Let $K$ be a number field with integral basis $\{\omega\_1,\ldots,\omega\_n\}$.
The affine variety $A\_K$ defined by
$$ N\_{K/{\mathbb Q}}(X\_1 \omega\_1 + \ldots + X\_n \omega\_n) = 1 $$
is an algebraic group, the group structure coming from multiplication 
of units with norm $1$; in fact, $A\_K$ is a norm-1 torus. For pure cubic extensions ${\mathbb Q}(\sqrt[3]{m})$ with $m \not \equiv \pm 1 \bmod 9$, the unit variety is defined by 
$$ X\_1^3 + mX\_2^3 + m^2X\_3^2 - 3mX\_1X\_2X\_3 = 1, $$
for example.


The affine part of the variety $A\_K$ is smooth; the affine part of its 
reduction modulo $p$ is smooth if and only if $p \nmid \Delta$, where
$\Delta$ denotes the discriminant of $K$.


For each prime $p \nmid \Delta$ let $N\_r$ denote the number of 
${\mathbb F}\_q$-rational points on $A\_K$, where $q = p^r$. Define the
Hasse-Weil zeta function
$$ Z\_p(T) = \exp\bigg( \sum\_{r=1}^\infty N\_r \frac{T^r}r \bigg). $$
This zeta function has the following properties:


1. The zeta function $Z\_p(T)$ is a rational function of $T$; the degrees 
of numerator and denominators are equal. More exactly, $Z\_p(T)$ can 
be written in the form
$$ Z\_p(T) = \begin{cases} 
 \frac{P\_0(T) P\_2(T) \cdots P\_{n-1}(T)}{P\_1(T)P\_3(T) \cdots P\_{n}(T)} 
 & \text{ if $n$ is odd}, \\\
 \frac{P\_1(T)P\_3(T) \cdots P\_{n-1}(T)}{P\_0(T) P\_2(T) \cdots P\_{n}(T)} 
 & \text{ if $n$ is even}, 
 \end{cases} $$
 where $P\_j(T)$ is a product of terms of the form $1 - \zeta p^{j}T$ for suitable roots of unity $\zeta$. The actual factors $P\_j(T)$ essentially 
 depend only on the prime ideal factorization of $p$ in $K$. 


Moreover, $Z\_p(\infty) = \lim\_{T \to \infty} Z\_p(T)$ exists and
 satisfies $Z\_p(\infty) = \epsilon\_p$, where $\epsilon\_p = \chi(p)/p$ for
 Pell conics (unit varieties for quadratic extensions) and 
 $\epsilon\_p = \pm 1$ in general.
2. The zeta function $Z\_p(T)$ admits a functional equation of the form
$$ Z\_p\Big(\epsilon\_p \frac1{p^nT}\Big) = \eta\_p Z\_p(T)^{(-1)^n} $$
for some constant $\eta\_p$ depending only on $p$.
3. The global zeta function $Z\_K(s)$ is constructed as follows: set 
$L\_p(s) = P\_{n-1}(p^{-s})$ and $Z(s) = \prod\_p L\_p(s)$. Then, up
to Euler factors at the ramified primes, $Z(s) = \zeta\_K(s+n-2)/\zeta(s+n-2)$, 
 where $\zeta\_K$ is the Dedekind zeta function of $K$.


My impression is that the case of Pell conics is slightly different from
the general case because Pell conics are smooth even at infinity.


I am unaware of almost any of the results on norm-1 tori obtained in the last
30 years, and my main question is:


 **Is all of this a special case of known results on algebraic tori, and if yes, what are the relevant references?** 


BTW, the rationality and the functional equation seem to be 
known in quite general situations. So a more precise question would be whether these unit varieties have received any special attention. 



Weil Conjectures for Number Fields
Some of the relevant results can be found in the book "Algebraic Groups and their birational invariants" by V. E. Voskresenskii (Translations of Math Monographs 179, American Math Society 1998). 
Specifically, Chapter 4, Section 9 is all about tori over a finite field, number of rational points, and zeta function; this proves the (local) results you mention for a general torus. Section 13 (same chapter) shows how the global zeta function of a general torus is expressible in terms of Artin L-functions. The case of norm-one tori is actually a simple but important case. Here is a link:


<http://books.google.com/books?id=O-R8m2qRS04C&lpg=PP1&ots=jnOsvxLq_B&dq=algebraic%20groups%20and%20their%20birational%20invariants&pg=PP1#v=onepage&q&f=false>


