Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 126 / 4

The Theorem of the square says that for an abelian variety $X$, a line bundle $L$, and $x,y\in X(k)$,where $T_x$ denotes translation by $x$.

For a line bundle $L$ on $X$, define the homomorphism associated to a line bundle on an abelian varietyThe Theorem of the square givesso ${\varphi }_L$ is a homomorphism. Pullback distributes over the tensor product of sheaves, and thereforeIterating the homomorphism law in $x$ gives

Suppose $L^{\otimes n}\in {\mathrm{Pic}}^0(X)$. Then ${\varphi }_L(nx)$ is trivial for every $x$. The multiplication-by-n morphism on an abelian variety is surjective, so ${\varphi }_L$ is trivial and $L\in {\mathrm{Pic}}^0(X)$. Thus the Néron-Severi groupis torsion-free.

Finally, put $M={\varphi }_L(x)$. For every $y$,which is trivial by the Theorem of the square. Hence

LetIf $L\in {\mathrm{Pic}}^0(X)$, then its restriction to $X\times \{y\}$ is $T_y^{\ast }L\otimes L^{\vee }={\varphi }_L(y)$, up to a constant one-dimensional factor, and is therefore trivial. Its restriction to $\{e\}\times X$ is also trivial. The Seesaw theorem now implies that $\Lambda (L)$ is trivial on $X\times X$.

Conversely, if $\Lambda (L)$ is trivial, restricting it to $X\times \{y\}$ shows that ${\varphi }_L(y)$ is trivial for every $y\in X(k)$. Thus

If $L\in {\mathrm{Pic}}^0(X)$, part (iii) makes $\Lambda (L)$ trivial. Pulling it back along $(f,g):Y\to X\times X$ givesTaking $Y=X$, $f={\mathrm{id}}_X$, and $g=i$, where $i$ is inversion, givesInduction with $f=[n]$ and $g={\mathrm{id}}_X$ proves $[n{]}^{\ast }L\simeq L^{\otimes n}$ for $n\ge 0$; combining this with inversion proves

Conversely, suppose $i^{\ast }L\simeq L^{\vee }$. For $M={\varphi }_L(x)$, part (ii) gives $M\in {\mathrm{Pic}}^0(X)$, so the result just proved yields $i^{\ast }M\simeq M^{\vee }$. On the other hand,whereas $i^{\ast }L\simeq L^{\vee }$ gives ${\varphi }_{i^{\ast }L}(x)={\varphi }_{L^{\vee }}(x)=M^{\vee }$. Hence $M^{\otimes 2}$ is trivial for every $x$, so ${\varphi }_{L^{\otimes 2}}$ is trivial and $L^{\otimes 2}\in {\mathrm{Pic}}^0(X)$. The torsion-freeness proved in part (ii) now implies