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For simplicity, let us assume we live in the field $\mathbb{C}$. Then all elliptic curves are hyperelliptic, so they admit a double cover of $\mathbb{P}^1$ branched over four points, whose preimage are the four 2-torsion points in the elliptic curve.

So does any one know how to visualize this double cover of a torus over a sphere topologically? Like how to construct a simplicial "complex" structure of the torus and sphere and the double cover maps $n$-simplex $0\leq n \leq2$ in the torus to $n$-simplex in the sphere.

Does the following simplex structure gives a double cover? The British national flag is a simplicial structure of a torus, its left half (right picture) with corresponding edges identified gives a sphere, similarly for the other half. a,b,c and d are four ramification points in the torus.

enter image description here

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Look at the last figure here (reproduced below). The double cover is quotienting the torus by a $180^{\circ}$ rotation through an axis skewering the torus, which has four fixed points. In elliptic curve terms you're quotienting by negation in the group law, whose fixed points are the four elements of order (dividing) $2$.

enter image description here

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  • $\begingroup$ This is nice, the rotation induces the right ramification picture! I guess my simplicial structure need the same rotation when quotient the left half and the right half. $\endgroup$ – Wenzhe Mar 3 '16 at 22:28

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