Notes on class field theory

To choose a character $X$ as in Lemma 6.6.1, we may precompose with a trace map to reduce to the case $K=\mathbb{Q}\text{.}$ In that case, for $v=\mathrm{\infty }$ we may take $X$ to be the character $t\mapsto e^{-2\pi it}\text{;}$ for $v=p\text{,}$ we may take it to be $t\mapsto e^{-2\pi i\lambda (t)}$ where $\lambda (t)\in {\mathbb{Z}}_{(p)}$ is congruent to $t$ modulo ${\mathbb{Z}}_p\text{.}$

Crucially, there is also a version of the Poisson summation formula in this context. In classical Fourier analysis, this involves summing a function and its Fourier transform over the lattice $\mathbb{Z}$ in $\mathbb{R}\text{.}$ In the adelic setup, the “lattice” is the subgroup $K$ of ${\mathbb{A}}_K\text{,}$ and the result can also be viewed as an analogue of the Riemann-Roch theorem in complex geometry! See [4], XV, Theorem 4.2.1.

The adelic zeta function of $K$ will be a function on the space of quasi-characters. Its restriction to the distinguished copy of $\mathbb{C}$ will give the usual zeta function. If we take the translate of this copy of $\mathbb{C}$ by some other quasi-character, we will end up computing the $L$-function associated to some Hecke character. The idea of the adelic setup is to package all of these Hecke $L$-functions together into a single object, which can be studied by an adelic analogue of the classical proof of analytic continuation for the Riemann zeta function. More on this below.

Theorem 6.6.9 looks a lot like what we are expecting except for the presence of the mysterious test function $f\text{.}$ To get back to more classical statements like Theorem 2.4.2 and Theorem 2.4.5, one must choose $f$ so that one can evaluate $\hat{f}$ and have it come out to be something similar to $f\text{.}$ See the very end of [4], XV for further discussion.