Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 125 / 4 / Solution / Hasse theorem and zeta functions

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