# Notes on the arithmetic of Hecke $L$-functions

A. Raghuram A. Raghuram: Department of Mathematics
Fordham University Lincoln Center, 113 West 60th Street, New York, NY 10023,
USA<EMAIL_ADDRESS>
###### Abstract.

This is an expository article that concerns the various related notions of
algebraic idèle class characters, the Größencharaktere of Hecke, and
cohomological automorphic representations of ${\rm GL}(1)$, all under the
general title of algebraic Hecke characters. The first part of the article
systematically lays the foundations of algebraic Hecke characters. The only
pre-requisites are: basic algebraic number theory, familiarity with the adelic
language, and basic sheaf theory. Observations that play a crucial role in the
arithmetic of automorphic $L$-functions are also discussed. The second part of
the article, on the ratios of successive critical values of the Hecke
$L$-function attached to an algebraic Hecke character, concerns certain
variations on a theorem of Harder [6], especially drawing attention to a
delicate signature that apparently has not been noticed before.

###### 1991 Mathematics Subject Classification:

11F67; 11F66, 11F70, 11F75, 20G05, 22E50, 22E55

###### Contents

  1. 1 Introduction
  2. 2 Hecke characters
    1. 2.1 Class groups
    2. 2.2 Basics of Hecke characters
    3. 2.3 Dirichlet characters: Hecke characters of finite order
    4. 2.4 Local components at finite places and conductor
    5. 2.5 The character at infinity of a Hecke character
  3. 3 Größencharaktere mod $\mathfrak{m}$
    1. 3.1 Definition
    2. 3.2 The correspondence between Hecke characters and Größencharaktere
    3. 3.3 The infinity type of a Größencharakter
  4. 4 Algebraic Hecke characters
    1. 4.1 Definition and purity
    2. 4.2 The $L$-function of an algebraic Hecke character
    3. 4.3 Existence of (critical) algebraic Hecke characters
  5. 5 Cohomological automorphic representations of ${\rm GL}_{1}$
    1. 5.1 Locally symmetric spaces for ${\rm GL}_{1}/F$
    2. 5.2 Sheaves on locally symmetric spaces
    3. 5.3 Cohomology of ${\rm GL}_{1}$ and algebraic Hecke characters of $F$ with coefficients in $E$
  6. 6 Ratios of critical values of Hecke $L$-functions
    1. 6.1 A result of Harder on ratios of critical $L$-values
    2. 6.2 An example: stability under base-change
    3. 6.3 Variations on a result of Harder
    4. 6.4 Compatibility with Deligne’s Conjecture

##  1\. Introduction

In [17] André Weil introduced a special class of characters on the group of
idèle-classes of an algebraic number field $F$ which he called characters of
type ($A_{0}$). The importance of these characters are best captured in Weil’s
words:

> “ As to the non-trivial characters of type ($A_{0}$), some of them arise
> with the theory of abelian varieties with complex multiplication … Taniyama
> has proved that the $L$-series attached to the characters of type ($A_{0}$)
> belonging to abelian varieties with complex multiplication are precisely
> those which occur in the zeta-functions of such varieties … all that can be
> said here is that they tend to emphasize the importance of the characters we
> have discussed and of their remarkable properties.”

With the hindsight afforded by decades of development, we can now say that the
modern arithmetic theory of automorphic forms has Weil’s landmark 1955 paper
as one of its points of origin.

In this article, we are primarily concerned with the arithmetic aspects of
automorphic forms and $L$-functions on ${\rm GL}(1).$ While working on the
special values of automorphic $L$-functions, especially in the context of [11]
and [12], the author was naturally confronted with the various notions
connecting Weil’s characters of type ($A_{0}$), the Größencharaktere of Hecke,
and cohomological automorphic representations of ${\rm GL}(1)$, all under the
general title of algebraic Hecke characters. The idea of writing these notes
is so that anyone with a similar interest in the arithmetic theory of
$L$-functions has a ready reference at hand with all the necessary details.
There is nothing new in the first five sections other than a reorganization of
one’s thoughts to wade through the various dictionaries; this part of the
article is based upon the references: Deligne [3], [4], Harder [6], Hida [8],
Neukirch [10], Serre [14], Schappacher [13], Waldspurger [16], and Weil [17].
Also included are observations that play a crucial role in the study of
special values of automorphic $L$-functions. The only pre-requisites to read
the first part are basic algebraic number theory, the language of adèles and
idèles (for example, the first seven chapters of Weil’s book [18] are
sufficient), and some basic theory of sheaf cohomology.

The last section (Sect. 6), which concerns the special values of Hecke
$L$-functions, is conceptually and technically much deeper. In a pioneering
paper, Günter Harder [6] laid the foundations of Eisenstein cohomology for
${\rm GL}(2)$ and as a consequence proved a rationality result on the ratios
of critical values of Hecke $L$-functions. To give an idea of the shape of
this result let $E$ and $F$ be number fields and $\chi$ an algebraic Hecke
character of $F$ with coefficients in $E$. For an embedding of fields
$\iota:E\rightarrow\mathbb{C}$ consider the $\mathbb{C}$-valued Hecke
$L$-function $L(s,{}^{\iota}\chi)$. Suppose $m\in\mathbb{Z}$ is such that both
$m$ and $m+1$ are critical for $L(s,{}^{\iota}\chi)$. This condition restricts
$F$ to be a totally imaginary field that contains a maximal CM subfield. Let
$\delta_{F/\mathbb{Q}}$ denote the absolute discriminant of $F$. Harder proved
([6]) that the complex number
$|\delta_{F/\mathbb{Q}}|^{1/2}\frac{L(m,{}^{\iota}\chi)}{L(m+1,{}^{\iota}\chi)}$
is algebraic (see (41)) and satisfies a reciprocity law that it is equivariant
under the Galois group ${\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ of
$\mathbb{Q}$ (see (42)); it follows then that the algebraicity statement can
be strengthened to assert that the above quantity is in $\iota(E)$ (see (43)).
The raison d’être for writing this article is that whereas the preliminary
algebraicity result (41) is correct, the reciprocity law (42) is not correct
as it stands by producing an example (see Sect. 6.2) when $F$ is a totally
imaginary field that is not of CM type that gives a counterexample to (43).
The real aim of this article is to state the correct reciprocity law which is
stated in Thm. 56. The missing signature, which is the term $\varepsilon_{{\bf
n},\iota}(\varsigma)\cdot\varepsilon_{\tilde{\bf n},\iota}(\varsigma)$ in Thm.
56, needed to rectify the reciprocity law is rather complicated, and turns out
to be trivial when $F$ is a CM field, and can be nontrivial for a general
totally imaginary field. The presence of this subtle signature can be finessed
using, interestingly enough, some other passages in [6]. Sect. 6 is an
exposition of some results in my paper [12] for Rankin–Selberg $L$-functions
for ${\rm GL}_{n}\times{\rm GL}_{n^{\prime}}$ over a totally imaginary base
field in the simple situation of $n=n^{\prime}=1.$

That there is a signature missing was suggested by calculations due to Pierre
Deligne (see Prop. 59), in the context of motivic $L$-functions, carried out
from the point of view afforded by his celebrated conjecture–which is recalled
in Conj. 60 in the context at hand. After relating the missing signature to
the signature predicted by Deligne, the main theorem is restated as a much
cleaner looking Thm. 70.

As a final comment to end the introduction, the results of this article show
that whereas the analytic theory of $L$-functions of automorphic forms on
${\rm GL}(1)$ is not sensitive to the inner structure of the base field, the
arithmetic theory of $L$-functions concerning the special values of
$L$-functions crucially depends on the arithmetic structure of the base field.

Acknowledgements: I thank Pierre Deligne, Günter Harder, and Haruzo Hida for
conversations and email correspondence on arithmetic aspects of Hecke
characters and special values of $L$-functions; some of these discussions have
found their way into this article. I acknowledge support from a MATRICS grant
MTR/2018/000918 of the Science and Engineering Research Board, Department of
Science and Technology, Government of India.

##  2\. Hecke characters

We set up some general notation for this article:

  1. $\mathbb{Q}$ is the rational number field, with closure $\bar{\mathbb{Q}}$ inside complex numbers $\mathbb{C}$;

  2. $F$ is a number field, i.e., a finite extension of $\mathbb{Q}$;
we do not assume that $F$ is contained in $\bar{\mathbb{Q}}\subset\mathbb{C}$;

  3. $d_{F}=d=[F:\mathbb{Q}]$;

  4. $\mathcal{O}_{F}$ is the ring of integers of $F$;

  5. $U_{F}=\mathcal{O}_{F}^{\times}$ is the group of units of $\mathcal{O}_{F}$;

  6. $\Sigma_{F}={\rm Hom}(F,\mathbb{C})$, all embeddings of $F$ into $\mathbb{C}$; note: ${\rm Hom}(F,\bar{\mathbb{Q}})={\rm Hom}(F,\mathbb{C});$

  7. $S_{\infty}$ = set of all archimedean places of $F$;

  8. $S_{\infty}=S_{r}\cup S_{c}$ the real and complex places, respectively;

  9. $r_{1}=\\#S_{r},$ $r_{2}=\\#S_{c}$; $d_{F}=r_{1}+2r_{2};$

  10. $\mathfrak{p}$ a finite prime ideal of $\mathcal{O}_{F}$ or an infinite place;

  11. $F_{\mathfrak{p}}$ the completion of $F$ at $\mathfrak{p};$

  12. $\mathcal{O}_{\mathfrak{p}}$ the ring of integers of $F_{\mathfrak{p}}$ at a finite prime $\mathfrak{p};$

  13. $\varpi_{\mathfrak{p}}$ is a uniformizer at a finite prime $\mathfrak{p};$ hence $\mathfrak{p}\mathcal{O}_{\mathfrak{p}}=\varpi_{\mathfrak{p}}\mathcal{O}_{\mathfrak{p}};$

  14. $U_{\mathfrak{p}}=\mathcal{O}_{\mathfrak{p}}^{\times}=\mathcal{O}_{\mathfrak{p}}-\varpi_{\mathfrak{p}}\mathcal{O}_{\mathfrak{p}}$ is the group of units of $\mathcal{O}_{\mathfrak{p}}$ for a finite prime $\mathfrak{p}.$

###  2.1. Class groups

We start with some preliminaries on class groups, towards which we need some
more notation:

  1. $\mathbb{J}_{F}$ is the group of all fractional ideals of $F;$

  2. $\mathbb{P}_{F}$ is the group of all principal fractional ideals $(x)=x\mathcal{O}_{F}$ for $x\in F^{\times};$

  3. $x\gg 0$ means that $x\in F$ is totally positive, i.e., $\rho(x)>0$ for all $\rho\in{\rm Hom}(F,\mathbb{R})$;

  4. $\mathbb{P}_{F}^{+}=\\{(x)\in\mathbb{P}_{F}\ :\ x\gg 0\\};$

  5. $\mathcal{C}l_{F}=\mathbb{J}_{F}/\mathbb{P}_{F}$ is the class group of $F$;

  6. $\mathcal{C}l_{F}^{+}=\mathbb{J}_{F}/\mathbb{P}_{F}^{+}$ is the narrow class group of $F$.

