The <zeta function of an elliptic curve over a finite field> is the formal power series
$$
Z_E(T)=\exp\left(\sum_{r\geq1}\#E(\mathbb F_{p^r})\frac{T^r}{r}\right).
$$
The 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> is
$$
\#E(\mathbb F_{p^r})=p^r+1-\alpha^r-\beta^r.
$$
Using $\exp(\sum_{r\geq1}u^rT^r/r)=(1-uT)^{-1}$ therefore gives
$$
Z_E(T)=\frac{(1-\alpha T)(1-\beta T)}{(1-T)(1-pT)}
=\frac{1-aT+pT^2}{(1-T)(1-pT)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 125 / 1 / b / Solution

{title2=$Z_E(T)$}
{wiki=Hasse–Weil_zeta_function}

If $a=q+1-\#E(\mathbb F_q)$, then
$$
Z_E(T)=\exp\left(\sum_{r\geq1}\#E(\mathbb F_{q^r})\frac{T^r}{r}\right)
=\frac{1-aT+qT^2}{(1-T)(1-qT)}.
$$


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