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Calculating the zeta function over a hypersurface in project

  1. May 4, 2015 #1
    1. The problem statement, all variables and given/known data
    Calculate the zeta function of [itex]x_0x_1-x_2x_3=0[/itex] in [itex]F_p[/itex]

    2. Relevant 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]

    3. 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!
     
  2. jcsd
  3. May 9, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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