The adjective ‘narrow’ is to suggest narrow conditions in $\mathbb{P}_{F}^{+}$
that we divide by to get $\mathcal{C}l_{F}^{+}$; the narrow class group
surjects onto the class group; we have the exact sequence:

(1)
$0\longrightarrow\frac{\mathbb{P}_{F}}{\mathbb{P}_{F}^{+}}\longrightarrow\frac{\mathbb{J}_{F}}{\mathbb{P}_{F}^{+}}\longrightarrow\frac{\mathbb{J}_{F}}{\mathbb{P}_{F}}\longrightarrow
0.$

Observe that

$\frac{\mathbb{P}_{F}}{\mathbb{P}_{F}^{+}}\ \simeq\
\frac{F^{\times}/U_{F}}{F^{\times}_{+}/U_{F}^{+}}\ \simeq\
\frac{F^{\times}}{F^{\times}_{+}U_{F}},$

where $F^{\times}_{+}=\\{x\in F^{\times}:x\gg 0\\}$ and $U_{F}^{+}=U_{F}\cap
F^{\times}_{+}$ is the group of totally positive units. We have an exact
sequence:

(2)
$0\longrightarrow\frac{U_{F}}{U_{F}^{+}}\longrightarrow\frac{F^{\times}}{F^{\times}_{+}}\longrightarrow\frac{F^{\times}}{F^{\times}_{+}U_{F}}\longrightarrow
0.$

Splicing the two exact sequences, we get a four-term exact sequence relating
the narrow class group to the class group:

(3)
$0\longrightarrow\frac{U_{F}}{U_{F}^{+}}\longrightarrow\frac{F^{\times}}{F^{\times}_{+}}\longrightarrow\mathcal{C}l_{F}^{+}\longrightarrow\mathcal{C}l_{F}\longrightarrow
0.$

The narrow class group surjects onto the class group and the ratio of their
orders is a power of $2$ since $[F^{\times}:F^{\times}_{+}]=2^{r_{1}}.$ (This
suggests an interesting exercise in basic number theory: given
$r_{1},t_{1}\in\mathbb{Z},$ with $r_{1}\geq 1$ and $0\leq t_{1}\leq r_{1}$,
construct a number field $F$ with exactly $r_{1}$ real embeddings and such
that the order of the group $U_{F}/U_{F}^{+}$ is $2^{t_{1}},$ equivalently,
the kernel of $\mathcal{C}l_{F}^{+}\rightarrow\mathcal{C}l_{F}$ is
$2^{r_{1}-t_{1}}.$)

To discuss the class groups in terms of idèles, here are some more notations:

  1. $\mathbb{A}_{F}$ is the adèle ring of $F;$

  2. $\mathbb{I}_{F}$ is the group of idèles of $F$;

  3. $C_{F}=\mathbb{I}_{F}/F^{\times}$ is the group of idèle-classes of $F$;

  4. $U_{\mathfrak{p}}:=\mathcal{O}_{\mathfrak{p}}^{\times}$ for finite $\mathfrak{p}$ as before, but moreover
$U_{\mathfrak{p}}:=\mathbb{R}^{\times}_{+}$ for $\mathfrak{p}\in S_{r},$ and
$U_{\mathfrak{p}}:=\mathbb{C}^{\times}$ for $\mathfrak{p}\in S_{c};$

  5. $\mathbb{I}_{F}^{S}$ = group of $S$-idèles for any finite set $S$ of places;

  6. $F_{\infty}\ :=\ F\otimes\mathbb{R}\ \simeq\ \prod_{v\in S_{r}}F_{v}\times\prod_{w\in S_{c}}F_{w}\ \simeq\ \prod_{v\in S_{r}}\mathbb{R}\times\prod_{w\in S_{c}}\mathbb{C};$

  7. $F_{\infty+}=\\{x=(x_{\lambda})\in F_{\infty}\ :\ x_{v}>0,\ \forall v\in S_{r}\\}.$

There is a canonical map
$\bm{\upiota}:\mathbb{I}_{F}\rightarrow\mathbb{J}_{F}$ which sends an idèle
$a=(a_{\mathfrak{p}})$ to the ideal
$\bm{\upiota}(a):=\prod_{\mathfrak{p}\notin S_{\infty}}\mathfrak{p}^{{\rm
ord}_{\mathfrak{p}}(a_{\mathfrak{p}})}.$ It is clear that ${\rm
Kernel}(\bm{\upiota})=F_{\infty}^{\times}\prod_{\mathfrak{p}\notin
S_{\infty}}U_{\mathfrak{p}}.$ The map $\bm{\upiota}$ induces an isomorphism
which gives an idèle-theoretic description of the class group of $F$:

(4)
$\frac{\mathbb{I}_{F}}{F^{\times}(F_{\infty}^{\times}\prod_{\mathfrak{p}\notin
S_{\infty}}U_{\mathfrak{p}})}\ \simeq\ \frac{\mathbb{J}_{F}}{\mathbb{P}_{F}}\
=\ \mathcal{C}l_{F}.$

Similarly, one has for the narrow class group:

$\frac{\mathbb{I}_{F}}{F_{+}^{\times}(F_{\infty}^{\times}\prod_{\mathfrak{p}\notin
S_{\infty}}U_{\mathfrak{p}})}\ \simeq\
\frac{\mathbb{J}_{F}}{\mathbb{P}_{F}^{+}}\ =\ \mathcal{C}l_{F}^{+}.$

Weak-approximation gives that $F^{\times}$ is dense in $F_{\infty}^{\times}$;
hence $F_{+}^{\times}F_{\infty}^{\times}=F^{\times}F_{\infty+}^{\times};$
whence:

(5)
$\frac{\mathbb{I}_{F}}{F^{\times}(F_{\infty+}^{\times}\prod_{\mathfrak{p}\notin
S_{\infty}}U_{\mathfrak{p}})}\ \simeq\
\frac{\mathbb{J}_{F}}{\mathbb{P}_{F}^{+}}\ =\ \mathcal{C}l_{F}^{+}.$

The four-term long exact sequence relating the class group and the narrow
class group described using idèles takes the form:

(6)
$0\longrightarrow\frac{U_{F}}{U_{F}^{+}}\longrightarrow\frac{F_{\infty}^{\times}}{F_{\infty+}^{\times}}\longrightarrow\frac{\mathbb{I}_{F}}{F^{\times}(F_{\infty+}^{\times}\prod_{\mathfrak{p}\notin
S_{\infty}}U_{\mathfrak{p}})}\longrightarrow\frac{\mathbb{I}_{F}}{F^{\times}(F_{\infty}^{\times}\prod_{\mathfrak{p}\notin
S_{\infty}}U_{\mathfrak{p}})}\longrightarrow 0.$

To compare the second term from the left in (3) with that in (6) we note that
weak-approximation says that the canonical map
$F^{\times}/F_{+}^{\times}\rightarrow
F_{\infty}^{\times}/F^{\times}_{\infty+}$ induced by the diagonal inclusion
$F^{\times}\hookrightarrow F_{\infty}^{\times}$ is an isomorphism.

###  2.2. Basics of Hecke characters

A Hecke character is a continuous homomorphism

$\chi\ :\ \mathbb{I}_{F}/F^{\times}\ \rightarrow\ \mathbb{C}^{\times}.$

We do not ask $\chi$ to be unitary. Some people prefer calling a unitary Hecke
character as a Hecke character, and what we call a Hecke character, as a Hecke
quasi-character. It is painful to keep writing ‘quasi-character’ all the time,
and so, with a possible abuse of terminology, we will work with the above
definition. This is not so serious as we now explain: Consider the norm of an
idèle $\alpha\in\mathbb{I}_{F}$ defined as
$|\\!|\alpha|\\!|:=\prod_{\mathfrak{p}}|\alpha_{\mathfrak{p}}|_{\mathfrak{p}}$,
where $\mathfrak{p}$ runs over all valuations (finite and archimedean), and
each valuation is normalized. Then $|\\!|\
|\\!|:\mathbb{I}_{F}\rightarrow\mathbb{R}^{\times}_{+}$ is a surjective
homomorphism and let ${}^{0}\mathbb{I}_{F}$ denote its kernel. The product
formula says that $|\\!|a|\\!|=1$ for all $a\in F^{\times}$, i.e.,
$F^{\times}\subset{}^{0}\mathbb{I}_{F}.$ We have the following exact sequence:

$0\longrightarrow\frac{{}^{0}\mathbb{I}_{F}}{F^{\times}}\longrightarrow\frac{\mathbb{I}_{F}}{F^{\times}}\longrightarrow\mathbb{R}^{\times}_{+}\longrightarrow
0.$

This sequence splits. There are several splittings; for example: map
$t\in\mathbb{R}^{\times}_{+}$ into the idèle which is $t$ at a particular real
infinite place and $1$ elsewhere, or map it to $t^{1/2}$ at a particular
complex infinite place and $1$ elsewhere. But this depends on a choice of an
infinite place. Instead, using $d=d_{F}=[F:\mathbb{Q}]$, we will always
consider the following splitting:

$t\mapsto(t^{1/d},\dots,t^{1/d};1,1,\dots),$

where the idèle has the positive $d$-th root of $t$ at every infinite place
and $1$ at every finite place. (Note that it is indeed a splitting, because
for any complex place, the normalized valuation satisfies
$|x|_{\mathbb{C}}=|x|_{\mathbb{R}}^{2}$ for $x\in\mathbb{R}.$) This splitting
gives

$\mathbb{I}_{F}/F^{\times}\ \simeq\
{}^{0}\mathbb{I}_{F}/F^{\times}\times\mathbb{R}^{\times}_{+}.$

It is a fundamental fact that ${}^{0}\mathbb{I}_{F}/F^{\times}$ is compact
(Neukirch [10, Theorem VI.1.6]). A continuous homomorphism of
${}^{0}\mathbb{I}_{F}/F^{\times}$ into $\mathbb{C}^{\times}$ has compact image
and so lands in $S^{1}.$ Further, any homomorphism
$\mathbb{R}^{\times}_{+}\rightarrow\mathbb{C}^{\times}$ is of the form
$x\mapsto|x|^{w}$ for a complex number $w=\sigma+i\varphi$. (See 2.5.1 below.)
Putting these remarks together, any Hecke character $\chi$ can be uniquely
factored as

