Geometry of Numbers

3 Unit theorem

To study the units of \(\cO _K\), we will consider the

log embedding \(\cO _K \to \RR ^{r+s}, x \mapsto [F_i:\RR ]\log (|\sigma _i(x)|)=l_i(x)\), where \(F_i\) is the field \(\sigma _i\) 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 \(\sum _i c_il_i=0\) 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 \(\RR ^r\times \CC ^s\) defined by \(|\sigma _i| \leq b_i\). 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 \(c=2^{r}\pi ^{-s}\sqrt {|\disc (K)|} \leq \prod b_i^\RR \times \prod (b_i^\CC )^2\). 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 \(1 \leq |N(x)| \leq \prod _i b_i^\RR \times \prod _i (b_i^\CC )^2\). On the other hand, for \(i<{r+s}\), \(|\sigma _i| = \frac {N(\sigma _i)}{\prod _{j\neq i}\sigma _j} \geq \frac {b_i}{c}\). Thus \(l_i\) is between \([F_i:\RR ]\log (b_i)-\log (c)\) and \([F_i:\RR ]\log (b_i)\). 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 \(\sum _i c_il_i\) 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. □