The main goal of this series of posts is to give a brief summary of Tate’s thesis [Tat67]. Given a number field \( K \) and a character \( \chi : \mathrm{Cl} _ \mathfrak{m}(K) \rightarrow S^1 \), one defines the Dirichlet \( L \)-function as \[ \displaystyle L(s, \chi) = \sum _ {\mathfrak{a} \subseteq \mathcal{O} _ K}^{} \frac{\chi(\mathfrak{a})}{(N\mathfrak{a})^s} = \prod _ {\mathfrak{p} \subseteq \mathcal{O} _ K}^{} \frac{\chi(\mathfrak{p})}{1 - (N\mathfrak{p})^{-s}}, \] where we regard \( \chi(\mathfrak{a}) = 0 \) if \( (\mathfrak{a}, \mathfrak{m}) \neq 1 \). Although this series converges only for \( \Re(s) > 1 \), Hecke proved that the function extends meromorphically to the entire complex plane and moreover found a functional equation relating \( L(s, \chi) \) and \( L(1-s, \overline{\chi}) \). It turns out that we can define an analogous function for a character \( \chi : \mathbb{A} _ K^\times / K^\times \rightarrow S^1 \).
Tate’s contribution to this theory is to prove the functional equation and meromorphic extension in a systematic, elegant way. The main tool he uses is Fourier analysis on the adele \( \mathbb{A} _ K \), in particular the Poisson summation formula. Because \( K \) is discrete inside \( \mathbb{A} _ K \) with finite quotient, \( K \) really can be thought of as a lattice inside \( \mathbb{A} _ K \) and the Poisson summation formula in this context can be written down. Imitating the proof of the functional equation for the Riemann zeta function in this context (with the right definition of the zeta function) then gives the functional equation.
The theory also gives new interpretations of classical knowledge on \( L \)-functions. For example, the functional equation for the Riemann zeta function can be written as \[ \displaystyle \pi^{-\frac{s}{2}} \Gamma\Bigl( \frac{s}{2} \Bigr) \zeta(s) = \pi^{-\frac{1-s}{2}} \Gamma\Bigl( \frac{1-s}{2} \Bigr) \zeta(1-s). \] Here, the additional factor \[ \displaystyle \pi^{-\frac{s}{2}} \Gamma\Bigl( \frac{s}{2} \Bigr) = \int _ {-\infty}^{\infty} e^{-\pi x^2} \lvert x \rvert^s \frac{dx}{\lvert x \rvert} \] can be interpreted as the contribution of infinity, so that the "completed" zeta function \[ \displaystyle \xi(s) = \pi^{-\frac{s}{2}} \Gamma\Bigl( \frac{s}{2} \Bigr) \prod _ {p}^{} \Bigl( 1 - \frac{1}{1 - p^{-s}} \Bigr) \] is the product of the local factors at each place, finite and infinite. The functional equation \( \xi(s) = \xi(1-s) \) follows from a neat application of the Poisson summation formula in the adelic setting.
1. Fourier analysis on topological groups#
We are going to need a theory of Fourier analysis on locally compact Hausdorff topological abelian groups. We will ultimately apply this to groups like \( K _ v^+ \), \( K _ v^\times \) or \( \mathbb{A} _ K^+ \), \( \mathbb{A} _ K^\times \). Let me state the duality theory for locally compact abelian groups.
Definition 1. Let \( G \) be an locally compact abelian topological group. A Haar measure on \( G \) is a regular, locally finite, translation-invariant Borel measure \( \mu \) on \( G \).
Any locally compact abelian topological group has a Haar measure. Moreover, the Haar measure is unique up to multiplication by a positive constant. For an locally compact abelian group \( G \), we define its dual group as \( \widehat{G} = \mathrm{Hom}(G, S^1) \), with the compact-open topology. We call an element of \( \widehat{G} \) a character of \( G \).
Theorem 2 (Pontryagin duality). If \( G \) is locally compact abelian, its dual group \( \widehat{G} \) is as well. The canonical homomorphism \( G \rightarrow (\widehat{G})^{\widehat{}} \) is an isomorphism of topological groups.
