Section 6.6 Adelic Fourier analysis after Tate
Reference.
The original source is [4], XV; note that the “valuation vectors” used therein are our adèles, as per Remark 6.0.1. For a modern (and much less terse) treatment, see [42].
As an aside, we describe another classic use of adèles in algebraic number theory: the derivation of the analytic continuation and functional equation of Dedekind zeta functions and Dirichlet \(L\)-functions via Fourier analysis on the adèles, as described by Tate in his PhD thesis. This is meant merely as a guide to the latter, so we omit essentially all proofs.
Subsection Additive characters
Lemma 6.6.1.
Let \(K\) be a number field and let \(v\) be a place of \(K\text{.}\) Then the dual group \((K_v^+)^\vee\) of continuous characters from \(K_v^+\) to \(\{z \in \CC: |z| = 1\}\) is a locally compact topological group. Moreover, for any nontrivial element \(X \in (K_v^+)^\vee\text{,}\) the map
defines a continuous isomorphism \(K_v^+ \to (K_v^+)^{\vee}\text{.}\)
Proof.
See [4], XV, Lemma 2.2.1.
Remark 6.6.2.
To choose a character \(X\) as in Lemma 6.6.1, we may precompose with a trace map to reduce to the case \(K = \QQ\text{.}\) In that case, for \(v = \infty\) we may take \(X\) to be the character \(t \mapsto e^{-2 \pi i t}\text{;}\) for \(v = p\text{,}\) we may take it to be \(t \mapsto e^{-2 \pi i \lambda(t)}\) where \(\lambda(t) \in \ZZ_{(p)}\) is congruent to \(t\) modulo \(\ZZ_p\text{.}\)
This discussion globalizes directly to the adèles, as long as we are careful about normalization.
Theorem 6.6.3.
Let \(K\) be a number field. For each place \(v\) of \(K\text{,}\) let \(X_v\) be a nontrivial additive character on \(K_v^+\) as in Remark 6.6.2, and define the additive character \(X\) on \(\AA_K^+\) via
Then the dual group \((\AA_K^+)^\vee\) is a locally compact topological group, and the map
defines a continuous isomorphism \(\AA_K^+ \to (\AA_K^+)^\vee\text{.}\)
Proof.
See [4], XV, Theorem 4.1.1.
Subsection Fourier inversion
Theorem 6.6.4. Local Fourier inversion.
Let \(K\) be a number field and let \(v\) be a place of \(K\text{.}\) Fix a Haar measure on \(K_v^+\) and a nontrivial character \(X \in (K_v^+)^\vee\text{.}\) For \(f \in L_1(K_v^+)\text{,}\) define the Fourier transform
If \(\hat{f} \in L_1(K_v^+)\) also, then
for some \(c > 0\) which depends only on the Haar measure and the character \(X\text{.}\) In particular, these can be normalized so that \(c = 1\text{.}\)
Proof.
See [4], XV, Theorem 2.2.2.
Theorem 6.6.5. Global Fourier inversion.
Let \(K\) be a number field, fix a Haar measure on \(\AA_K\text{,}\) and define the additive character \(X\) on \(\AA_K^+\) as in Theorem 6.6.3. For \(f \in L_1(\AA_K^+)\text{,}\) define the Fourier transform
If \(\hat(f) \in L_1(K_v^+)\) also, then
for some \(c > 0\) which depends only on the Haar measure and the character \(X\text{.}\) In particular, these can be normalized so that \(c = 1\text{.}\)
Proof.
See [4], XV, Theorem 4.1.2.
Remark 6.6.6.
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 \(\ZZ\) in \(\RR\text{.}\) In the adelic setup, the “lattice” is the subgroup \(K\) of \(\AA_K\text{,}\) 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.
Subsection The space of quasi-characters
Definition 6.6.7.
Let \(K\) be a number field. By a quasi-character on the idèle class group \(C_K\text{,}\) we will mean any continuous homomorphism from this group into \(\CC^*\text{.}\) By contrast, a character is required to map into the unit circle.
For each quasi-character \(c\text{,}\) there exists a unique real number \(s\) such that \(|c(\alpha)| = |\alpha|^s\) for all \(\alpha \in I_K\) (where \(|\alpha|\) is defined as in Definition 6.2.7). We call \(s\) the exponent of \(c\text{.}\)
The space of quasicharacters on \(C_K\) contains a distinguished copy of \(\CC\text{:}\) each complex number \(s\) corresponds to the character \(\alpha \mapsto |\alpha|^s\text{.}\) The exponent of this character is precisely the real part of \(s\text{.}\)
Remark 6.6.8.
The adelic zeta function of \(K\) will be a function on the space of quasi-characters. Its restriction to the distinguished copy of \(\CC\) will give the usual zeta function. If we take the translate of this copy of \(\CC\) 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.
Subsection Zeta functions and \(L\)-functions
The classical approach to deriving the analytic continuation and functional equation for a Dedekind zeta function, or for Dirichlet \(L\)-functions, is to interpret via an integral representation (technically, as a Mellin transform of a theta series). The functional equation then follows from Poisson summation. Something similar is possible in the adelic situation, with the additional advantage of admitting a “local-global compatibility”.
Theorem 6.6.9.
For suitable functions \(f: \AA_K \to \CC\text{,}\) define the associated zeta function as the following function of quasicharacters on \(C_K\) with exponent greater than \(1\text{:}\)
This function is single-valued and holomorphic except at the points corresponding to \(s=0\) and \(s=1\) where it has simple poles with residues \(-\kappa f(0)\) and \(\kappa \hat{f}(0)\text{,}\) respectively, where
(with \(r_1\) the number of real places, \(r_2\) the number of complex places, \(h\) the class number, \(R\) the unit regulator, \(\Delta_K\) the discriminant, and \(\omega_K\) the order of the group of roots of unity). Moreover, we have the functional equation
where \(\hat{c}(\alpha) = \alpha c(\alpha)^{-1}\) (so in particular \(s \mapsto 1-s\)).
Proof.
See [4], XV, Theorem 4.4.1.
Remark 6.6.10.
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.