Geometry of Numbers

3 Unit theorem

To study the units of

O_{K}

, we will consider the

log embedding $O_{K} \to R^{r + s}, x \mapsto [F_{i} : R] \log (| σ_{i} (x) |) = l_{i} (x)$ , where $F_{i}$ is the field $σ_{i}$ is an embedding into.

Lemma 3.1. The kernel of the log embedding consists of roots of unity.

Proof. Certainly they are in the kernel. Conversely, note that anything in the kernel has bounded coefficients of the characteristic polynomial, so that there are finitely many possibilities. But its powers are in the kernel, so it must be a root of unity. □

Lemma 3.2. The image of the units via the log embedding is discrete.

Proof. Any thing whose image is near zero has bounded coefficients of the characteristic polynomial, so there are only finitely many such things. □

The units lie in the hyperplane $\sum l_{i} = 0$ . It turns out that they are full rank, and hence free of rank $r + s - 1$ .

Theorem 3.3. The units form a lattice in the hyperplane $\sum l_{i} = 0$ .

Proof. Suppose that we have a nonzero linear form on $\sum_{i} c_{i} l_{i} = 0$ on $\sum l_{i} = 0$ . We will show there is a unit such that the form on that unit is nonzero. We can assume WLOG that $c_{n} = 0, c_{1} > 0$ . Consider the region in $R^{r} \times C^{s}$ defined by $| σ_{i} | \leq b_{i}$ . For fixed $b_{i}$ , $r + s > i$ , we want to choose $b_{r + s}$ large enough such that the region has area contains a lattice point. This will happen when $c = 2^{r} π^{- s} \sqrt{| disc (K) |} \leq \prod b_{i}^{R} \times \prod (b_{i}^{C})^{2}$ . Now choose any values of $b_{1}, \dots, b_{r + s - 1}$ , and a sufficiently large $b_{r + s}$ will make this an equality. Now if $x$ is a nonzero lattice point in the region, then $1 \leq | N (x) | \leq \prod_{i} b_{i}^{R} \times \prod_{i} (b_{i}^{C})^{2}$ . On the other hand, for $i < r + s$ , $| σ_{i} | = \frac{N (σ_{i})}{\prod_{j \neq i} σ_{j}} \geq \frac{b_{i}}{c}$ . Thus $l_{i}$ is between $[F_{i} : R] \log (b_{i}) - \log (c)$ and $[F_{i} : R] \log (b_{i})$ . Thus we can manufacture each $l_{i}$ to be in any interval of our choosing of a fixed size, so we can produce infinitely many $x$ such with norm at most $c$ such that the $\sum_{i} c_{i} l_{i}$ takes distinct values on each. Since there are finitely many ideals of a given norm, two must differ by a unit on which the linear form doesn’t vanish. □