Let us fix an locally compact abelian group \( G \), and a Haar measure \( \mu \). For a function \( f \in L^1(G) \), we can define its Fourier transform as \[ \displaystyle \hat{f}(\xi) = \int _ {G}^{} f(x) \overline{\xi(x)} d\mu(x). \]
There exists a Haar measure \( \nu \) on \( \widehat{G} \) such that if \( f \in L^1(G) \) and \( \hat{f} \in L^1(\widehat{G}) \) then \[ \displaystyle f(x) = \int _ {\widehat{G}}^{} \hat{f}(\xi) \xi(x) d\nu(\xi). \] If \( f \in L^1(G) \cap L^2(G) \), then we have \[ \displaystyle \lVert f \rVert _ {L^2(G)} = \lVert \hat{f} \rVert _ {L^2(\widehat{G})}. \] Because \( L^1(G) \cap L^2(G) \) is dense in \( L^2(G) \), we can continuously extend Fourier transform to an isometry \( L^2(G) \cong L^2(\widehat{G}) \).
However, sometimes it is not enough to have functions that are defined only up to a measure zero set. There is a generalization of Schwartz functions, called Schwartz–Bruhat functions that are defined on an arbitrary locally compact abelian group. Denote by \( \mathcal{S}(G) \) the space of complex-valued Schwartz–Bruhat functions on \( G \). Here are some properties they satisfy:
- On \( G = \mathbb{R}^n \times T^m \times F \) where \( F \) is a finite group, Schwartz–Bruhat functions are functions \( f \) with \( \sup _ G \lvert x^\alpha D^\beta f \rvert < \infty \) for all multi-indices \( \alpha \) on the \( \mathbb{R}^n \) component and \( \beta \) on the \( \mathbb{R}^n \times T^m \) component. So this gives a description of Schwartz–Bruhat functions on \( \mathbb{R}^\times \) and \( \mathbb{C}^\times \) as well.
- On \( G = K _ v \) or \( G = K _ v^\times \) with \( v \) finite, Schwartz–Bruhat functions are compactly supported locally constant functions.
- On \( G = \mathbb{A} _ K \), Schwartz–Bruhat functions are finite linear combinations of \( \prod _ {v}^{} f _ v \) where \( f _ v : K _ v \rightarrow \mathbb{C} \) are Schwartz–Bruhat functions described above, and \( f _ v = \mathbf{1} _ {\mathcal{O} _ v} \) for almost all \( v \).
- Fourier transform sends \( \mathcal{S}(G) \) to \( \mathcal{S}(\widehat{G}) \).
We can even define tempered distributions in the Schwartz–Bruhat sense, by considering the dual space \( \mathcal{S}^\prime(G) \). A Fourier transform of a tempered distribution will be a tempered distribution.
2. Characters and measures on local fields#
In this section, we classify all the additive characters of a local field. It turns out that, given a local field \( K _ v \), there is a (non-canonical) isomorphism \( K _ v^+ \cong \widehat{K} _ v^+ \) as topological groups. This allows us to talk about the Fourier transform as an isometry \( L^2(K _ v^+) \cong L^2(K _ v^+) \).
Definition 3. A local field is a field that is either a finite extension of \( \mathbb{R} \) (which is either \( \mathbb{R} \) or \( \mathbb{C} \)) or a finite extension of \( \mathbb{Q} _ p \) for some prime \( p \).
Let \( K _ v \) be a local field, with absolute value \( \lvert - \rvert _ v \) that is normalized so that \[ \displaystyle \mu(a S) = \mu(S) \lvert a \rvert _ v \] for all Borel \( S \subseteq K _ v \), \( a \in K _ v^\times \), and Haar measures \( \mu \) on \( K _ v^+ \). (If \( K _ v \cong \mathbb{R} \), it is usual absolute value, if \( K _ v \cong \mathbb{C} \) it is square of the absolute value, and if \( K _ v \) is non-archemedian, it is such that \( \lvert \pi _ v \rvert _ v = (N \pi _ v)^{-1} \) for \( N \pi _ v \) the size of the residue field.) With this absolute value, we can construct a Haar measure \( \mu^\times \) on \( K _ v^\times \) from a Haar measure \( \mu \) of \( K _ v^+ \) by taking \[ \displaystyle \mu _ 1^\times(S) = \int _ {S}^{} \frac{1}{\lvert x \rvert} d\mu(x). \]
Now let us try to construct characters on \( K _ v^+ \) for a local field \( K _ v \). Here are some examples:
For \( G = K _ v = \mathbb{R} \), we consider the character \[ \displaystyle \chi _ v(x) = e^{-2 \pi i x} \in S^1. \]
For \( G = K _ v = \mathbb{Q} _ p \), we consider the character \[ \displaystyle \chi _ v(x) = \{ x \} = (\text{fractional part of } x) \in \mathbb{R}/\mathbb{Z} \cong S^1. \] This could be equivalently described as the unique rational number \( \chi(x) \in \mathbb{Z}[\frac{1}{p}] / \mathbb{Z} \) such that \( x - \chi(x) \in \mathbb{Z} _ p \). This character \( K _ v \rightarrow S^1 \) is even locally constant.