(7) $\chi\ =\ {}^{0}\chi\otimes|\\!|\ |\\!|^{\sigma}$

for a unitary Hecke character ${}^{0}\chi:\mathbb{I}_{F}/F^{\times}\rightarrow
S^{1}$ and $\sigma\in\mathbb{R};$ sometimes ${}^{0}\chi$ is denoted also as
$\chi^{u}.$

###  2.3. Dirichlet characters: Hecke characters of finite order

A continuous homomorphism
$\chi:\mathbb{I}_{F}/F^{\times}\rightarrow\mathbb{C}^{\times}$ with finite
image and which is unramified everywhere (i.e.,
$\chi_{\mathfrak{p}}:=\chi|_{F_{\mathfrak{p}}^{\times}}$ is trivial on the
units $U_{\mathfrak{p}}$ for all finite $\mathfrak{p}$) gives a character of
$\frac{\mathbb{I}_{F}}{F^{\times}(F_{\infty+}^{\times}\prod_{\mathfrak{p}\notin
S_{\infty}}U_{\mathfrak{p}})},$ and by (5), is a character of the narrow class
group $\mathcal{C}l_{F}^{+}.$ This generalizes by introducing some level
structure giving us a description of Hecke characters of finite order in terms
of characters of narrow ray class groups with level structure. Here is some
more notation:

  1. $\mathfrak{m}$ will be an integral ideal with prime-factorization $\mathfrak{m}=\prod_{\mathfrak{p}\notin S_{\infty}}\mathfrak{p}^{m_{\mathfrak{p}}};$

  2. $U_{\mathfrak{p}}(m_{\mathfrak{p}})$ will be defined as:

$U_{\mathfrak{p}}(m_{\mathfrak{p}})=\left\\{\begin{array}[]{ll}1+\mathfrak{p}^{m_{\mathfrak{p}}},&{\rm
if}\ \mathfrak{p}\notin S_{\infty},\ \mathfrak{p}\,|\,\mathfrak{m},\\\
U_{\mathfrak{p}},&{\rm if}\ \mathfrak{p}\notin S_{\infty},\
\mathfrak{p}\not|\,\mathfrak{m},\\\ \mathbb{R}^{\times}_{+},&{\rm if}\
\mathfrak{p}\in S_{r},\\\ \mathbb{C}^{\times},&{\rm if}\ \mathfrak{p}\in
S_{c}.\end{array}\right.$

  3. $\mathbb{U}_{F}(\mathfrak{m}):=\prod_{\mathfrak{p}}U_{\mathfrak{p}}(m_{\mathfrak{p}});$

  4. $\mathbb{U}_{F,f}(\mathfrak{m}):=\prod_{\mathfrak{p}\notin S_{\infty}}U_{\mathfrak{p}}(m_{\mathfrak{p}});$

  5. $C_{F}(\mathfrak{m}):=\mathbb{U}_{F}(\mathfrak{m})F^{\times}/F^{\times}$ is called the congruence subgroup mod $\mathfrak{m}$ in $C_{F};$

  6. $C_{F}/C_{F}(\mathfrak{m}):=\mathbb{I}_{F}/\mathbb{U}_{F}(\mathfrak{m})F^{\times}$ is the idèle-theoretic narrow ray class group mod $\mathfrak{m}$;

  7. $\mathbb{J}_{F}(\mathfrak{m})$ is the group of all fractional ideals relatively prime to $\mathfrak{m};$

  8. $\mathbb{P}_{F}(\mathfrak{m})$ is the group of all principal fractional ideals $(x)$ with $x\equiv 1\pmod{\mathfrak{m}}.$

  9. $\mathbb{P}_{F}^{+}(\mathfrak{m})$ is the subgroup of $\mathbb{P}_{F}(\mathfrak{m})$ consisting of those $(x)$ with $x\gg 0;$

  10. $\mathcal{C}l_{F}(\mathfrak{m}):=\mathbb{J}_{F}(\mathfrak{m})/\mathbb{P}_{F}(\mathfrak{m})$ is the ray class group mod $\mathfrak{m}$;

  11. $\mathcal{C}l_{F}^{+}(\mathfrak{m}):=\mathbb{J}_{F}(\mathfrak{m})/\mathbb{P}_{F}^{+}(\mathfrak{m})$ is the narrow ray class group mod $\mathfrak{m}$.

######  Proposition 8.

The canonical homomorphism $w:\mathbb{I}_{F}\rightarrow\mathbb{J}_{F}$ induces
an isomorphism:

$\frac{C_{F}}{C_{F}(\mathfrak{m})}\ \simeq\
\frac{\mathbb{J}_{F}(\mathfrak{m})}{\mathbb{P}_{F}^{+}(\mathfrak{m})}.$

###### Proof.

Define:

$\mathbb{I}_{F}(\mathfrak{m})\ :=\ \\{\alpha\in\mathbb{I}_{F}\ :\
\alpha_{\mathfrak{p}}\in U_{\mathfrak{p}}(m_{\mathfrak{p}}),\
\forall\mathfrak{p}|\mathfrak{m}\cdot\infty\\}.$

Weak approximation implies:
$\mathbb{I}_{F}=F^{\times}\cdot\mathbb{I}_{F}(\mathfrak{m}).$ Hence

$C_{F}=\mathbb{I}_{F}/F^{\times}=\mathbb{I}_{F}(\mathfrak{m})/(F^{\times}\cap\mathbb{I}_{F}(\mathfrak{m})).$

Note that $\bm{\upiota}$ maps $\mathbb{I}_{F}(\mathfrak{m})$ onto
$\mathbb{J}_{F}(\mathfrak{m})$ and similarly, maps
$F^{\times}\cap\mathbb{I}_{F}(\mathfrak{m})$ onto
$\mathbb{P}_{F}^{+}(\mathfrak{m}).$ Further, ${\rm
Ker}\left(\bm{\upiota}:\mathbb{I}_{F}(\mathfrak{m})\rightarrow\mathbb{J}_{F}(\mathfrak{m})\right)\
=\ \mathbb{U}_{F}(\mathfrak{m})$. The rest is easy. ∎

A fundamental property of these congruence subgroups $C_{F}(\mathfrak{m})$ is
captured by

######  Proposition 9.

For any integral ideal $\mathfrak{m}$, the congruence subgroup
$C_{F}(\mathfrak{m})$ is a subgroup of the idèle class group $C_{F}$ of finite
index. Conversely, any subgroup of finite index of $C_{F}$ contains
$C_{F}(\mathfrak{m})$ for some $\mathfrak{m}.$

###### Proof.

See Neukirch [10, Proposition VI.1.8]. ∎

The moral of the above discussion is that Hecke characters of finite order are
exactly the characters of narrow ray class groups:

######  Corollary 10 (Finite order Hecke characters).

Consider a continuous homomorphism

$\chi\ :\ \mathbb{I}_{F}/F^{\times}\ \longrightarrow\ \mathbb{C}^{\times}$

whose image is of finite order. Then there is an integral ideal $\mathfrak{m}$
such that $\chi$ factors as

---
$\textstyle{\mathbb{I}_{F}/F^{\times}=C_{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi}$$\textstyle{\mathbb{C}^{\times}}$$\textstyle{C_{F}/C_{F}(\mathfrak{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

The smallest such $\mathfrak{m}$ is called the conductor of $\chi$ and will be
denoted $\mathfrak{f}_{\chi}.$

A Dirichlet character of $F$ is a (necessarily unitary) character of a narrow
ray class group of $F$ with level structure, i.e., it is a homomorphism

$\chi:\mathbb{J}_{F}(\mathfrak{m})/\mathbb{P}_{F}(\mathfrak{m})^{+}\
\rightarrow\ S^{1},$

for some integral ideal $\mathfrak{m}.$ By Prop. 8, Prop. 9, and Cor. 10 it
follows that Dirichlet characters are exactly the Hecke characters of finite
order. The algebraic Hecke characters that we want to understand are in
general not of finite order.

An easy exercise is to take $F=\mathbb{Q}$, $\mathfrak{m}=N\mathbb{Z}$ to see
that a Dirichlet character for $\mathbb{Q}$ modulo $N\mathbb{Z}$ is exactly a
homomorphism
$(\mathbb{Z}/N\mathbb{Z})^{\times}\rightarrow\mathbb{C}^{\times};$ such
characters were considered by Dirichlet in his proof of infinitude of primes
in arithmetic progressions; this exercise justifies the terminology.

###  2.4. Local components at finite places and conductor

Fix any place $\mathfrak{p}$ of $F$. Then $F_{\mathfrak{p}}^{\times}$ embeds
into $\mathbb{I}_{F}$ as
$x_{\mathfrak{p}}\mapsto(1,\dots,1,x_{\mathfrak{p}},1\dots),$ the idele having
$x_{\mathfrak{p}}$ at the place corresponding to $\mathfrak{p}$ and $1$
elsewhere. The restriction $\chi_{\mathfrak{p}}$ of $\chi$ to this copy of
$F_{\mathfrak{p}}^{\times}$ is called the local character at $\mathfrak{p}$.
Sometimes we use $v$ for a place instead of $\mathfrak{p}.$ For almost all
finite $\mathfrak{p}$, the character $\chi_{\mathfrak{p}}$ is unramified,
i.e., it is trivial on $U_{\mathfrak{p}}$. This may be seen by considering the
restriction of $\chi$ to $\prod_{\mathfrak{p}\notin
S_{\infty}}U_{\mathfrak{p}}$, a compact and totally disconnected group the
image of which under a continuous homomorphism into $\mathbb{C}^{\times}$ must
be finite by the so-called ‘no small-subgroups argument’; see, for example,
Bump [2, Exercise 3.1.1]. One can write $\chi=\otimes^{\prime}_{v}\chi_{v}$ as
a restricted tensor product. By the same token $\chi$ is trivial on
$\mathbb{U}_{F,f}(\mathfrak{m})$ for some integral ideal $\mathfrak{m}.$ The
smallest ideal $\mathfrak{m}$ such that $\chi$ is trivial on
$\mathbb{U}_{F,f}(\mathfrak{m})$ is called conductor of $\chi$ and will be
denoted $\mathfrak{f}_{\chi}.$

###  2.5. The character at infinity of a Hecke character

####  2.5.1. Characters of $\mathbb{R}^{\times}$

Any continuous homomorphism
$\chi:\mathbb{R}^{\times}\rightarrow\mathbb{C}^{\times}$ is of the form

$\chi(x)\ =\ {\rm sgn}(x)^{n}\,|x|^{w}\ =\
\left(\frac{x}{|x|}\right)^{n}\,|x|^{w},$

with $n\in\\{0,1\\}$ and $w\in\mathbb{C}.$ Such a character $\chi$ is unitary
if and only if $w=\mathfrak{i}\varphi\in\mathfrak{i}\mathbb{R}.$

