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I am a math undergrad, I know some basics of algebraic geometry and hence, elliptic curves. I was attending a talk today which was about Poncelet's theorem. In the end of the talk, the speaker discussed complex elliptic curves and wrote

$$\Bbb{C}/\Lambda \simeq \text{torus}$$ i.e. complex elliptic curves are tori. Where $\Lambda$ is a lattice in $\Bbb{C}$.

Is there an intuitive explanation to this? I know some basics of quotient topology and elliptic curves.

Siddharth Acharya
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  • What do you know about elliptic curves? Have you seen them defined, for example, by a polynomial in Weierstrass form? – Lazzaro Campeotti Oct 22 '19 at 13:55
  • One intuitive explanation is that points of $\mathbf C/\Lambda$ have a unique representative in a fundamental parallelogram of $\Lambda$, except that opposite edges of the parallelogram give the same point in the quotient. So you can form the quotient space by taking the parallelogram and glueing its opposite edges together. If you do this, you get a torus. Is that the kind of explanation you are looking for? – Lazzaro Campeotti Oct 22 '19 at 14:21
  • Something of that sort. Could you explain your answer in a bit more detail in the answer section. – Siddharth Acharya Oct 22 '19 at 14:27
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    Is your problem understanding why an elliptic curve, is the quotient of $\mathbb{C}$ by a lattice, or understanding why such a quotient is a torus? Or both? – Ahr Oct 22 '19 at 15:15
  • Both. Precisely. Yes, that's my question. – Siddharth Acharya Oct 22 '19 at 15:20

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