If \( G = L _ w \) is a finite extension of \( K _ v \), we may consider a character \[ \displaystyle \chi _ w(x) = \chi(\mathrm{Tr} _ {L _ w/K _ v} x) \in S^1. \] Because \( \mathrm{Tr} _ {L _ w/K _ v} : L _ w \rightarrow K _ v \) is additive, it is going to be a character.
It turns out that once we have one character, we can get all the characters from it by composing with multiplication.
Theorem 4. For \( G = K _ v \) a local field, the map \[ \displaystyle \Phi : K _ {v} \rightarrow \widehat{K} _ v; \quad x \mapsto (y \mapsto \chi _ v(xy)) \] is an isomorphism of topological groups.
Proof. This is continuous because \( K _ v \times K _ v \rightarrow S^1 \) given by \( (x, y) \mapsto \chi _ v(xy) \) is continuous. It is injective because \( \chi _ v(x(-)) = 0 \) and \( \chi _ v \neq 0 \) implies \( x = 0 \). For surjectivity, denote by \( I \subseteq \widehat{G} \) the closure of the image of the map \( \Phi \). Then \( \widehat{G} / I \) is a locally compact Hausdorff group as well, so by Pontryagin duality, there exists a nontrivial character \( \widehat{G} \rightarrow S^1 \) vanishing on \( I \), unless \( I = \widehat{G} \). Any such character \( \widehat{G} \rightarrow S^1 \) should be given by evaluation at some \( y \), again by Pontryagin duality, and then \( \chi _ v(xy) = 0 \) for all \( x \) implies that \( y = 0 \). Therefore \( \widehat{G} = I \).
So far we know that this map \( \Phi \) is continuous and injective with dense image. We want to show that this is a homeomorphism onto its image. Given any ball \( B _ \epsilon(0) \subseteq K _ v \), one can show that \( \chi _ v(x(-)) \) sending \( B _ N(0) \) into \( (-\epsilon^\prime, \epsilon^\prime) \subseteq S^1 \) implies \( x \in B _ \epsilon(0) \), if \( N \) is sufficiently large and \( \epsilon^\prime \) is sufficiently small. (You can try out in both the archemedian and non-archemedian cases.) Once we have this, completeness of \( G \) shows that the image of \( \Phi \) is closed, and hence the entire group \( \widehat{G} = \widehat{K} _ v \). Then \( \Phi \) is an isomorphism of topological groups. ▨
This identification is not really "canonical", in the sense that we have somewhat arbitrarily chosen the character \( \chi \). However, this choice of identification will be important because later in the global setting we will be able to put them together to an identification \( \mathbb{A} _ K \cong \widehat{\mathbb{A}} _ K \) in which \( K \) is self-dual.
A Haar measure \( \mu \) on \( G \) naturally induces a Haar measure \( \nu \) on \( \widehat{G} \) in such a way that the Fourier transform is an \( L^2 \)-isometry. If we scale \( \mu \) by a constant \( \alpha \), we should scale \( \nu \) by \( \alpha^{-1} \). This means that given an identification \( G \cong \widehat{G} \), there should be a unique Haar measure \( \mu \) on \( G \) such that the induced measure \( \nu \) on \( \widehat{G} \) coincides with \( \mu \) push-forwarded along the isomorphism \( G \rightarrow \widehat{G} \). It will be nice to find precisely this normalization.
