Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 125 / 4

Both alternatives are given.

Let $E$ be an elliptic curve over ${\mathbb{F}}_q$ and let $\pi$ be its Frobenius isogeny. The fixed points of $\pi$ are exactly $E({\mathbb{F}}_q)$, and separability of $1-\pi$ givesHere Trace of Frobenius is the integer $a$. Since $\mathrm{deg}\pi =q$, the quadraticity of degree gives, for every pair of integers $m,n$,If the discriminant $a^2-4q$ were positive, this homogeneous quadratic would be negative for some real ratio $m/n$, hence for a nearby rational ratio and then for some pair of integers. ThereforeThis is the Hasse theorem for elliptic curves,

The Frobenius isogeny satisfiesIf $\alpha ,\beta$ are the roots of $X^2-aX+q$, then $|\alpha |=|\beta |=\sqrt{q}$ andThe zeta function of an elliptic curve over a finite field isSubstitution of the point-count formula and $-\log (1-z)={\sum }_{n\ge 1}{z}^n/n$ gives the rational functionThe bounds $|\alpha |=|\beta |=\sqrt{q}$ are the Riemann hypothesis for an elliptic curve over a finite field. The relations $\alpha \beta =q$ and the displayed formula also give the functional equation

For a discrete module $M$ over the absolute Galois group $G_K$, the first Galois cohomology group iswhere a one-cocycle satisfies $c(\sigma \tau )=c(\sigma )+\sigma c(\tau )$ and a coboundary has the form $c(\sigma )=\sigma m-m$.

For an integer $n\ge 2$, the Kummer exact sequence of an elliptic curveproduces the injective Kummer map of an elliptic curveExplicitly, if $nQ=P$, then $\delta (P)$ is represented by $\sigma \mapsto \sigma Q-Q$.

For every completion $K_v$ there is a local Kummer map. The n-Selmer group isIt fits intowhere $\mathrm{Sha}(E/K)$ is the Tate–Shafarevich group.

Only finitely many places divide $n$, are places of bad reduction, or are Archimedean. Outside this finite set $S$, every Selmer class is unramified. Since the finite Galois module $E[n]$ has finite order, there are only finitely many $E[n]$-valued cohomology classes unramified outside $S$; equivalently, the relevant finite extensions have bounded degree and ramification, and their number is finite by the Hermite–Minkowski theorem. Hence ${\mathrm{Sel}}^{(n)}(E/K)$ is finite, and its subgroup $E(K)/nE(K)$ is finite. This is the Weak Mordell-Weil theorem. Combined with height descent, which chooses representatives of bounded height in the finitely many cosets modulo $nE(K)$, it yields the finite generation asserted by the Mordell-Weil theorem.