####  2.5.2. Characters of $\mathbb{C}^{\times}$

For $z=x+\mathfrak{i}y\in\mathbb{C}$, define
$|z|:=|z|_{\mathbb{R}}:=\sqrt{x^{2}+y^{2}},$ and the normalised valuation on
$\mathbb{C}$ as $|z|_{\mathbb{C}}:=|z|_{\mathbb{R}}^{2}=x^{2}+y^{2}.$ A
continuous homomorphism
$\chi:\mathbb{C}^{\times}\rightarrow\mathbb{C}^{\times}$ is of the form

$\chi(z)\ =\ \left(\frac{z}{|z|}\right)^{n}\,|z|_{\mathbb{C}}^{w},$

with $n\in\mathbb{Z}$ and $w\in\mathbb{C}.$ If $z=re^{\mathfrak{i}\theta}$
with $r\in\mathbb{R}^{\times}_{+}$ and $\theta\in\mathbb{R}$, then
$\chi(z)=r^{2w}e^{\mathfrak{i}n\theta}.$ Such a character $\chi$ is unitary if
and only if $w=\mathfrak{i}\varphi\in\mathfrak{i}\mathbb{R}.$

####  2.5.3. Description of $\chi_{\infty}$

For a Hecke character $\chi$, let
$\chi_{\infty}=\chi\,|_{F_{\infty}^{\times}}$ where
$F_{\infty}^{\times}\hookrightarrow\mathbb{I}_{F}.$ To describe this character
at infinity explicitly, we introduce some more notation:

  1. $\lambda$ is any infinite place; $\lambda\in S_{\infty};$

  2. $v$ is any real place; $v\in S_{r}$ and $F_{v}\simeq\mathbb{R}$ canonically;

  3. $w$ is any complex place; $w\in S_{c}$ and $F_{w}\simeq\mathbb{C}$ non-canonically;

  4. $|x_{\infty}|_{\infty}=\prod_{\lambda}|x_{\lambda}|_{\lambda}$ for $x_{\infty}\in F_{\infty}$ is the product of normalized valuations.

Keeping in mind that a Hecke character factorizes as $\chi={}^{0}\chi\otimes|\
|^{\sigma}$ (see (7)), one can write the character at infinity $\chi_{\infty}$
on $x_{\infty}\in F_{\infty}^{\times}$ as

(11) $\chi_{\infty}(x_{\infty})=\left(\prod_{\lambda\in
S_{\infty}}\left(\frac{x_{\lambda}}{|x_{\lambda}|}\right)^{n_{\lambda}}|x_{\lambda}|_{\lambda}^{i\varphi_{\lambda}}\right)\,|x_{\infty}|_{\infty}^{\sigma}.$

where $n_{v}\in\\{0,1\\}$, $n_{w}\in\mathbb{Z}$,
$\varphi_{\lambda}\in\mathbb{R},$ and $\sigma\in\mathbb{R}$.

##  3\. Größencharaktere mod $\mathfrak{m}$

###  3.1. Definition

We follow the treatment in Neukirch [10, Section VII.6], except that for us
these characters can take values in $\mathbb{C}^{\times}$, i.e., they need not
be unitary. Before getting started, some further notation that will be useful:

  1. $U_{F}^{1}(\mathfrak{m}):=\\{u\in U_{F}\ |\ u\equiv 1\pmod{\mathfrak{m}}\\}$ are the units congruent to $1$ mod $\mathfrak{m}$;

  2. $U_{F}^{1}(\mathfrak{m})^{+}:=\\{u\in U_{F}^{1}(\mathfrak{m})\ |\ u\gg 0\\};$

  3. $\mathcal{O}_{F}(\mathfrak{m}):=\\{a\in\mathcal{O}_{F}\ |\ (a,\mathfrak{m})=1\\};$ nonzero integers in $F$ relatively prime to $\mathfrak{m};$

  4. $F(\mathfrak{m}):=\\{x\in F\ |\ (x,\mathfrak{m})=1\\};$ nonzero elements of $F$ relatively prime to $\mathfrak{m};$

  5. $F^{1}(\mathfrak{m}):=\\{x\in F\ |\ x\equiv 1\pmod{\mathfrak{m}}\\};$ elements of $F$ congruent to $1$ mod $\mathfrak{m}$.

By a Größencharakter mod $\mathfrak{m}$ we mean a homomorphism
$\psi:\mathbb{J}_{F}(\mathfrak{m})\rightarrow\mathbb{C}^{\times}$ for which
there exists a pair of characters $(\psi_{f},\psi_{\infty})$ with

$\psi_{f}:(\mathcal{O}_{F}/\mathfrak{m})^{\times}\rightarrow\mathbb{C}^{\times},\
\ {\rm and}\ \
\psi_{\infty}:F_{\infty}^{\times}\rightarrow\mathbb{C}^{\times},$

such that for all $a\in\mathcal{O}_{F}(\mathfrak{m})$, we have

(12) $\psi((a))\ =\ \psi_{f}(a\ {\rm mod}\ \mathfrak{m})\,\psi_{\infty}(a),$

where in the left hand side $(a):=a\mathcal{O}_{F}$ is the principal ideal
generated by $a$. We call $\psi_{f}$ the finite part and $\psi_{\infty}$ the
infinite part of $\psi$; they satisfy the compatibility condition: if $u\in
U_{F}=\mathcal{O}_{F}^{\times}$ is a unit then $\psi((u))=1$ and so we
require:

$\psi_{f}(u\ {\rm mod}\ \mathfrak{m})\,\psi_{\infty}(u)=1.$

Further, if $\epsilon\in U_{F}^{1}(\mathfrak{m}):=\\{u\in U_{F}\ |\ u\equiv 1\
{\rm mod}\ \mathfrak{m}\\}$ then the above requirement implies that
$\psi_{\infty}(\epsilon)=1.$ The following proposition explains the relation
between $\psi$ and $(\psi_{f},\psi_{\infty})$.

######  Proposition 13.

A Größencharakter $\psi$ mod $\mathfrak{m}$ uniquely determines $\psi_{f}$ and
$\psi_{\infty}$; indeed, the restriction of $\psi$ to the group
$\mathbb{P}_{F}(\mathfrak{m})$ uniquely determines $\psi_{f}$ and
$\psi_{\infty},$ and these satisfy the above compatibility conditions.

Conversely, given homomorphisms
$\psi_{f}^{\prime}:(\mathcal{O}_{F}/\mathfrak{m})^{\times}\rightarrow\mathbb{C}^{\times}$
and $\psi_{\infty}^{\prime}:F_{\infty}^{\times}\rightarrow\mathbb{C}^{\times}$
satisfying the compatibility condition $\psi_{f}^{\prime}(u\ {\rm mod}\
\mathfrak{m})\,\psi_{\infty}^{\prime}(u)=1$ for all $u\in U_{F}$ (and so
necessarily $\psi_{\infty}^{\prime}(\epsilon)=1$ for all $\epsilon\in
U_{F}^{1}(\mathfrak{m})$), there exists a Größencharakter $\psi$ mod
$\mathfrak{m}$ such that $\psi_{f}=\psi_{f}^{\prime}$ and
$\psi_{\infty}=\psi_{\infty}^{\prime}.$

Finally, if $\xi$ is another Größencharakter mod $\mathfrak{m}$ such that
$\xi_{f}=\psi_{f}$ and $\xi_{\infty}=\psi_{\infty}$, then $\psi\xi^{-1}$ is a
character of $\mathcal{C}l_{F}(\mathfrak{m})$–the ray class group mod
$\mathfrak{m}.$

###### Proof.

Using $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
16.36531pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{F^{1}(\mathfrak{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
40.36531pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern-3.0pt\lower
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
40.36531pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbb{P}_{F}(\mathfrak{m})}$}}}}}}}\ignorespaces}}}}\ignorespaces\subset\mathbb{J}_{F}(\mathfrak{m})$
we see that a Größencharakter $\psi$ mod $\mathfrak{m}$ determines via
restriction a character of $F^{1}(\mathfrak{m}).$ By weak approximation,
$F^{1}(\mathfrak{m})$ is dense in $F_{\infty}^{\times}$; thus $\psi$
restricted to $\mathbb{P}_{F}(\mathfrak{m})$ uniquely determines
$\psi_{\infty}.$ Clearly, $\psi$ and $\psi_{\infty}$ uniquely determine
$\psi_{f}$ by (12). For the converse, given $\psi_{f}^{\prime}$ and
$\psi_{\infty}^{\prime}$, define $\tilde{\psi}^{\prime}$ on
$\mathcal{O}_{F}(\mathfrak{m})$ by

$\tilde{\psi}^{\prime}((a))\ :=\ \psi_{f}^{\prime}(a\ {\rm mod}\
\mathfrak{m})\,\psi_{\infty}^{\prime}(a).$

The compatibility conditions say that $\tilde{\psi}^{\prime}(a)$ is well-
defined. This $\tilde{\psi}^{\prime}$ extends uniquely to $F(\mathfrak{m})$,
and being trivial on $U_{F}^{1}(\mathfrak{m}),$ we get a character
$\psi^{\prime}:\mathbb{P}_{F}(\mathfrak{m})\rightarrow\mathbb{C}^{\times}$ as
$\psi^{\prime}((a))=\tilde{\psi}^{\prime}(a).$ Now induce $\psi^{\prime}$ to
$\mathbb{J}_{F}(\mathfrak{m})$, i.e., consider the induced representation

${\rm
Ind}_{\mathbb{P}_{F}(\mathfrak{m})}^{\mathbb{J}_{F}(\mathfrak{m})}(\psi^{\prime}).$

An easy exercise with Mackey theory says that this representation is a
multiplicity free direct sum of characters $\psi$ of
$\mathbb{J}_{F}(\mathfrak{m})$ such that
$\psi|_{\mathbb{P}_{F}(\mathfrak{m})}=\psi^{\prime}$. Moreover, fix such a
$\psi$, then

${\rm
Ind}_{\mathbb{P}_{F}(\mathfrak{m})}^{\mathbb{J}_{F}(\mathfrak{m})}(\psi^{\prime})\
=\ \psi\otimes{\rm
Ind}_{\mathbb{P}_{F}(\mathfrak{m})}^{\mathbb{J}_{F}(\mathfrak{m})}(1\\!\\!1)\
=\ \bigoplus_{\chi}\psi\otimes\chi$

with $\chi$ running over the characters of
$\mathbb{J}_{F}(\mathfrak{m})/\mathbb{P}_{F}(\mathfrak{m})=\mathcal{C}l_{F}(\mathfrak{m})$
the ray class group modulo $\mathfrak{m}$. Hence, if $\xi$ is a
Größencharakter with same finite and infinite parts as $\psi$ then
$\xi=\psi\otimes\chi.$ ∎

Following the German language, the plural of Größencharakter will be
Größencharaktere.

