"Any function addition law is due to an elliptic curve lurking in the background."
When I was reading about the origin of the concept of a genus, I came across a quote along these lines, I believe the quote was attributed to Weierstrass. I have been unable to find it again!
What is the source/precise wording of this quote?
This is an informal statement of the Weierstrass theorem: If $F$ is a polynomial in three variables, and $f$ solves $F(f(x+y),f(x),f(y))=0$, then $f$ is an elliptic function, possibly degenerate. Weierstrass did not publish it: apparently it comes from his lectures. Unfortunately, many of his lectures were not recorded, or no record survives. The first published proof that I know is due to Phragmen. But it is widely known as Weierstrass theorem. There was a lot of generalizations. A distant descendent is the Chevalley Structure theorem on algebraic groups https://en.wikipedia.org/wiki/Chevalley%27s_structure_theorem
