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The Hasse–Weil zeta function of an algebraic variety over a number field is assembled from local point-counting factors at finite places. At a good place with residue field of size $q$, the factor is $Z(V_q,q^{-s})$, where $Z(V_q,T)=\exp ({\sum }_{n\ge 1}\#V_q({\mathbb{F}}_{q^n})T^n/n)$. Defining all bad-place factors requires a specified integral model or convention. The zeta function of an elliptic curve over a finite field is a local factor, rather than this global product.

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

The zeta function of an elliptic curve over a finite field is the formal power seriesThe proof of Hasse's theorem gives the characteristic equation ${\pi }^2-[a]\pi +[p]=0$. If $\alpha ,\beta$ are the roots of $X^2-aX+p$, then the elliptic-curve point count over a finite field isUsing $\exp ({\sum }_{r\ge 1}{u}^r{T}^r/r)=(1-uT{)}^{-1}$ therefore gives