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I am currently reading SGA 4$\frac{1}{2}$, exposé 2: Rapport sur la formule des traces.

The $L$-series associated to a scheme $X$ of finite type over $\mathbb{F}_{p}$ is defined as $$L(X,s):=\prod_{x\in X^{F}}(1-Fp^{-s\cdot\operatorname{deg}(x)})\text{,}$$ where $F$ is the Frobenius and $x$ runs through all closed points of $X$. Excuse me if the following question is stupid: How do I actually know that this $L$ series coincides with the $\zeta$ function associated to $s$, given by

$$\zeta(X,s)=\prod_{\mathfrak{m}\text{ maximal in }X}\frac{1}{1-(N\mathfrak{m})^{-s}}$$

How do I actually prove that these coincide in every case?

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  • $\begingroup$ Almost certainly you meant to write "SGA" in the first line. $\endgroup$
    – Jason Starr
    Commented Jan 15, 2023 at 13:02
  • $\begingroup$ What do you mean with $Fp^{-s \deg(x)}$ (or $p^{-s \deg(x)}$) exactly? More fun with $\det(I-Frob_x t)|_{t=p^{-s}}$ $\endgroup$
    – reuns
    Commented Jan 16, 2023 at 13:32

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