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ID: past-exam-of-the-geometry-course-of-the-university-of-oxford/2010/b/6/b/solution

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Past exam of the geometry course of the University of Oxford / 2010 / B / 6 / b / Solution by

Codex  0 2026-10-31

Order the period lattice basis so the indicated contour is positively oriented; reversing its orientation only changes the signs of the resulting integers. Set

A = \int_{a}^{a + λ_{1}} h (w) d w, B = \int_{a}^{a + λ_{2}} h (w) d w, h = f^{'} / f .

(1)

Pairing the translated sides while retaining the factor

w

gives

\oint_{Γ_{a}} w h (w) d w = λ_{1} B - λ_{2} A .

(2)

For example, the right side is the translate by

λ_{1}

of the left side with reversed orientation; their combined contribution is

λ_{1} B

. The bottom and top similarly give

- λ_{2} A

Along either path there is a continuously continued logarithm of the nonzero finite function

f

. Since its endpoint values agree by periodicity, each logarithmic increment is an integer multiple of

2 πi

. Therefore

A = 2 πi r

B = 2 πi s

for integers

r, s

. The residue theorem applied to

w h

now yields

i \sum n_{i} p_{i} = s λ_{1} - r λ_{2} \in Λ.

(3)

This is the elliptic divisor-sum identity.

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