Characters of a number-field torus

Statement

For $X=(K_\infty/\mathcal O_K)^d$ , all continuous unitary characters are

\chi_a([x])=\exp\bigl(2\pi i\operatorname{Tr}_\infty(a\cdot x)\bigr), \qquad a\in(\mathcal O_K^\vee)^d.

Thus $\widehat X\cong(\mathcal O_K^\vee)^d$ . Here the ring of integers and codifferent are identified with their Minkowski images.

Why the formula descends

Adding $\ell\in\mathcal O_K^d$ changes the exponent divided by $2\pi i$ by the integer $\operatorname{Tr}(a\cdot\ell)$ . Hence the value is independent of representatives.

Choose an integral basis and its trace-dual basis. In these real coordinates the displayed functions are precisely the integer Fourier modes of a real $md$ -torus, so they include every character. They form an orthonormal basis for Haar $L^2$ .

References

Keith Conrad, The Different Ideal. §3, trace-dual bases; the torus statement follows by the coordinate calculation above.