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Let $t=q+1-\#E({\mathbb{F}}_q)$. The stated characteristic equation for the Frobenius isogeny isTake $\alpha ,\beta$ to be the complex roots of $T^2-tT+q$. They satisfy $\alpha +\beta =t$ and $\alpha \beta =q$. By the Hasse bound, $t^2\le 4q$. If the inequality is strict, the roots are complex conjugates and have squared modulus $q$. In the equality case both roots are the same real number $\pm \sqrt{q}$. Thus in every caseTo obtain every extension-field count without assuming distinct roots, set $t_n=\mathrm{tr}({\pi }^n)$ and $t_0=2$. The dual isogeny satisfies $\hat{\pi }\pi =[q]$ and $\pi +\hat{\pi }=[t]$. Consequently the sums ${\pi }^n+{\hat{\pi }}^n=[t_n]$ satisfyAlternatively, apply additivity of the trace of an elliptic-curve endomorphism directly to the characteristic equation multiplied by ${\pi }^{n-2}$. The sequence ${\alpha }^n+{\beta }^n$ has the same recurrence and initial values, so $t_n={\alpha }^n+{\beta }^n$ by induction. This is the Frobenius trace recurrence, valid also at a repeated root.

The fixed points of ${\pi }^n$ are $E({\mathbb{F}}_{q^n})$, and $1-{\pi }^n$ is again separable. Since $\mathrm{deg}{\pi }^n=q^n$, the trace-degree identity givesFactoring and using $\beta =q/\alpha$ yieldsThis is the elliptic-curve point count over a finite field formula and the Riemann hypothesis for an elliptic curve over a finite field. In particular the powers on both factors are necessary; they are damaged in the converted TeX but present in the original PDF.

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