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The question title says it all.

I am looking for a geometric proof for the fact that the group law defined on elliptic curves is associative.

I've heard somewhere about something on the internet about 2 cubics intersecting at 8 points must have a ninth point in common but I've never understood what that meant. Maybe this is what I'm looking for.

If a well-known reference has this result and I could easily find it in a library (or better yet over the internet, which I did not manage to find), then I'd accept this as an answer too.

Thanks in advance,

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    Take a look to page 6 of http://web.math.ku.dk/~kiming/lecture_notes/2000-2001-elliptic_curves/grouplaw.pdf2018-12-11
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    @JeanMarie : Thanks!2018-12-12

2 Answers 2

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There is a geometric proof of associativity in the elementary undergraduate book by Silverman and Tate Rational Points on Elliptic Curves.
The proof there is indeed along the lines you suggest of considering a pencil of cubics with nine base points, and is illustrated by a nice drawing.
The textbook derives from 1961 lectures by Tate, one of the best specialists ever in elliptic curves (he received the prestigious Abel prize in 2010).

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    I was writing my answer while Adam posted his: the book I refer to is *not* the one by Silverman alone mentioned by Adam.2011-10-28
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    Ah, I see there was a modification to Adam's post: so it *is* the same book after all! The confusing point is that Silverman wrote alone the more advanced book *The Arithmetic of Elliptic Curves*, where however he proves associativity only algebraically, not geometrically.2011-10-28
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    Yeah, Silverman's more advanced book is beautiful, but not at all appropriate for undergraduates. His undergraduate book with Tate, however, is amazingly accessible. I've never taught a course with it, but I have a colleague who has several times, and the (not very well prepared) students really liked it.2011-10-29
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This is contained in any intro book on elliptic curves. Here are two ones which are accessible to undergraduates:

  1. McKean and Moll, "Elliptic Curves: Function Theory, Geometry, Arithmetic".
  2. Silverman and Tate, "Rational points on elliptic curves"
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    Whoops, that's an embarrassing lacunae!2011-10-28