Proposition 5. This correct normalization is
- for \( K _ v = \mathbb{R} \), the usual Lebesgue measure,
- for \( K _ v = \mathbb{Q} _ p \), the measure \( \mu \) with \( \mu(\mathbb{Z} _ p) = 1 \),
- for \( L _ w \) a finite extension of \( K _ v \) with degree \( n \), choose a basis \( x _ 1, \ldots, x _ n \) of \( L _ w \) over \( K _ v \). Then consider the isomorphism \[ \displaystyle K _ v^{\oplus n} \cong L _ w; \quad (\alpha _ i) \mapsto \alpha _ 1 x _ 1 + \cdots + \alpha _ n x _ n \] and then we push-forward the measure \[ \displaystyle d\mu(\alpha _ 1, \ldots, \alpha _ n) = \sqrt{\lvert \det(\sigma _ i x _ j) _ {1 \le i, j \le n}^2 \rvert _ v} \cdot d\mu(\alpha _ 1) \cdots d\mu(\alpha _ n) \] along this map, where \( \sigma _ i \) are the Galois embeddings \( L _ w \hookrightarrow \overline{K} _ v \). (This is independent of the choice of basis \( x _ 1, \ldots, x _ d \).)
In particular, on \( \mathbb{C} \) this is \( 2 \) times the normal Euclidean measure \( dx dy \), and on an archemedian local field \( K _ v \) this is such that \( \mu(\mathcal{O} _ v) = (N\mathcal{D} _ v)^{-1/2} \) where \( \mathcal{D} _ v^{-1} = \{ x \in K _ v : \mathrm{Tr} _ {K _ v/\mathbb{Q} _ p}(x) \in \mathcal{O} _ v \} \) and \( N\mathcal{D} _ v \) is the index of \( \mathcal{D} _ v \) in \( \mathcal{O} _ v \).
To prove this, it suffices to check that with this measure, there is one nonzero function \( f \) such that \( \lVert f \rVert _ {L^2} = \lVert \hat{f} \rVert _ {L^2} \). We can explicitly compute the following:
For \( K _ v = \mathbb{R} \) and \( f(x) = e^{-\pi x^2} \), we have \[ \displaystyle \hat{f}(\xi) = \int _ {\mathbb{R}}^{} f(x) e^{2 \pi i x \xi} dx = e^{-\pi \xi^2}. \]
For \( K _ v = \mathbb{Q} _ p \) and \( f(x) = \mathbf{1} _ {\mathbb{Z} _ p} \), we have \[ \displaystyle \hat{f}(\xi) = \int _ {\mathbb{Q} _ p}^{} f(x) \overline{\chi(\xi x)} dx = \frac{1}{\lvert \xi \rvert} \int _ {\xi \mathbb{Z} _ p}^{} \overline{\chi(x)} dx = \mathbf{1} _ {\mathbb{Z} _ p}(\xi). \]
For \( L _ w \) a finite extension of \( K _ v \) of degree \( n \), take an arbitrary basis \( x _ 1, \ldots, x _ n \) of \( L _ w \) over \( K _ v \) and take \[ \displaystyle f(\alpha _ 1 x _ 1 + \cdots + \alpha _ n x _ n) = g(\alpha _ 1) g(\alpha _ 2) \cdots g(\alpha _ n) \] for an nonzero arbitrary Schwartz–Bruhat function \( g : K _ v \rightarrow \mathbb{C} \). Then we compute \[ \displaystyle \begin{aligned} \hat{f}(\xi) & \displaystyle= \int _ {L}^{} g(\alpha _ 1) \cdots g(\alpha _ n) \overline{\chi(\mathrm{Tr}(x \xi))} d\mu(\alpha _ 1 x _ 1 + \cdots + \alpha _ n x _ n) \\ & \displaystyle= \sqrt{\lvert (\det (\sigma _ i x _ j))^2 \rvert _ v} \cdot \hat{g}(\mathrm{Tr}(x _ 1 \xi)) \cdots \hat{g}(\mathrm{Tr}(x _ n \xi)). \end{aligned} \] Here, we note that \( (\det (\sigma _ i x _ j))^2 = \det(\mathrm{Tr}(x _ i x _ j)) \) and this implies that \( \lVert f \rVert _ {L^2} = \lVert \hat{f} \rVert _ {L^2} \).