###  3.2. The correspondence between Hecke characters and Größencharaktere

The domain of definition of a Größencharakter surjects onto the domain of
definition of a Hecke character as in the following exact sequence:

######  Proposition 14.

We have an exact sequence

$1\ \ \longrightarrow\ \frac{F(\mathfrak{m})}{U_{F}^{1}(\mathfrak{m})}\ \
\stackrel{{\scriptstyle\varkappa}}{{\longrightarrow}}\ \
\mathbb{J}_{F}(\mathfrak{m})\times(\mathcal{O}_{F}/\mathfrak{m})^{\times}\times\frac{F_{\infty}^{\times}}{U_{F}^{1}(\mathfrak{m})}\
\ \stackrel{{\scriptstyle\varrho}}{{\longrightarrow}}\ \
\frac{\mathbb{I}_{F}}{F^{\times}\cdot\mathbb{U}_{F,f}(\mathfrak{m})}\ \
\longrightarrow\ \ 1,$

where

  1. (1)

$\varkappa(a)=((a)^{-1},\ a\ {\rm mod}\ \mathfrak{m},\ a\ {\rm mod}\
U_{F}^{1}(\mathfrak{m}))$ for all $a\in F(\mathfrak{m})$;

  2. (2)

$\varrho=\alpha\otimes\beta\otimes\gamma^{-1}$ where

    1. (a)

$\alpha:\mathbb{J}_{F}(\mathfrak{m})\rightarrow\mathbb{I}_{F}/(F^{\times}\cdot\mathbb{U}_{F,f}(\mathfrak{m}))$
is induced by the map $\mathbb{J}_{F}\rightarrow\mathbb{I}_{F}$ that sends a
prime ideal $\mathfrak{q}$ to the idèle that has $\varpi_{\mathfrak{q}}$ at
the place corresponding to $\mathfrak{q}$ and $1$’s elsewhere;

    2. (b)

$\beta:(\mathcal{O}_{F}/\mathfrak{m})^{\times}\rightarrow\mathbb{I}_{F}/(F^{\times}\cdot\mathbb{U}_{F,f}(\mathfrak{m}))$
is induced by the map that sends $a\in\mathcal{O}_{F}(\mathfrak{m})$ to the
idèle that has $a$ at all infinite places and also at all finite places
$\mathfrak{p}$ not dividing $\mathfrak{m}$, and is $1$ at all
$\mathfrak{p}|\mathfrak{m}$; and

    3. (c)

$\gamma:F_{\infty}^{\times}/U_{F}^{1}(\mathfrak{m})\rightarrow\mathbb{I}_{F}/(F^{\times}\cdot\mathbb{U}_{F,f}(\mathfrak{m}))$
is induced by the inclusion $F_{\infty}^{\times}\subset\mathbb{I}_{F}$; check
that $U_{F}^{1}(\mathfrak{m})$ is mapped into
$F^{\times}\cdot\mathbb{U}_{F,f}(\mathfrak{m}).$

See Neukirch [10, Prop. VII.6.13] for a proof but it is an instructive
exercise to fix a proof for oneself. Let us denote an $a\in\mathbb{I}_{F}$ as
$(a_{\infty};\dots a_{\mathfrak{p}}\dots;\dots a_{\mathfrak{q}}\dots)$ for
$\mathfrak{p}|\mathfrak{m}$ and $\mathfrak{q}\\!\not|\,\mathfrak{m}$, and
$[a]$ the class of the idèle $a$ in
$\mathbb{I}_{F}/(F^{\times}\cdot\mathbb{U}_{F,f}(\mathfrak{m})).$ Note that:

  1. (1)

$\alpha(\mathfrak{q})=[(\dots 1\dots;\ \dots 1\dots;\ 1,\dots
1,\varpi_{\mathfrak{q}},1,\dots)]$ for any prime ideal
$\mathfrak{q}\\!\not|\,\mathfrak{m}$.

  2. (2)

$\beta(a)=[((\dots 1\dots;\ a^{-1},\dots,a^{-1};\ \dots 1\dots)].$

  3. (3)

$\gamma(x_{\infty})=[(x_{\infty};\ \dots 1\dots;\ \dots 1\dots)].$

The rest of the details are left to the reader.

A Hecke character $\chi$ trivial on $\mathbb{U}_{F,f}(\mathfrak{m})$ will be
called a Hecke character mod $\mathfrak{m}$; necessarily the conductor of
$\chi$ divides $\mathfrak{m}.$ From the exact sequence, the dictionary between
Größencharaktere mod $\mathfrak{m}$ and Hecke characters mod $\mathfrak{m}$ is
clear: Given a Größencharakter $\psi$ mod $\mathfrak{m}$, with $\psi_{f}$ and
$\psi_{\infty}$ its finite and infinite parts, we get a character
$\psi\otimes\psi_{f}\otimes\psi_{\infty}$ on
$\mathbb{J}_{F}(\mathfrak{m})\times(\mathcal{O}_{F}/\mathfrak{m})^{\times}\times(F_{\infty}^{\times}/U_{F}^{1}(\mathfrak{m}))$
and (12) says that this is trivial on the image of $\varkappa$ and hence we
get a character $\chi=\chi_{\psi}$ on
$\mathbb{I}_{F}/(F^{\times}\cdot\mathbb{U}_{F,f}(\mathfrak{m})),$ i.e., $\chi$
is a Hecke character mod $\mathfrak{m}$. Conversely, given such a $\chi$, we
inflate it via $\varrho$ to get
$\psi\otimes\psi_{f}\otimes\psi_{\infty}=\chi\circ\varrho.$

###  3.3. The infinity type of a Größencharakter

Given a Größencharakter $\psi=(\psi_{f},\psi_{\infty})$ mod $\mathfrak{m}$, by
using the above dictionary and (11), one sees that the character at infinity,
which is a continuous character
$\psi_{\infty}:F_{\infty}^{\times}\rightarrow\mathbb{C}^{\times},$ is of the
form:

$\psi_{\infty}(x_{\infty})=\left(\prod_{\lambda\in
S_{\infty}}\left(\frac{x_{\lambda}}{|x_{\lambda}|}\right)^{n_{\lambda}}|x_{\lambda}|_{\lambda}^{i\varphi_{\lambda}}\right)\,|x_{\infty}|_{\infty}^{\sigma},$

where $n_{v}\in\\{0,1\\}$, $n_{w}\in\mathbb{Z}$,
$\varphi_{\lambda}\in\mathbb{R}$ and $\sigma\in\mathbb{R}$; furthermore,
$\psi_{\infty}$ being trivial on $U_{F}^{1}(\mathfrak{m})$ imposes certain
restrictions on the infinity type, i.e., the collection
$((n_{\lambda},\varphi_{\lambda})_{\lambda},\sigma)$ of exponents, that can
occur.

######  Lemma 15.

Let $\psi$ be a Größencharakter mod $\mathfrak{m}$ with the above notations
for $\psi_{\infty}$. Then:

  1. (i)

there are no restrictions on $n_{\lambda}$ for any $\lambda\in S_{\infty}$,
and on $\sigma$; however,

  2. (ii)

$2\varphi_{v}=\varphi_{w}=\varphi$ (say), for all $v\in S_{r}$ and all $w\in
S_{c}$.

Hence, the character at infinity takes the shape:

$\psi_{\infty}(x_{\infty})=\left(\prod_{\lambda\in
S_{\infty}}\left(\frac{x_{\lambda}}{|x_{\lambda}|}\right)^{n_{\lambda}}\right)\,|x_{\infty}|_{\infty}^{\sigma+i\varphi}.$

We adumbrate how one proves this lemma. Write the character at infinity as:

$\psi_{\infty}(x_{\infty})\ =\ \prod_{v\in S_{r}}{\rm
sgn}(x_{v})^{n_{v}}|x_{v}|^{\sigma+i\varphi_{v}}\cdot\prod_{w\in
S_{c}}x_{w}^{\sigma+n_{w}/2+i\varphi_{w}/2}\,\bar{x}_{w}^{\,\sigma-
n_{w}/2+i\varphi_{w}/2}$

For brevity, denote $f_{v}:=\sigma+i\varphi_{v},\
f_{w}:=\sigma+n_{w}/2+i\varphi_{w}/2,\ f_{\bar{w}}:=\sigma-
n_{w}/2+i\varphi_{w}/2.$ As $\psi_{\infty}$ is trivial on any $u\in
U_{F}^{1}(\mathfrak{m})^{+}$ (totally positive global units that are $1$ mod
$\mathfrak{m}$) we get

$\prod_{v\in S_{r}}v(u)^{f_{v}}\prod_{w\in
S_{c}}w(u)^{f_{w}}\bar{w}(u)^{f_{\bar{w}}}=1.$

We will go through the rest of the proof, which uses the ingredients of the
proof in Dirichlet’s unit theorem, in detail in the proof of Lem. 17 below
(the main focus of this article is the notion of an algebraic Hecke character
to which that lemma particularly applies) giving us the constraint:
$2f_{v}=f_{w}$, from which the above lemma follows.

