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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.

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