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I ask why adjective 'elliptic' for elliptic curves?

I have read that for the calculus of length of arc of ellipse founds a integral of type $\displaystyle\int_0^a\sqrt{\frac{1-k^2t^2}{1-t^2}} dt$ (elliptic integral).

Then call $u$ the integrand I have $u^2(1-t^2)=1-k^2t^2$.

This curve (in a $t/u$-plane) have a connection with elliptic curves (i.e. with a Weierstrass equation)?

I have also read that elliptic curve on $\mathbb{C}$ can be parametrized with elliptic functions, in particular with Weierstrass elliptic function.

Can someone please synthesize the story in between elliptic curves, elliptic integral and elliptic function?

Bernard
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user791759
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1 Answers1

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An elliptic curve admits a parametrization through the Weierstrass $\wp$ function, whose periods are given by elliptic integrals of the first kind. The perimeter of an ellipse is given by an elliptic integral of the second kind, and I believe this is the reason for the adjective elliptic in the description of objects like $\{(x,y)\in\mathbb{C}^2: y^2=x^3+px+q\}$ (with suitable constraints for the discriminant).

Jack D'Aurizio
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