Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 126 / 4 / ii

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- Solution ii

Solution

Define the homomorphism associated to a line bundle on an abelian variety

ϕ_{L} : X (k) \to Pic X, ϕ_{L} (x) = T_{x}^{*} L \otimes L^{- 1} .

The Theorem of the square gives

ϕ_{L} (x + y) = ϕ_{L} (x) \otimes ϕ_{L} (y)

, so this is a homomorphism.

Tensor products satisfy

ϕ_{L \otimes M} = ϕ_{L} \otimes ϕ_{M}

and

ϕ_{L^{- 1}} = ϕ_{L}^{- 1}

. Therefore

Pic^{0} X = {L : ϕ_{L} = 0}

is a subgroup of the Picard group. If

M = ϕ_{L} (x)

, translations commute and the theorem of the square gives

T_{y}^{*} M \otimes M^{- 1} ≅ O_{X}

for every

y

. Hence

M \in Pic^{0} X

, proving

im ϕ_{L} \subseteq Pic^{0} X

.

Solved by gpt-5.6-sol high.

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