For \( K _ v \) a non-archemedian local field, consider \( f = \mathbf{1} _ {\mathcal{O} _ v} \). Its Fourier transform is explicitly computed as \[ \displaystyle \hat{f} = \mu(\mathcal{O} _ v) \mathbf{1} _ {\mathcal{D} _ v^{-1}}. \] Then \( \lVert f \rVert _ {L^2} = \lVert \hat{f} \rVert _ {L^2} \) implies \( \mu(\mathcal{O} _ v) = \mu(\mathcal{O} _ v)^2 \mu(\mathcal{D} _ v^{-1}) \) and so \( \mu(\mathcal{O} _ v) = (N\mathcal{D} _ v)^{-1/2} \).
We now look at multiplicative characters of \( K _ v \), i.e., characters of \( K _ v^\times \). But for purposes of defining the zeta function, we are going to look at a more general class of characters.
Definition 6. For a topological group \( G \), a quasi-character on \( G \) is continuous group homomorphism \( c : G \rightarrow \mathbb{C}^\times \).
Note that for any local field \( K _ v \), the subgroup \[ \displaystyle \mathcal{U} = \{ x \in K _ v^\times : \lvert x \rvert _ v = 1 \} \subseteq K _ v^\times \] is compact. Thus for any quasi-character \( c \), the image \( c(\mathcal{U}) \) is going to be compact and hence bounded. But it is a subgroup of \( \mathbb{C}^\times \), and therefore \( c(\mathcal{U}) \subseteq S^1 \). Then \( x \mapsto \lvert c(x) \rvert \) is trivial on \( \mathcal{U} \) and thus factors as \( K _ v^\times / \mathcal{U} \rightarrow \mathbb{R} _ {>0} \). The group \( K _ v^\times \) is either \( \mathbb{Z} \) or \( \mathbb{R} \). Therefore any quasi-character \( c \) is actually going to look like \[ \displaystyle c(x) = \tilde{c}(x) \lvert x \rvert^\sigma \] for some \( \sigma \in \mathbb{R} \) and an actual character \( \tilde{c} : K _ v^\times \rightarrow S^1 \). Such a representation is unique, and we call \( \sigma \) the exponent of the quasi-character \( c \).
Recall that we defined the norm on \( K _ v \) so that \[ \displaystyle d\mu _ 1^\times(x) = \frac{1}{\lvert x \rvert} d\mu(x) \] becomes a Haar measure on \( K _ v^\times \). However, for global purposes, we want \( \mathcal{U} \) to have unit measure in most cases. This will allow us to define a measure on \( \mathbb{A} _ K^\times \) just by multiplying all the local measures. Thus we renormalize this measure and define \[ \displaystyle d\mu^\times(x) = \begin{cases} d\mu _ 1^\times(x) & \displaystyle K _ v \text{ archemedian} \\ \frac{N \pi _ v}{N \pi _ v - 1} d\mu _ 1^\times(x) & \displaystyle K _ v \text{ non-archemedian}. \end{cases} \] It follows that \[ \displaystyle \mu^\times(\mathcal{O} _ v^\times) = \frac{N \pi _ v}{N \pi _ v - 1} \mu(\mathcal{O} _ v^\times) = \mu(\mathcal{O} _ v) = (N \mathcal{D})^{-1/2}. \] in the non-archemedian case.
3. Local zeta factors#
In some sense, we are going to define the zeta function by multiplying together the local factors. We define the local factor and investigate some properties in this section.
Definition 7. For \( f \in \mathcal{S}(K _ v) \) and \( c \) a quasi-character on \( K _ v^\times \), we define \[ \displaystyle \zeta(f, c) = \int _ {K _ v^\times}^{} f(x) c(x) d\mu^\times(x) = \int _ {K _ v^\times}^{} f(x) c(x) \frac{(\mathrm{factor}) d\mu(x)}{\lvert x \rvert}. \]
One simple example is the case when \( v \) is finite, \( f = \mathbf{1} _ {\mathcal{O} _ v} \), and \( c = \lvert - \rvert^s \). Then we easily compute \[ \displaystyle \zeta(f, c) = \int _ {K _ v^\times}^{} \mathbf{1} _ {\mathcal{O} _ v}(x) \lvert x \rvert^s d\mu^\times(x) = \sum _ {k \ge 0}^{} \mu^\times(\pi _ v^k \mathcal{O} _ v^\times) (N \pi _ v)^{-ks} = \frac{1}{1 - (N \pi _ v)^{-s}}. \]
Note that this does not even necessarily converge. But if the exponent of \( c \) is greater than \( 0 \), we will have decay in \( c(\alpha) \) and so the integral converges. Moreover, the function \[ \displaystyle s \mapsto \zeta(f, c \lvert - \rvert^s) \] for some quasi-character \( c \) will be holomorphic function on \( \Re(s) > -\sigma \), where \( \sigma \) is the exponent of \( c \). The fun fact is that we can meromorphically extend it to the entire plane, and this is where we’re heading for.
