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I'm reading about group structures in Elliptic curves (Joseph H. Silverman The Arithmetic of Elliptic Curves; chapter III and VI).

I can not understand the introduction of Weierstrass P-function for the case of elliptic curves over $\mathbb{C}$.

I would like other references.

Thank you

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  • $\begingroup$ See also the answer to this question. $\endgroup$ – Dietrich Burde Dec 9 '16 at 21:19
  • $\begingroup$ You have to show $\wp(z)$ is doubly periodic and meromorphic with a single pole of order $2$ at $z=0$, so that $\wp'(z)^2-4 \wp(z)^3-g_2 \wp(z)$ is entire and doubly periodic and hence constant. The same kind of argument shows that $\wp(z+y)=\frac14 (\frac{\wp'(z)-\wp'(y)}{\wp(z)-\wp(y)})^2-\wp(z)-\wp(y)$ proving the Riemann surface and group isomorphism $(\mathbb{C}/\Lambda,+) \to E(\mathbb{C})$ where $E(\mathbb{C}) = \{ (x,y) \in \mathbb{C}^2, y^2 = 4 x^3-g_2 x-g_3\}$ with the usual group law $\endgroup$ – reuns Dec 9 '16 at 21:22
  • $\begingroup$ See Section 1.4 of Diamond and Shurman's "A First Course in Modular Forms" as well $\endgroup$ – Matt B Dec 9 '16 at 21:25
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    $\begingroup$ I can't tell if you mean you don't see the motivation, but the point of the construction is to show that elliptic curves are in one-to-one correspondence with complex tori $\endgroup$ – Alex Mathers Dec 9 '16 at 21:37
  • $\begingroup$ @AlexMathers the point is what I wrote : a complex torus is isomorphic to a complex elliptic curve as a Riemann surface and as an algebraic group (as an abelian variety). $\endgroup$ – reuns Dec 9 '16 at 21:47

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