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

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.

Embed $E$ as a smooth plane cubic with $O_E$ an inflection point. A line through $P,Q$, using the tangent when $P=Q$, has a third intersection $S$ counted with multiplicity. The chord-and-tangent group law defines $P+Q=-S$, where $-S$ is the third point on the line through $S$ and $O_E$.

The divisor cut out by the first line isand the line through $S$ and $O_E$ gives $(S)+(-S)-2(O_E)$. Their quotient therefore showsUnder the bijection from part (a), the chord-and-tangent operation is exactly addition in the abelian group ${\mathrm{Pic}}^0(E)$. It is consequently associative and commutative, has identity $O_E$, and has the geometrically defined point $-P$ as inverse. Thus it makes $E$ an abelian group.

Because $\varphi$ is a separable isogeny,The degree-zero divisorcorresponds under $E_1\simeq {\mathrm{Pic}}^0(E_1)$ to the sum of all elements of the finite abelian group $E_1[\varphi ]$. Pairing every $T$ with $-T$ shows that this sum is the sum of the elements in $E_1[\varphi ]\cap E_1[2]$. It vanishes when that intersection has one element and also when it has four elements, since the sum of the four elements of $(\mathbb{Z}/2\mathbb{Z}{)}^2$ is zero. The principal divisor criterion on an elliptic curve therefore gives a rational function $g$ with $\mathrm{div}(g)=D$.

The divisor $D$ is invariant under $[-1]$, so $[-1{]}^{\ast }g/g$ has zero divisor and is constant. Applying $[-1]$ twice shows that this constant has square one; hence $[-1{]}^{\ast }g=\pm g$.

For a short Weierstrass equation of an elliptic curvethe multiplication-by-two isogeny hasThus one may take $g=y$, and $[-1{]}^{\ast }y=-y$. For multiplication by three, the third division polynomial of an elliptic curvevanishes simply at the eight nonzero points of $E[3]$ and has a pole of order eight at $O_E$. Hence $g={\psi }_3(x)$ has divisor $[3{]}^{\ast }(O_E)-9(O_E)$ and satisfies $[-1{]}^{\ast }g=g$. Both signs occur.