Proposition 8. For \( f, g \in \mathcal{S}(K _ v) \), we have \[ \displaystyle \zeta(f, c) \zeta(\hat{g}, \hat{c}) = \zeta(g, c) \zeta(\hat{f}, \hat{c}) \] for any quasi-character \( c \) of exponent \( 0 < \sigma < 1 \) and \( \hat{c} = \lvert - \rvert / c \).
Proof. Note that the condition \( 0 < \sigma < 1 \) is needed to make sure that both \( c \) and \( \hat{c} \) has positive exponent. We can directly compute \[ \displaystyle \begin{aligned} \zeta(f, c) \zeta(\hat{g}, \hat{c}) & \displaystyle= \int _ {K _ v^\times}^{} f(x) c(x) d\mu^\times(x) \int _ {K _ v^\times}^{} \hat{g}(y) \frac{\lvert y \rvert}{c(y)} d\mu^\times(y) \\ & \displaystyle= \int _ {K _ v^\times}^{} \int _ {K _ v^\times}^{} f(x) c(x) \hat{g}(x y) \frac{\lvert x y \rvert}{c(x y)} d\mu^\times(x) d\mu^\times(y) \\ & \displaystyle= \int _ {K _ v^\times}^{} \biggl( \int _ {K _ v^\times}^{} f(x) \hat{g}(xy) \lvert x \rvert d\mu^\times(x) \biggr) \frac{\lvert y \rvert}{c(y)} d\mu^\times(y) \end{aligned} \] by Fubini and change of variables \( y \mapsto xy \). Then the part in the parenthesis is symmetric in \( f \) and \( g \) because \[ \displaystyle \begin{aligned} \int _ {K _ v^\times}^{} f(x) & \displaystyle{} \biggl( \int _ {K _ v^\times}^{} g(z) \overline{\chi(xyz)} \lvert z \rvert d\mu^\times(z) \biggr) \lvert x \rvert d\mu^\times(x) \\ & \displaystyle= \int _ {K _ v^\times}^{} \int _ {K _ v^\times}^{} \lvert x z \rvert f(x) g(z) \overline{\chi(x y z)} d\mu^\times(x) d\mu^\times(z) \end{aligned} \] again by Fubini. ▨
Suppose \( c \) is a quasi-character with \( 0 < \sigma < 1 \). If we choose a generic enough \( f \), we will get both \( \zeta(f, c) \) and \( \zeta(\hat{f}, \hat{c}) \) nonzero. This means that there exists a constant \( \rho(c) \neq 0 \) such that \[ \displaystyle \zeta(f, c) = \rho(c) \zeta(\hat{f}, \hat{c}). \] This is the local version of the functional equation for the zeta function.
Now \( \rho(c) \) seems to be a pretty interesting number. It is not necessary to compute the number to develop the global theory, but it will be useful to compute \( \rho \) to get explicit functional equations. Because \( \rho(c) \) is independent of \( f \), we can compute it by using whatever convenient \( f \) we choose.
- \( K _ v = \mathbb{R} \). In this case, a quasi-character looks like \( \lvert - \rvert^s \) or \( \mathrm{sign}(-) \lvert - \rvert^s \) for some complex number \( s \).