##  4\. Algebraic Hecke characters

###  4.1. Definition and purity

There is a canonical map $\Sigma_{F}\rightarrow S_{\infty}.$ Let $\lambda\in
S_{\infty}$. If $\lambda=v\in S_{r}$, i.e., $\lambda$ is a real place $v$ then
it corresponds to a unique real embedding $\tau_{v}:F\rightarrow\mathbb{R}$,
and if $\lambda=w\in S_{c}$, i.e., $\lambda$ is a complex place $w$ then it
corresponds to a conjugate pair of complex embeddings
$\\{\tau_{w},\bar{\tau}_{w}\\}$ with the understanding that the choice of
$\tau_{w}:F\rightarrow\mathbb{C}$ is not canonical; we use $\tau_{w}$ to
identify $F_{w}\simeq\mathbb{C}.$ Recall, as a reminder of our notational
artifice, that $F_{\infty}=F\otimes\mathbb{R}\simeq\prod_{\lambda\in
S_{\infty}}F_{\lambda}\simeq\prod_{v\in S_{r}}\mathbb{R}\times\prod_{w\in
S_{c}}\mathbb{C}.$ One may write $x_{\infty}\in F_{\infty}$ simply as
$x_{\infty}=(x_{\lambda})_{\lambda\in S_{\infty}},$ or more elaborately as
$x_{\infty}=((x_{v})_{v\in S_{r}},(z_{w})_{w\in S_{c}})$; both notations have
their intrinsic virtue. Let $\chi$ be Hecke character of $F$, and
$\chi_{\infty}$ its character at infinity; see (11). We say that $\chi$ is an
algebraic Hecke character if for every $\tau\in\Sigma_{F}$ there exists an
integer $n_{\tau}$ such that for $x_{\infty}\in F_{\infty}^{\times}$ we have:

(16) $\chi_{\infty}(x_{\infty})\ =\ \prod_{v\in
S_{r}}x_{v}^{n_{\tau_{v}}}\prod_{w\in
S_{c}}z_{w}^{n_{\tau_{w}}}\bar{z}_{w}^{\,n_{\bar{\tau}_{w}}}.$

By separating, the right hand side into unitary and non-unitary parts, while
noting that
$z^{p}\bar{z}^{q}=\left(\tfrac{z}{|z|}\right)^{p-q}|z|_{\mathbb{C}}^{(p+q)/2}$,
we have:

$\chi_{\infty}(x_{\infty})\ =\ \left(\prod_{v\in
S_{r}}\left(\frac{x_{v}}{|x_{v}|}\right)^{n_{\tau_{v}}}\prod_{w\in
S_{c}}\left(\frac{z_{w}}{|z_{w}|}\right)^{n_{\tau_{w}}-n_{\bar{\tau}_{w}}}\right)\left(\prod_{v\in
S_{r}}|x_{v}|_{v}^{n_{\tau_{v}}}\prod_{w\in
S_{c}}|z_{w}|_{w}^{(n_{\tau_{w}}+n_{\bar{\tau}_{w}})/2}\right).$

If we compare this with (11), we see $v\in S_{r},$ $w\in S_{c},$ and
$\lambda\in S_{\infty}$ that:

$n_{v}\equiv n_{\tau_{v}}\ {\rm mod}\ 2,\ \
n_{w}=n_{\tau_{w}}-n_{\bar{\tau}_{w}},\ \ \varphi_{\lambda}=0,\ \
2\sigma=2n_{\tau_{v}}=n_{\tau_{w}}+n_{\bar{\tau}_{w}}$

Especially note the constraint:
$2n_{\tau_{v}}=n_{\tau_{w}}+n_{\bar{\tau}_{w}}$ for all $v\in S_{r},$ $w\in
S_{c}.$ Now, let us suppose the algebraic Hecke character has modulus
$\mathfrak{m}.$ Then this constraint gets significantly tighter by the
following:

######  Lemma 17 (Purity).

For each $\tau\in\Sigma_{F}$, suppose we are given $n_{\tau}\in\mathbb{Z}$.
Suppose for some integral ideal $\mathfrak{m}$ of $F$ we have

(18) $\prod_{\tau\in\Sigma_{F}}\tau(u)^{n_{\tau}}\ =\ 1,\quad\forall u\in
U_{F}^{1}(\mathfrak{m}).$

Then, there exists ${\sf w}\in\mathbb{Z}$ such that

  1. (i)

if $S_{r}\neq\emptyset$ then $n_{\tau}={\sf w}$ for all $\tau\in\tau_{F},$ and

  2. (ii)

if $S_{r}=\emptyset$ then $n_{\gamma\circ\tau}+n_{\gamma\circ\bar{\tau}}={\sf
w}$ for all $\tau\in\Sigma_{F}$ and $\gamma\in{\rm
Gal}(\bar{\mathbb{Q}}/\mathbb{Q}).$

###### Proof.

The proof uses the ingredients of the proof in Dirichlet’s unit theorem. Note
that $U_{F}^{1}(\mathfrak{m})$ has finite index in $U_{F}$. Enumerate the real
embeddings of $F$ as $\\{v_{1},\dots,v_{r_{1}}\\}$ and the complex embeddings
as $\\{w_{1},\bar{w}_{1},\dots,w_{r_{2}},\bar{w}_{r_{2}}\\}.$ With a minor
abuse of notation, for brevity, write $\tau_{v_{i}}=v_{i}$, and
$\tau_{w_{j}}=w_{j}.$ Let $\mathcal{H}\subset\mathbb{R}^{r_{1}+r_{2}}$ be the
hyperplane given by the sum of all coordinates being $0.$ The Minkowski map
$\mathfrak{l}:U_{F}\rightarrow\mathcal{H}$ is given by:

$\mathfrak{l}(u)\ =\ ({\rm log}|v_{1}(u)|_{\mathbb{R}},\,\dots,\,{\rm
log}|v_{r_{1}}(u)|_{\mathbb{R}},\,{\rm
log}|w_{1}(u)|_{\mathbb{C}},\dots,\,{\rm log}|w_{r_{2}}(u)|_{\mathbb{C}}).$

In (18), apply ${\rm log}|\ |_{\mathbb{C}}$ to get:

(19) $2n_{v_{1}}{\rm log}|v_{1}(u)|_{\mathbb{R}}+\cdots+2n_{v_{r_{1}}}{\rm
log}|v_{r_{1}}(u)|_{\mathbb{R}}+\\\ (n_{w_{1}}+n_{\bar{w}_{1}}){\rm
log}|w_{1}(u)|_{\mathbb{C}}+\cdots+(n_{w_{r_{2}}}+n_{\bar{w}_{r_{2}}}){\rm
log}|w_{r_{2}}(u)|_{\mathbb{C}}\ =\ 0,$

for all $u\in U_{F}^{1}(\mathfrak{m})^{+}.$ The proof of Dirichlet’s unit
theorem gives that $\Gamma=\mathfrak{l}(U_{F})$ is a lattice in $\mathcal{H}$;
hence $\Gamma_{F}^{1}(\mathfrak{m}):=\mathfrak{l}(U_{F}^{1}(\mathfrak{m}))$ is
also a lattice in $\mathcal{H}$; whence, there exists $u_{1},\dots,u_{t-1}\in
U_{F}^{1}(\mathfrak{m})$ such that
$\\{v_{1}:=\mathfrak{l}(u_{1}),\dots,v_{t-1}:=\mathfrak{l}(u_{t-1})\\}$ is an
$\mathbb{R}$-basis for $\mathcal{H},$ where, for brevity, we denote
$t=r_{1}+r_{2}.$ Let us write $v_{i}=(a_{i1},\dots,a_{it})$ as a vector in
$\mathbb{R}^{t},$ and consider the $(t-1)\times t$ sized matrix $A=[a_{ij}].$
The rank of $A$ is $t-1$ and hence its nullity is $1.$ Since each
$v_{i}\in\mathcal{H}$ we know $\sum_{j}a_{ij}=0$, we deduce that for any
$X\in\mathbb{R}^{t\times 1}$, if $AX=0$ then all the coordinates of $X$ are
equal. Now (19) applied to $u_{1},\dots,u_{t-1}$ gives a solution to the
system of equations $AX=0$ from which we get
$2n_{v_{1}}=\cdots=2n_{v_{1}}=n_{w_{1}}+n_{\bar{w}_{1}}=\cdots=n_{w_{r_{2}}}+n_{\bar{w}_{r_{2}}}.$
(The reader may pause here to note that the above argument proves Lem. 15.)
Let us denote the common value by ${\sf w}^{\prime}.$ Now, take any
$\gamma\in{\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$, apply $\gamma^{-1}$ to (18)
to get

$1\ =\ \prod_{\tau\in\tau_{F}}\gamma^{-1}(\tau(u))^{n_{\tau}}\ =\
\prod_{\tau\in\tau_{F}}\tau(u)^{n_{\gamma\circ\tau}},$

and we go through the name argument to get
$n_{\gamma\circ\tau}+n_{\gamma\circ\overline{\tau}}={\sf w}^{\prime}$ for all
$\tau\in\tau_{F}$ and $\gamma\in{\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q}).$ If
$S_{r}\neq\emptyset$, then since ${\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ acts
transitively on $\Sigma_{F}$ we see that $n_{\tau}=n_{v_{1}}$ for all $\tau$;
then take ${\sf w}=n_{v_{1}}={\sf w}^{\prime}/2$. If $S_{r}=\emptyset$ then
take ${\sf w}={\sf w}^{\prime}.$ ∎

Let $\chi$ be an algebraic Hecke character of $F$ mod $\mathfrak{m}$ and with
infinity type $(n_{\tau})_{\tau\in\Sigma_{F}}.$ Then the exponents $n_{\tau}$
satisfy the conditions of Lem. 17. The integer ${\sf w}$ is called the purity
weight of $\chi$. It is worth noting that when $F$ is totally real or totally
imaginary then there is no constraint on the parity of the purity weight ${\sf
w}$, but when $F$ has both real and complex embeddings then the purity weight
${\sf w}$ is necessarily an even integer. Note the following easy consequence
of purity:

######  Corollary 20.

Let $\chi$ be an algebraic Hecke character of $F$ mod $\mathfrak{m}$ and with
infinity type $(n_{\tau})_{\tau\in\Sigma_{F}}$ and purity weight ${\sf w}.$

  1. (i)

If $S_{r}\neq\emptyset$ then $\chi=\chi^{\circ}|\\!|\ |\\!|^{\sf w}$ for a
Dirichlet character $\chi^{\circ}$ of $F$ mod $\mathfrak{m}$.

  2. (ii)

If $S_{r}=\emptyset$ then $\chi=\chi^{u}|\\!|\ |\\!|^{{\sf w}/2}$ for a
unitary Hecke character $\chi^{u}$ of $F$ mod $\mathfrak{m}$.

In case (ii), let us observe that the character at infinity of $\chi^{u}$ has
the following shape:

$\chi^{u}_{\infty}(x_{\infty})\ =\ \prod_{w\in
S_{c}=S_{\infty}}\left(x_{w}/\bar{x}_{w}\right)^{n_{w}-{\sf w}/2}.$

###  4.2. The $L$-function of an algebraic Hecke character

####  4.2.1. Definition(s) of Hecke $L$-function

Hecke considered the class of Größencharaktere mod integral ideals as the
optimal class of characters for which one may study the analytic properties of
$L$-functions, called Hecke $L$-functions, that we now define. Let $\chi$ be
an algebraic Hecke character, and $\psi=(\psi_{f},\psi_{\infty})$ the
associated Großencharakter. Suppose that $\mathfrak{m}$ is the conductor of
$\chi$, and so also $\mathfrak{m}$ is the conductor of $\psi$. The (finite
part of the) Hecke $L$-function of $\chi$ may be defined in two possible ways:

  1. (1)

As defined by Hecke in the form of a Dirichlet series:

$L_{f}(s,\chi)\ =\
\sum_{\mathfrak{a}}\frac{\psi(\mathfrak{a})}{N_{F/\mathbb{Q}}(\mathfrak{a})^{s}},$

the summation running over integral ideals $\mathfrak{a}$ of $\mathcal{O}_{F}$
that are relatively prime to $\mathfrak{m}$.

