Problem Show that the curve $C \subset \mathbb{A}^2(\mathbb{C})$ cut out by $y^2 =x^3+x$ is actually isomorphic to $\mathbb{C} / \mathbb{Z}[i]$.
Now I have shown that $C$ is not isomoprhic to $(\mathbb{C}, 0)$ since their coordinate rings $$\mathbb{C}[x, y]/(y^2 - x^3-x) \ncong \mathbb{C}[x, y]/y \cong \mathbb{C}[x]$$.
I also see that $\mathbb{C}/\mathbb{Z}[i]$ is a torus.
But I don't clearly understand the question because this torus is not a subset of $\mathbb{C}^2$ so in what sense do I need to show the isomorphism? Which $\mathbb{C}^n$ do I view $\mathbb{C}/\mathbb{Z}[i]$ as sitting in? Basically how do I view it as an algebraic variety?
Please help.
The claim is false since (for instance) the complex torus is compact (in the complex toplogy) and the affine curve isn't. What's true is that the projective closure of the affine curve is isomorphic to the complex torus.
The isomorphism uses the Weierstrass $\wp$-function, and I sort of doubt it's appropriate for an exercise if you haven't heard of that function. See here: https://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functions#Relation_to_elliptic_curves