 Solution

ID: past-exam-of-the-geometry-course-of-the-university-of-oxford/2011/2/c/i/solution

Multiplication by two on the complex torus is a holomorphic unramified covering

[2] : T \to T, [z] \mapsto [2 z] .

Its kernel is

(\frac{1}{2} Λ) /Λ

, of order four; its derivative is the nonzero constant two. Therefore

de g [2] = 4

. Composition with the degree-two Weierstrass elliptic function gives

de g ([z] \mapsto ℘ (2 z)) = 4 \cdot 2 = 8.

It is well-defined because changing

z

ω \in Λ

changes

2 z

by the period

2 ω

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