  2. (2)

As defined by Tate in the form of an Euler product:

$L_{f}(s,\chi)\ =\
\prod_{\mathfrak{p}}\,\left(1-\chi_{\mathfrak{p}}(\varpi_{\mathfrak{p}})N_{F/\mathbb{Q}}(\mathfrak{p})^{-s}\right)^{-1},$

the product is over prime ideals $\mathfrak{p}$ of $\mathcal{O}_{F}$ that are
relatively prime to $\mathfrak{m}.$

Here, $N_{F/\mathbb{Q}}(\mathfrak{p})$ is defined as
$|\mathcal{O}_{F}/\mathfrak{p}|,$ and is multiplicatively extended to define
$N_{F/\mathbb{Q}}(\mathfrak{a}).$ One has the usual analytic properties:
absolute convergence in a half-plane, analytic continuation, and functional
equation. In the form of Hecke’s definition, we refer the reader to Neukirch
[10]; in the form of Tate’s definition, we refer the reader to Tate’s thesis
[15] and Weil’s book [18].

####  4.2.2. The coefficients of the $L$-function of an algebraic Hecke
character

The main reason to consider algebraic Hecke characters (called characters of
type $A_{0}$ by Weil [17]) is the following proposition which puts us in the
right context to study the arithmetic properties of Hecke $L$-functions.

######  Proposition 21.

Let $\chi$ be an algebraic Hecke character, and
$\psi=(\psi_{f},\psi_{\infty})$ the associated Großencharakter of modulus,
say, $\mathfrak{m}.$ Then the coefficients of the Dirichlet series giving the
finite part of the Hecke $L$-function $L_{f}(s,\chi)$ are contained in a
number field, i.e., there is a finite extension $E$ of $\mathbb{Q}$ such that
$\psi(\mathfrak{a})\in E$ (resp.,
$\chi_{\mathfrak{p}}(\varpi_{\mathfrak{p}})\in E$) for all integral ideals
$\mathfrak{a}$ (resp., prime ideals $\mathfrak{p}$) relatively prime to
$\mathfrak{m}.$

###### Proof.

By (12), it follows that for $a\in\mathcal{O}_{F}(\mathfrak{m})$, and hence
also for $a\in F(\mathfrak{m})$, $\psi((a))$ takes values in a finite
extension $E_{1}$ of $\mathbb{Q}$ inside $\bar{\mathbb{Q}}$, since
$(\mathcal{O}_{F}/\mathfrak{m})^{\times}$ is a finite group, and since
$\psi_{\infty}$ is algebraic (16) we see that $\psi_{\infty}(a)$ takes values
in the compositum of all the conjugates of $F$ inside $\bar{\mathbb{Q}}.$
Hence the values of $\psi$ on $\mathbb{P}_{F}(\mathfrak{m})$ lie in $E_{1}$.
Since, $\mathbb{P}_{F}(\mathfrak{m})$ has finite index in
$\mathbb{J}_{F}(\mathfrak{m})$ we see that the values of $\psi$ are in a
finite extension $E$ of $E_{1}$ contained inside $\bar{\mathbb{Q}}.$ By Prop.
14, $\psi(\mathfrak{p})=\chi_{\mathfrak{p}}(\varpi_{\mathfrak{p}})$ for all
prime ideals $\mathfrak{p}\\!\not|\,\mathfrak{m}$. ∎

####  4.2.3. The rationality field of an algebraic Hecke character

The smallest subfield of $\bar{\mathbb{Q}}$ that contains all the ‘values’ of
an algebraic Hecke character $\chi$, denoted $\mathbb{Q}(\chi)$, is called its
rationality field. More precisely, with notations as in the above proposition,
suppose $\chi$ has conductor $\mathfrak{m}$, we may define:

(22) $\mathbb{Q}(\chi)\ =\ \mathbb{Q}(\\{\chi(\mathfrak{p}):\mbox{for all
prime ideals $\mathfrak{p}$ relatively prime to $\mathfrak{m}$}\\})\\\
=\mathbb{Q}(\\{\psi(\mathfrak{a}):\mbox{for all integral ideals $\mathfrak{a}$
relatively prime to $\mathfrak{m}$}\\}).$

As Weil comments in [17], $\mathbb{Q}(\chi)$ need not contain $F$; consider,
for example, a quadratic character of any number field $F$.

####  4.2.4. The critical values of the $L$-function of an algebraic Hecke
character

Let us recall the shape of the functional equation satisfied by
$L_{f}(s,\chi),$ in as much as we recall the precise recipe for the
$\Gamma$-factors $L_{\infty}(s,\chi)$ necessary to define the completed
$L$-function:

$L(s,\chi)\ :=\ L_{\infty}(s,\chi)L_{f}(s,\chi),$

where, if $\chi$ has infinity type $(n_{\tau})_{\tau\in\Sigma_{F}}$,
$n_{\tau}\in\mathbb{Z}$, satisfying the purity condition of Lem. 17, then we
define:

$L_{\infty}(s,\chi)\ =\ \prod_{v\in
S_{r}}\Gamma_{\mathbb{R}}(s+n_{v}+\epsilon_{v})\cdot\prod_{w\in
S_{c}}\Gamma_{\mathbb{C}}\left(s+\frac{n_{w}+n_{\bar{w}}}{2}+\frac{|n_{w}-n_{\bar{w}}|}{2}\right),$

where $\Gamma_{\mathbb{R}}(s)=\pi^{-s/2}\Gamma(s/2),$
$\Gamma_{\mathbb{C}}(s)=2(2\pi)^{-s}\Gamma(s)$, and $\epsilon_{v}\in\\{0,1\\}$
with $\epsilon_{v}\equiv n_{v}$ mod $2$. When $\chi$ is nontrivial, the
completed $L$-function admits an analytic continuation to an entire function
of $s\in\mathbb{C}$, and a beautiful aspect of $L$-functions is the functional
equation:

$L(s,\chi)\ =\ \varepsilon(s,\chi)L(1-s,\chi^{-1}),$

where $\varepsilon(s,\chi)$ is an exponential function. (See [10] or [18].) An
integer $m$ is said to be critical for the Hecke $L$-function $L(s,\chi)$ if
both $L_{\infty}(s,\chi)$ and $L_{\infty}(1-s,\chi^{-1})$ are regular (i.e.,
finite) at $s=m.$ We may say that the $\Gamma$-factors on either side of the
functional equation do not have poles at $s=m$. The set of all critical
integers for $L(s,\chi)$, denoted ${\rm Crit}(L(s,\chi))$, is given by the
following

######  Proposition 23.

Let $\chi$ be an algebraic Hecke character of a number field $F$ with infinity
type $(n_{\tau})_{\tau\in\Sigma_{F}}$, $n_{\tau}\in\mathbb{Z}$, satisfying the
purity condition of Lem. 17 with purity weight ${\sf w}$. Then the critical
set of integers for $L(s,\chi)$ is given by:

  1. (i)

$F$ is totally real ($S_{c}=\emptyset$): Recall that $n_{\tau}={\sf w}$ for
all $\tau$.

    1. (a)

If there exists $v_{1},v_{2}\in S_{r}$, such that
$\epsilon_{v_{1}}\neq\epsilon_{v_{2}}$ then ${\rm Crit}(L(s,\chi))=\emptyset.$

    2. (b)

If $\epsilon_{v}=0$ for all $v\in S_{r}$ then ${\rm Crit}(L(s,\chi))$ is given
by:

$\\{\dots,\,-{\sf w}+1-2k,\,\dots,\,-{\sf w}-3,\,-{\sf w}-1;\,-{\sf
w}+2,\,-{\sf w}+4,\,\dots,\,-{\sf w}+2k,\,\dots\\}_{k\in\mathbb{Z}_{\geq 0}}.$

The critical set is centered at $\tfrac{1}{2}-{\sf w}$.

    3. (c)

If $\epsilon_{v}=1$ for all $v\in S_{r}$ then ${\rm Crit}(L(s,\chi))$ is given
by:

$\\{\dots,-{\sf w}-2k,\dots,-{\sf w}-2,-{\sf w};\,-{\sf w}+1,-{\sf
w}+3,\dots,-{\sf w}+1+2k,\dots\\}_{k\in\mathbb{Z}_{\geq 1}}.$

The critical set is centered at $\tfrac{1}{2}-{\sf w}$.

  2. (ii)

$F$ is totally imaginary ($S_{r}=\emptyset$): Define the width of $\chi$ as
the non-negative integer: $\ell=\ell(\chi)=\min\\{|n_{w}-n_{\bar{w}}|\,:\,w\in
S_{c}\\}.$ Then ${\sf w}\equiv\ell\ {\rm mod}\ 2.$ We have:

${\rm Crit}(L(s,\chi))\ =\ \left\\{m\in\mathbb{Z}\ :\ 1-\frac{\sf
w}{2}-\frac{\ell}{2}\ \leq\ m\ \leq\ -\frac{\sf
w}{2}+\frac{\ell}{2}\right\\}.$

The critical set, centered at $\tfrac{1-{\sf w}}{2}$, is a finite set of
cardinality $\ell$.

  3. (iii)

If $F$ has real and complex places ($S_{r}\neq\emptyset\neq S_{c}$), then
${\rm Crit}(L(s,\chi))=\emptyset.$

We omit the tedious proof of the above proposition and just add a few comments
after which it is an extended exercise that is left to the reader. Recall that
the poles of $\Gamma(s)$ are simple and located at non-positive integers, and
furthermore that $\Gamma(s)\neq 0$ for all $s$. In case (i)(a), one sees the
presence of a product $\Gamma(s/2)\Gamma((1+s)/2)$ because of which no integer
can be critical. In case (i)(b), up to shifting by $-{\sf w},$ it is the same
as ${\rm Crit}(\zeta(s))$, the critical set of the Riemann zeta function. In
case (i)(c), up to shifting by $-{\sf w},$ it is the same as the critical set
of an odd classical Dirichlet character. For case (ii), the reader is referred
to [12, Sect. 3.1] where the details are explicated. Case (iii), is like the
intersection of cases (i) and (ii); by Lem. 17, $n_{\tau}={\sf w}$ for all
$\tau$; $\ell=0$ where $\ell$ is defined using only $S_{c}$, forcing no
critical points if there are real and complex places.

There has been a huge amount of work on the nature of the critical values of
$L(m,\chi)$. Deligne’s conjecture [4] on the special values of motivic
$L$-functions sets the stage for the sort of theorem one seeks. The interested
reader is referred to Blasius [1], Harder [6], and Schappacher [13], and all
the references therein. My reason for writing this article is that while I was
working on the critical values of automorphic $L$-functions ([11] and [12]), I
had to work through the results of this article frequently; and these results,
although well-known to experts, did not seem to be readily available in the
literature in the way I needed them. There are some very deep sign problems,
apparently hitherto not well-appreciated, that appear in these works which
have their roots – to paraphrase Weil [17] – in the remarkable properties of
such algebraic Hecke characters.

