When it comes to general number fields beyond $\mathbb{Q}$, the litterature is not so abundant in analytic number theory. For instance over $\mathbb{Q}$, for primitve Dirichlet characters modulo $q$, we know the associated L-function $L(s, \chi)$ can be completed into $$\Lambda(s,\chi) = \left(\frac{\pi}{q}\right)^{-\frac{1}{2}(s+a)} \Gamma\left(\frac{s+a}{2}\right)L(s, \chi)$$

where $a$ depends of the value of $\chi(-1)$. This completed $L$-function can be holomorphically continued the the whole complex plane (execept for the principal character) and satisfies the functional equation $$\Lambda(s, \chi) = \frac{\tau(\chi)}{i^a \sqrt{q}}\Lambda(1-s, \overline{\chi})$$

For general number fields $F$, denoting $d$ the degree of the extension over $\mathbb{Q}$, $r_1$ and $r_2$ the real and complex embeddings respectively, we can introduce analogously the completed L-function \begin{equation} \Lambda(s, \chi) = \left( \frac{2^{r_1}|d_F|}{(2\pi)^d} \right)^{s/2} \Gamma\left( \frac{s}{2} \right) \Gamma(s)^{r_2} L(s, \chi), \end{equation}

and it satisfies the functional equation, if $\chi$ is a character attached to the ideal $q$: \begin{equation} \Lambda(s, \chi) = N(q)^{\frac{1}{2}-s} \Lambda(1-s, \chi). \end{equation}

I would like to have a precise reference for L-functions and functional equations in the case of automorphic representations. Let $\pi$ be a self-contragredient cuspidal automorphic representation of $GL(n)$, over a general number field $F$.

Where can I find the precise functional equation satisfies by $L(s, \pi)$, making appear the dependency on the field as well as on the representation $\pi$?

  • 1
    $\begingroup$ This amounts to knowing archimedean L-factor as well as the global epsilon factor. Have you looked in Godement-Jacquet? $\endgroup$ – Kimball Jan 14 '19 at 21:31

As Kimball mentioned, this is all in Godement-Jacquet's monograph "Zeta Functions of Simple Algebras". For the case $n = 1$, this is just Tate's thesis. When the field is $\mathbb{Q}$, a good reference is Goldfeld-Hundley "Automorphic Representations and $L$-Functions for the General Linear Group".

The story is roughly the following. There is a functional equation of the following form: \[\Lambda(s,\pi) = \epsilon(s,\pi) \Lambda(1 - s,\widetilde{\pi}).\] Here $\pi$ is a unitary cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_F)$, where $F$ is a number field, $\widetilde{\pi}$ denotes the contragredient, and $\Lambda(s,\pi) = \prod_v L_v(s,\pi_v)$ is the completed $L$-function of $\pi$ (including the archimedean factors). The epsilon factor $\epsilon(s,\pi)$ factorises as \[\epsilon(s,\pi) = \epsilon(s,\pi,\psi) = \prod_v \epsilon_v(s,\pi_v,\psi_v),\] where $\psi_v$ is an additive character of $F_v$ that is the local component of an additive character of $\mathbb{A}_F$. Each local epsilon factor may depend on $\psi_v$, but the global epsilon factor is independent of $\psi$.

