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:

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.

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 $p$.

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.