Calculating the zeta function over a hypersurface in project

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Homework Statement

Calculate the zeta function of $x_0x_1-x_2x_3=0$ in $F_p$

Homework 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 $P^n(F_p)$

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!

 

[mighty2000]

Hello, thank you for your post. Let's break down the problem and see if we can find a solution.

First, let's define the zeta function of a hypersurface in F_p. The zeta function is defined as:

\exp(\sum_{s=1}^\infty \frac{N_s u^s}{s})

Where N_s is the number of solutions to the hypersurface in P^n(F_p). In this case, our hypersurface is defined by the equation x_0x_1-x_2x_3=0. So, we need to find the number of solutions to this equation in P^3(F_p).

To find the number of solutions, we can use the following formula:

N_s = p^{2s} + \sum_{i=1}^s (-1)^i Tr(Frob^i, H^i)

Where p is the characteristic of the field, s is the degree of the hypersurface, Tr is the trace operator, and Frob is the Frobenius automorphism. In our case, p is the characteristic of F_p, s=3, and H^i is the i-th cohomology group of the hypersurface.

Now, let's break down the equation x_0x_1-x_2x_3=0. This is a degree 2 hypersurface in P^3(F_p). Using the formula above, we can find the number of solutions to this equation in P^3(F_p) by calculating the cohomology groups of the hypersurface.

To do this, we can use the Lefschetz hyperplane theorem, which states that the cohomology groups of a hypersurface in P^n(F_p) are isomorphic to the cohomology groups of a hyperplane in P^{n-1}(F_p). In our case, this means that the cohomology groups of x_0x_1-x_2x_3=0 in P^3(F_p) are isomorphic to the cohomology groups of x_0x_1=0 in P^2(F_p).

The cohomology groups of x_0x_1=0 in P^2(F_p) are given by:

H^0 = 1
H^1 = 0
H^2 = 1

Substituting these values into the formula for N_s, we get: