# Calculating the zeta function over a hypersurface in project

1. May 4, 2015

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1. The problem statement, all variables and given/known data
Calculate the zeta function of $x_0x_1-x_2x_3=0$ in $F_p$

2. Relevant equations
Zeta function of the hypersurface defined by f:
$\exp(\sum_{s=1}^\infty \frac{N_s u^s}{s})$
$N_s$ is the number of zeros of f in $P^n(F_p)$

3. The attempt at a solution
My biggest struggle is finding $N_s$, here's what I've thought so far:

$N_s$ is composed of finite points and "points at infinity". The points at infinity are solutions on the form $(0, 1, \frac{a_2}{a_1}, \frac{a_3}{a_1})$ which gives: $(\frac{a_2}{a_1})(\frac{a_3}{a_1})=0$, the first factor has p possibilities (second factor 0), or first factor is 0 and second factor has p possibilities minus when both iz 0 (overcounting) which gives $2p-1$ points at infinity. Not sure what to say about the amount of finite points.

According to my book the should be $2p+1$ points at infinity and not $2p-1$, and the number of finite points should be $p^2$ making $N_s=p^{2s}+2p^s+1$, (replacing p with $p^s$). What have I've done wrong? And is the number of finite points all possible combinations of $\frac{a_2}{a_1}$ and $\frac{a_3}{a_1}$, $p^2$?

Any help much appreciated!

2. May 9, 2015