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186 9. Elliptic and Hypergeometric Functions

2 2
wy = (x − e 1 w) x − e w (x − e 3 w)
to the projective line is a 2 to 1 map, in general, which can be pictured roughly with
the following

The projection mapping has four values over which there is double ramification
e 1 , e 2 , e 3 ,and ∞.
If the elliptic curve is viewed as the parallelogram identified on the opposite
edges with certain paths on the curve represented in the parallelogram, then we can
draw two images in the complex plane C contained in the Reimann sphere P 1 (C).

The first image is of the left-hand parallelogram under ℘(z) and the second of
the right-hand parallelogram under ℘(z).