- For \( c = \lvert - \rvert^s \), use \( f(x) = e^{-\pi x^2} \). Then we get
\[ \displaystyle \rho(\lvert - \rvert^s) = 2^{1-s} \pi^{-s} \cos\Bigl( \frac{\pi s}{2} \Bigr) \Gamma(s). \] - For \( c = \mathrm{sign}(-) \lvert - \rvert^s \), use \( f(x) = x e^{-\pi x^2} \) and we get
\[ \displaystyle \rho(\mathrm{sign}(-) \lvert - \rvert^s) = -i 2^{1-s} \pi^{-s} \sin\Bigl( \frac{\pi s}{2} \Bigr) \Gamma(s). \]
- \( K _ v = \mathbb{C} \). Denote \( c _ n(-) : K _ v^\times \rightarrow S^1 \) as \( c _ n(r e^{i \theta}) = e^{i n \theta} \). Then all quasi-characters look like \( c = c _ n \lvert - \rvert^s \) for some \( n \in \mathbb{Z} \) and \( s \in \mathbb{C} \).
- For \( c = c _ n \lvert - \rvert^s \), we can use \( f(z) = (\bar{z})^{\lvert n \rvert} e^{-2 \pi \lvert z \rvert^2} \) for \( n \ge 0 \) and \( f(z) = z^{\lvert n \rvert} e^{-2 \pi \lvert z \rvert^2} \) for \( n \le 0 \). (This \( \lvert z \rvert \) is the ordinary absolute value, not the normalized absolute value for the local field \( \mathbb{C} \).) Then
\[ \displaystyle \rho(c _ n \lvert - \rvert^s) = (-i)^{\lvert n \rvert} \frac{(2\pi)^{1-s} \Gamma(s + \frac{\lvert n \rvert}{2})}{(2 \pi)^{s} \Gamma(1 - s + \frac{\lvert n \rvert}{2})}. \]
- \( K _ v \) non-archemedian. For an arbitrary quasi-character \( c \), the image of \( \mathcal{O} _ v^\times \) is going to be a discrete subgroup of \( S^1 \). Then the kernel of \( c : \mathcal{O} _ v^\times \rightarrow S^1 \) contains some set of the filtration \[ \displaystyle \mathcal{U} _ 0 = \mathcal{O} _ v^\times \supseteq \mathcal{U} _ 1 = 1 + \pi _ v \mathcal{O} _ v \supseteq \mathcal{U} _ 2 = 1 + \pi _ v^2 \mathcal{O} _ v \supseteq \cdots. \] Define the conductor of \( c \) as the ideal \( \mathcal{F} = (\pi _ v^n) \) when \( n \) is minimal with \( c(\mathcal{U} _ n) = 1 \). We can use \( f(x) = \chi(x) \mathbf{1} _ {\pi _ v^{-n} \mathcal{D}^{-1}} \) to compute \( \rho \).
- If \( n = 0 \), then
\[ \displaystyle \rho(\lvert - \rvert^s) = (N\mathcal{D})^{s-\frac{1}{2}} \frac{1 - (N \pi _ v)^{s-1}}{1 - (N\pi _ v)^{-s}}. \] - If \( n \ge 1 \) and \( c(\pi _ v) = 1 \), then
\[ \displaystyle \rho(c \lvert - \rvert^s) = (N \pi _ v^n \mathcal{D})^{s-\frac{1}{2}} \rho _ 0(c), \quad \rho _ 0(c) = (N \pi _ v^n)^{-\frac{1}{2}} \sum _ {\epsilon}^{} c(\epsilon) \chi\biggl( \frac{\epsilon}{\pi _ v^{\mathrm{ord}(\mathcal{D}) + n}} \biggr), \] where \( \epsilon \) runs over \( \mathcal{O} _ v^\times / \mathcal{U} _ n \). This constant \( \rho _ 0(c) \), which has absolute value \( 1 \), is sometimes called the root number.
From all these computations, we immediately see that \( \rho(c \lvert - \rvert^s) \) meromorphically extends to the entire plane, for every quasi-character \( c \). As a corollary, \( \zeta(f, c \lvert - \rvert^s) \) meromorphically extends to the entire plane as well because \( \zeta(f, c \lvert - \rvert^s) \) makes sense for \( \Re(s) > -\sigma \) and \( \rho(c \lvert - \rvert^s) \zeta(\hat{f}, \hat{c} \lvert - \rvert^{-s}) \) makes sense for \( \Re(s) < 1-\sigma \).
References#
[Tat67] J. T. Tate, Fourier analysis in number field, and Hecke’s zeta-functions, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 305–347.