This question may be too elementary for this forum, but I have asked it on math stackexchange and didn't get an answer. I have now deleted it so there wouldn't be duplicates... Here is the question as it appeared on math stackexchange:

I know that this question is pretty elementary, but I'm still trying to grasp the basics of number theory, and I want to make sure I'm thinking about things correctly. Most likely, I am thinking about things in a way that is slightly off. Since it's so complicated, I would appreciate if you can point out for me any misconceptions that I have.

Let $X_{\mathbb{Q}}$ be a variety over $\mathbb{Q}$. I am trying to understand how to associate to $X_{\mathbb{Q}}$ an $L$-function. As I understand it (please correct any mistakes I make in my narrative) there's an $L$-function for every $i=0,1,2,...$ (for convenience call them $L_0, L_1,L_2, ...$). Starting at $2dim(X)+1$ the $L_i$'s are just $1$. The $L$-function'' associated to all of $X_{\mathbb{Q}}$, by which I mean the Hasse-Weil Zeta Function, is $\prod L_i$.

Let's fix $i$. The function $L_i$ is defined to be the product of $L$-functions $L_{i,p}$, where $p$ runs over all primes in $\mathbb{Z}$. Here things get murky again. We choose a model $X_{\mathbb{Z}}$ of $X_{\mathbb{Q}}$ over $\mathbb{Z}$. (How? It can't be that any model would work, can it?) Now we fix a Weil Cohomology theory (as I understand it, it shouldn't matter which. Is that true? Is it only a conjecture that it's true?). For example we can look at $l$-adic cohomology where $l$ is a prime coprime to $p$. Then we define $L_{i,p}(x)$ to be $P_i(x)^{(-1)^i}$ where $P_i(x)$ is the characteristic polynomial of the action of the Frobenius element on $H^i(X_{\mathbb{Z}/p},\mathbb{Q}_l)$ (or whichever Weil Cohomology theory we chose).

### Questions

First of all I would like to know if the narrative above is accurate. If not, please tell me where. Particular things I am vague about are:

$1.$ How does one choose the model of $X_{\mathbb{Q}}$ over $\mathbb{Z}$?

$2.$ Is it true that we can choose any Weil Cohomology? Is that conjectural, or proven?

$3.$ As I understand it, it should be true that $\prod_{p \in Spec(\mathbb{Z})} L_{i,p}= e^{\sum_{l=1}^{\infty} |X_{\mathbb{Z}/p}(\mathbb{F}_p)|\frac{x^l}{l}}$. Is that right?

$4.$ Does this construction differ in any substantial way if we defined a Hasse-Weil Zeta function for a variety defined over a number field different from $\mathbb{Q}$?

notthe product, it's the alternating product $(L_0)^{-1}L_1(L_2)^{-1}\ldots$ -- think about the fixed point formula in topology, it's the same idea. But as for your bad factors, I don't think the picture is clear. There are lots of models over $Z$ – user30035 Feb 3 '13 at 20:46notthe global $L$-function (which is their product), and – user30035 Feb 3 '13 at 20:48