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Homework Statement
Calculate the zeta function of [itex]x_0x_1-x_2x_3=0[/itex] in [itex]F_p[/itex]
Homework Equations
Zeta function of the hypersurface defined by f:
[itex]\exp(\sum_{s=1}^\infty \frac{N_s u^s}{s})[/itex]
[itex]N_s[/itex] is the number of zeros of f in [itex]P^n(F_p)[/itex]
The Attempt at a Solution
My biggest struggle is finding [itex]N_s[/itex], here's what I've thought so far:
[itex]N_s[/itex] is composed of finite points and "points at infinity". The points at infinity are solutions on the form [itex](0, 1, \frac{a_2}{a_1}, \frac{a_3}{a_1})[/itex] which gives: [itex](\frac{a_2}{a_1})(\frac{a_3}{a_1})=0[/itex], 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 [itex]2p-1[/itex] points at infinity. Not sure what to say about the amount of finite points.
According to my book the should be [itex]2p+1[/itex] points at infinity and not [itex]2p-1[/itex], and the number of finite points should be [itex]p^2[/itex] making [itex]N_s=p^{2s}+2p^s+1[/itex], (replacing p with [itex]p^s[/itex]). What have I've done wrong? And is the number of finite points all possible combinations of [itex]\frac{a_2}{a_1}[/itex] and [itex]\frac{a_3}{a_1}[/itex], [itex]p^2[/itex]?
Any help much appreciated!