Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 125 / 1 / a

For a smooth projective curve $C$ of geometric genus one, the Riemann-Roch theorem saysThe canonical divisor has degree zero and is principal because a nonzero regular differential has no zeros. Hence $K_C\sim 0$, so equivalentlyIn particular, $\ell (D)=\mathrm{deg}D$ when $\mathrm{deg}D>0$.

The group ${\mathrm{Pic}}^0(E)$ is the group of degree-zero divisor classes on $E$, with addition induced by addition of divisors. ConsiderFor surjectivity, let $D$ have degree zero. Since $\mathrm{deg}(D+(O_E))=1$, Riemann--Roch gives $\ell (D+(O_E))=1$. A nonzero element of this space makes $D+(O_E)$ linearly equivalent to an effective divisor of degree one, necessarily $(P)$ for some point $P$. Thus $[D]=[(P)-(O_E)]$.

For injectivity, suppose $(P)-(O_E)$ is a principal divisor. If $P\ne O_E$, its defining function would be nonconstant and would have at most one simple pole, whereas Riemann--Roch gives $\ell ((O_E))=1$, so every such function is constant. Therefore $P=O_E$, and $\iota$ is a bijection.