" so much as about "basic psychology of the voter" :). As to why many people downvote so many questions, it's because StackExchange is secretly a giant reenactment of the Milgram experiment. – Stef Jan 07 '24 at 22:39
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    I up-voted this for exactly the reasons @Stef described. I never down-vote questions, so I don't have any insights there. – andypea Jan 07 '24 at 23:10

2 Answers2

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They have to do with Elliptic integrals, not much to do with ellipses other than the fact that these integrals are used to find the perimeter of an ellipse. See the paper "Why Ellipses Are Not Elliptic Curves".

kodlu
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Studying the arc length of parts of an ellipse led to certain complicated integrals called, naturally, elliptic integrals. It was eventually realized that inverses of these integrals had much nicer properties than the original arc length integrals, just as arc length integrals along part of a circle lead to inverse trig functions ($\arcsin x$) and we know the inverses of the inverse trig functions are much more interesting. Inverses of elliptic integrals, from the viewpoint of complex analysis, are doubly periodic and together with their derivative parametrize certain algebraic curves that are now called elliptic curves, but these curves are in no way the original ellipses that gave rise to all of this.

So elliptic curves are several steps removed from ellipses: ellipses lead to arc length integrals (elliptic integrals), which lead to elliptic functions through inversion, which lead to elliptic curves as the curves defined by algebraic equations linking elliptic functions and their derivatives.

KCd
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