# Trying to understand a certain form of zeta function

A month ago I have asked a question about a certain form of the zeta function. I will now try to be more accurate.

facts:

Let $$N_s$$ be the number of points on the projective hypersurface $$\bar{H}_f(F_s)$$.

Under the condition $$N_s = \sum_j \beta_j - \sum_i \alpha_i$$ for $$\beta_j, \alpha_i \in \mathbb{C}$$ we can say that the zeta function is rational with $$Z(u) = \frac{\prod_i 1-\alpha_iu}{\prod_j 1-\beta_ju}.$$

First Question:

Now I have $$N_s = \sum_{k=0}^{n-1} q^{ks} -(-1)^n \sum_{\chi_0,...,\chi_n} [ \frac{(-1)^{n+1}}{q} \chi_0(a_o^{-1}) \cdots \chi_n(a_n^{-1})g(\chi_0) \cdots g(\chi_n)]^s .$$ the $$\chi_i$$ are characters of $$F$$ with the following condition: $$\chi_i^m = \varepsilon, \chi_i \neq \varepsilon$$ and $$\chi_0\chi_1 \cdots \chi_n = \varepsilon$$. And here comes my question: why does the zeta function has the following form?

$$Z(u) = \frac{P(u)^{(-1)^n}}{(1-u)(1-qu) \cdots (1-q^{n-1}u)},$$ where $$P(u) = \prod_{\chi_0,...,\chi_n} (1-(-1)^{n+1}\frac{1}{q}\chi_0(a_o^{-1}) \cdots \chi_n(a_n^{-1})g(\chi_0) \cdots g(\chi_n)u)$$.

Thoughts

It is clear to me that

$$\sum_j \beta_j^s = \sum_{k=0}^{n-1} q^{ks}$$ and $$\sum_i \alpha_i^s =\sum_{\chi_0,...,\chi_n} [ \frac{(-1)^{n+1}}{q} \chi_0(a_o^{-1}) \cdots \chi_n(a_n^{-1})g(\chi_0) \cdots g(\chi_n)]^s$$ but the term $$(-1)^n$$ disturbs me. Without this term I would have $$Z(u) = \frac{P(u)}{(1-u)(1-qu) \cdots (1-q^{n-1}u)}$$. That's totally fine. Just using the fact above. But like I said I don't get it why I get $$(-1)^n$$ in the exponent of $$P(u)$$ if I have the term $$(-1)^n$$.

Second question:

Do you see why $$\deg (P(u)) = m^{-1}[(m-1)^{n+1}+(-1)^{n+1}(m-1)]$$ ? Sorry that I am clueless, here.