6
$\begingroup$

"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?

$\endgroup$
4
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.