Let $v$ be a nonarchimedean place of $F$ with associated local field $F_v$ having ring of integers $\mathcal{O}_v$, maximal ideal $\mathfrak{p}_v$, and whose residue field $\mathcal{O}_v / \mathfrak{p}_v$ has cardinality $q_v$. Let $\psi_v$ be an additive character of $F_v$. The conductor of $\psi_v$ is $\mathfrak{p}_v^{c(\psi_v)}$, where $c(\psi_v)$ is the least integer (possibly negative) for which $\psi_v$ is trivial on $\mathfrak{p}_v^{c(\psi_v)}$. The conductor of $\pi_v$ is $\mathfrak{p}_v^{c(\pi_v)}$, where $c(\pi_v)$ is the least integer (necessarily nonnegative) for which $\pi_v$ contains a nonzero vector fixed by the congruence subgroup $$ K_0(\mathfrak{p}^{c(\pi_v)}) = \left\{ \begin{pmatrix} a & b \\\ c & d \end{pmatrix} \in \mathrm{GL}_n(\mathcal{O}_v) : c \in \mathrm{Mat}_{1 \times (n - 1)}(\mathfrak{p}_v^{c(\pi_v)}), \ d - 1 \in \mathfrak{p}_v^{c(\pi_v)} \right\}.$$ Then the local epsilon factor is \[\epsilon_v(s,\pi_v,\psi_v) = \epsilon_v\left(\frac{1}{2},\pi_v,\psi_v\right) q_v^{(n c(\psi_v) - c(\pi_v))\left(s - \frac{1}{2}\right)}.\] Suppose that $F_v$ is an extension of $\mathbb{Q}_p$. If we choose $\psi_v$ to be the composition of the standard unramified additive character of $\mathbb{Q}_p$ with the trace map $\mathrm{Tr}_{F_v/\mathbb{Q}_p}$, then $\epsilon_v(1/2,\pi_v,\psi_v)$ is a complex number of absolute value $1$. For example, if $n = 1$, so that $\pi_v$ is a character, $\epsilon_v(1/2,\pi_v,\psi_v)$ is a normalised Gauss sum; in general, it can be quite complicated. Moreover, $\mathfrak{p}_v^{-c(\psi_v)}$ is the different ideal $\mathfrak{D}_{F_v/\mathbb{Q}_p}$ and $q_v^{-c(\psi_v)}$ is the absolute discriminant $\Delta_{F_v/\mathbb{Q}_p} = N_{F_v/\mathbb{Q}_p}(\mathfrak{D}_{F_v/\mathbb{Q}_p})$ of the extension $F_v/\mathbb{Q}_p$.

Note that at all but finitely many places, the additive character $\psi_v$ is unramified ($c(\psi_v) = 0$) and the representation $\pi_v$ is unramified ($c(\pi_v) = 0$).

I haven't yet discussed the archimedean factors. The local $L$-function at an archimedean place $v$ of $F$ will be something of the form \[L_v(s,\pi_v) = \prod_{j = 1}^{n} \zeta_v(s + t_j),\] for some complex numbers $t_j$ (with some restrictions on the possible vertical lines that $t_j$ lies on, since I am assuming that $\pi$ is unitary). Here $\zeta_v(s) = \pi^{-s/2} \Gamma(s/2)$ for a real place $v$ and $\zeta_v(s) = 2(2\pi)^{-s} \Gamma(s)$ for a complex place $v$. Finally, the local epsilon factor at an archimedean place will just be something of the form $\epsilon_v(s,\pi_v,\psi_v) = i^k$ for some integer $k$ (in particular, this is independent of $s$). For more details on the archimedean $L$-functions and epsilon factors, see Knapp's paper "Local Langlands Correspondence: the Archimedean Case".

So the global epsilon factor is \[\prod_v \epsilon_v\left(\frac{1}{2},\pi_v, \psi_v\right) \prod_{v \text{ nonarchimedean}} q_v^{n c(\psi_v) \left(s - \frac{1}{2}\right)} \prod_{v \text{ nonarchimedean}} q_v^{- c(\pi_v) \left(s - \frac{1}{2}\right)}.\] The first term is $\epsilon(1/2,\pi)$, the global root number, which is some complex number of absolute value $1$. The second is \[\Delta_{F/\mathbb{Q}}^{-n\left(s - \frac{1}{2}\right)},\] where $\Delta_{F/\mathbb{Q}}$ is the absolute discriminant of the extension $F/\mathbb{Q}$; this is the norm $N_{F/\mathbb{Q}}(\mathfrak{D}_{F/\mathbb{Q}})$ of the different $$\mathfrak{D}_{F/\mathbb{Q}} = \prod_{v \text{ nonarchimedean}} \mathfrak{p}_v^{-c(\psi_v)}.$$ The third is \[N_{F/\mathbb{Q}}(\mathfrak{q})^{-\left(s - \frac{1}{2}\right)},\] where \[\mathfrak{q} = \prod_{v \text{ nonarchimedean}} \mathfrak{p}_v^{c(\pi_v)}\] is the (relative) conductor of $\pi$ and $N_{K/\mathbb{Q}}$ denotes the norm.

  • 2
    $\begingroup$ Also I would mention Neukrich, algebraic number theory does the full proof of the functional equation of Dedekind and Hecke and Artin L-functions (the latter requires class field theory, so 100 more pages) using the traditional approach of theta series (it takes more than 100 pages) $\endgroup$ – reuns Jan 16 '19 at 1:13
  • $\begingroup$ Thanks! Great answer with all the details, thanks a lot for providing such a synthesis of perhaps standard facts yet not easily accessible in a single place of the litterature! $\endgroup$ – TheStudent Jan 16 '19 at 3:59

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.