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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.

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up vote 10 down vote accepted

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:

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