Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 21 / 2 / b / Solution

Let $L$ be any degree-zero line bundle on the genus-one smooth projective curve. The twist $M=L\otimes \mathcal{O}(p_0)$ has degree one. The Riemann-Roch theorem givesThe second line bundle has degree $2g-2-1=-1$, so it has no nonzero section: the zero divisor of such a section would be effective of negative degree. Hence $h^0(M)=1$. A nonzero section has an effective zero divisor of degree one, which is a single point $p$. It is a $k$-rational point even over a general field, since a degree-one effective divisor has residue degree one. Thus $M\cong \mathcal{O}(p)$ and $L\cong \mathcal{O}(p-p_0)=\alpha (p)$.

This proves surjectivity; part (a) gives injectivity. Hence $\alpha$ is a bijection, as in the Abel-Jacobi map of a genus-one curve. DefineTensor product of line bundles is associative and commutative, so this is an abelian group law on $X(k)$. Its identity is $p_0$, since $\alpha (p_0)={\mathcal{O}}_X$. Equivalently, $p\oplus q$ is the unique point $r$ with $\mathcal{O}(r-p_0)\cong \mathcal{O}(p+q-2p_0)$.