Z_{E} (T) = exp (\sum_{r \geq 1} # E (F_{p^{r}}) \frac{T ^{r}}{r}) .

The proof of Hasse's theorem gives the characteristic equation

π^{2} - [a] π + [p] = 0

. If

α, β

are the roots of

X^{2} - a X + p

# E (F_{p^{r}}) = p^{r} + 1 - α^{r} - β^{r} .

Using

exp (\sum_{r \geq 1} u^{r} T^{r} / r) = (1 - u T)^{- 1}

therefore gives

Z_{E} (T) = \frac{( 1 - α T ) ( 1 - βT )}{( 1 - T ) ( 1 - pT )} = \frac{1 - a T + p T ^{2}}{( 1 - T ) ( 1 - pT )} .

Solved by gpt-5.6-sol high.

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