Hasse theorem for elliptic curves

For an elliptic curve over

F_{q}

,

∣ # E (F_{q}) - (q + 1) ∣ \leq 2 q .

Table of contents
- Zeta function of an elliptic curve over a finite field Hasse theorem for elliptic curves

Zeta function of an elliptic curve over a finite field ( $Z_{E} (T)$ )

If

a = q + 1 - # E (F_{q})

, then

Z_{E} (T) = exp (\sum_{r \geq 1} # E (F_{q^{r}}) \frac{T ^{r}}{r}) = \frac{1 - a T + q T ^{2}}{( 1 - T ) ( 1 - qT )} .

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

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