The above proposition begets the terminology of a critical algebraic Hecke
character by which we mean an algebraic Hecke character $\chi$ such that ${\rm
Crit}(L(s,\chi))\neq\emptyset.$ Existence of a critical algebraic Hecke
character $\chi$ of a number field $F$ implies first of all that $F$ is
totally real or totally imaginary, and furthermore, when $F$ is totally real
then the parities $\epsilon_{v}$ of all the local archimedean characters are
equal. We will now discuss the existence of critical algebraic Hecke
characters with prescribed infinity type subject to the purity condition.

###  4.3. Existence of (critical) algebraic Hecke characters

For a given number field $F$ we discuss the problem of existence of algebraic
Hecke characters that may or may not be critical. The purity lemma naturally
leads us to consider two cases (i) $S_{r}\neq\emptyset$ and (ii)
$S_{r}=\emptyset$; we will see that case (ii) when $F$ is totally imaginary is
in some sense far more interesting.

####  4.3.1. When $F$ has a real place ($S_{r}\neq\emptyset$)

From Cor. 20, (i), we know that an algebraic Hecke character $\chi$ is
necessarily of the form

$\chi\ =\ \chi^{\circ}|\\!|\ |\\!|^{\sf w},$

for a character $\chi^{\circ}$ of finite order and for an integer ${\sf w}.$
We may fine-tune our question a little further to note that we can arrange for
$\chi^{\circ}$ to have any prescribed signature at infinite places: for any
(possibly empty) $S\subset S_{r}$, by weak approximation, we can find $x\in
F^{\times}$ such that $v(x)>0$ for $v\in S$, and $v(x)<0$ for $v\in
S_{r}\setminus S.$ Take $E=F(\sqrt{x})$, and $\chi=\omega_{E/F}$ the quadratic
character of $F$ corresponding to $E$ by class field theory. Then
$\chi_{v}(-1)=1$ for $v\in S$ and $\chi_{v}(-1)=-1$ for $v\in S_{r}\setminus
S.$ In particular, when $F$ is a totally real field, by taking $S=\emptyset$
or $S=S_{r}=S_{\infty}$, we see that there exists critical algebraic Hecke
characters of any prescribed parity (totally even or totally odd) and any
prescribed purity weight ${\sf w}.$

####  4.3.2. When $F$ is totally imaginary ($S_{r}=\emptyset$)

Consider an infinity type: ${\bf n}:=(n_{\tau})_{\tau\in\Sigma_{F}},$
$n_{\tau}\in\mathbb{Z},$ satisfying the purity condition
$n_{\gamma\circ\tau}+n_{\gamma\circ\bar{\tau}}={\sf w}$ for all
$\tau\in\Sigma_{F}$ and all $\gamma\in{\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$.
The question we wish to address is whether there exists an algebraic Hecke
character $\chi$ of $F$ with infinity type ${\bf n}.$ As Weil says in [17],
Artin pointed out to him that this is essentially an exercise in Galois
theory.

First, consider the case when $F$ is a CM field, by which one means that $F$
is a totally imaginary quadratic extension of a totally real field $F^{+}.$
Let us note an easy

######  Lemma 24.

Let $F$ be a totally imaginary quadratic extension of a totally real field
$F^{+}$. Let $\tau:F\rightarrow\bar{\mathbb{Q}}$ and $\gamma\in{\rm
Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ and let $\mathfrak{c}$ stand for complex
conjugation as an element of ${\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$. Then
$\gamma\circ\mathfrak{c}\circ\tau\ =\ \mathfrak{c}\circ\gamma\circ\tau,$ i.e.,
complex conjugation and any automorphism of $\bar{\mathbb{Q}}$ commute on the
image of a CM field. This also holds for a totally real field.

###### Proof.

Let $\tau_{1}=\gamma\circ\mathfrak{c}\circ\tau$ and
$\tau_{2}=\mathfrak{c}\circ\gamma\circ\tau.$ Then it is easy to that
$\tau_{1}|_{F^{+}}=\tau_{2}|_{F^{+}}$ (since $F^{+}$ is totally real). This
means that $\tau_{1}=\tau_{2}$ or $\tau_{1}=\mathfrak{c}\circ\tau_{2}$; if the
latter, then $\gamma\circ\mathfrak{c}\circ\tau=\gamma\circ\tau$, whence
$\mathfrak{c}\circ\tau=\tau$ or that $\tau$ lands inside $\mathbb{R}$ which is
not possible. Hence $\tau_{1}=\tau_{2}.$ The last assertion for a totally real
field is obvious. ∎

For a CM field $F$, consider an infinity type ${\bf
n}:=(n_{\tau})_{\tau\in\Sigma_{F}},$ $n_{\tau}\in\mathbb{Z},$ satisfying the
purity condition $n_{\gamma\circ\tau}+n_{\gamma\circ\bar{\tau}}={\sf w}$. The
above lemma gives that
$n_{\gamma\circ\bar{\tau}}=n_{\overline{\gamma\circ\tau}}$. Hence, purity for
a CM field is the same as asking $n_{\tau}+n_{\bar{\tau}}={\sf w}$ for all
$\tau\in\Sigma_{F}.$

######  Proposition 25.

Let $F$ be a totally imaginary quadratic extension of a totally real field
$F^{+}$. Let ${\bf n}:=(n_{\tau})_{\tau\in\Sigma_{F}},$
$n_{\tau}\in\mathbb{Z},$ satisfying the purity condition
$n_{\tau}+n_{\bar{\tau}}={\sf w}$ for all $\tau\in\Sigma_{F}.$ Then there
exists an algebraic Hecke character (of some modulus) with that infinity type.

###### Proof.

If such a character exists, then the character at infinity would have the
shape:

$\chi_{\infty}(x_{\infty})\ =\ \left(\prod_{w\in
S_{c}}\left(\frac{x_{w}}{\bar{x}_{w}}\right)^{\tfrac{n_{\tau_{w}}-n_{\bar{\tau}_{w}}}{2}}\right)|x_{\infty}|_{\infty}^{{\sf
w}/2},$

and also for some integral ideal $\mathfrak{m}$, $\chi_{\infty}$ needs to be
trivial on $U_{F}^{1}(\mathfrak{m}).$ Put
$p_{w}=n_{\tau_{w}}-n_{\bar{\tau}_{w}}$, and write $x_{w}$ in polar form as
$r_{w}e^{i\theta_{w}}.$ Then we need to construct a unitary Hecke character
$\chi^{u}:\mathbb{I}_{F}/F^{\times}\rightarrow S^{1}$, such that
$\chi^{u}_{\infty}(x_{\infty})=\prod_{w}e^{ip_{w}\theta_{w}},$ and
$\chi^{u}_{\infty}$ needs to be trivial on $U_{F}^{1}(\mathfrak{m}).$ We can
then take $\chi=\chi_{u}|\\!|\ |\\!|^{{\sf w}/2}.$ Note that
$F_{\infty}^{\times}=(S^{1}\times\cdots\times
S^{1})\times(\mathbb{R}^{\times}_{+}\times\cdots\times\mathbb{R}^{\times}_{+}).$
The units $U_{F}$ (via the proof of Dirichlet’s unit theorem) intersects with
the compact part in roots of unity $\mu_{F}$ in $F^{\times}$. Take an integral
ideal $\mathfrak{m}$ such that $U_{F}^{1}(\mathfrak{m})\cap\mu_{F}=\\{1\\}$;
hence, $U_{F}^{1}(\mathfrak{m})\cap(S^{1}\times\cdots\times S^{1})=\\{1\\}$.
Define a unitary character $\psi^{u}_{\infty}$ that maps $x_{\infty}$ to
$\prod_{w}e^{ip_{w}\theta_{w}}.$ Then $\psi^{u}_{\infty}$ defines a character
of $F_{\infty}^{\times}/U_{F}^{1}(\mathfrak{m}).$ Take any character
$\psi^{u}_{f}$ of $(\mathcal{O}_{F}/\mathfrak{m})^{\times}$, and by Prop. 13
we get a Größencharakter $\psi^{u}$ mod $\mathfrak{m}$ with the necessary
infinity type. Take $\chi^{u}$ to be the corresponding Hecke character as in
the paragraph after Prop. 14. ∎

Next, consider the case of a general totally imaginary field $F$. There is a
maximal CM or totally real subfield $F_{1}$ of $F$ that plays an important
role. Following Weil [17], let $F_{0}$ be the largest totally real subfield of
$F$. Then $F_{0}$ admits at most one totally imaginary quadratic extension
contained inside $F_{0}$. If $F$ does admit such a CM subfield, denote it as
$F_{1}$; and if there is no such extension inside $F$, then put $F_{1}=F_{0}.$
A basic property of a pure infinity type of a totally imaginary field is that
it is the base-change of a pure infinity type of $F_{1}$ as explained by the
following

######  Proposition 26.

Let $F$ be a totally imaginary field. Consider an infinity type ${\bf
n}:=(n_{\tau})_{\tau\in\Sigma_{F}},$ $n_{\tau}\in\mathbb{Z},$ satisfying the
purity condition $n_{\gamma\circ\tau}+n_{\gamma\circ\bar{\tau}}={\sf w}$ for
all $\tau\in\Sigma_{F}$ and all $\gamma\in{\rm
Gal}(\bar{\mathbb{Q}}/\mathbb{Q}).$ Then $n_{\tau}$, as a function of $\tau$
depends only on the restriction of $\tau$ to $F_{1},$ i.e., there exists a
necessarily pure infinity type ${\bf
m}:=(m_{\tau_{1}})_{\tau_{1}\in\Sigma_{F_{1}}}$ for the field $F_{1}$, such
that if $\tau\in\Sigma_{F}$ with $\tau_{1}=\tau|_{F_{1}}$ then
$n_{\tau}=m_{\tau_{1}}.$ (In this situation, we may say that ${\bf n}$ is the
base-change of ${\bf m}$.)

###### Proof.

(I learnt of the following proof from Deligne; it appears in a different
context in [5].) The purity condition
$n_{\gamma\circ\tau}+n_{\gamma\circ\bar{\tau}}={\sf w},$ for all
$\tau\in\Sigma_{F}$ and all $\gamma\in{\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q}),$
also implies that $n_{\gamma\circ\tau}+n_{\overline{\gamma\circ\tau}}={\sf
w}$, from which deduce

$n_{\gamma\circ\bar{\tau}}\ =\
n_{\overline{\gamma\circ\tau}},\quad\forall\tau,\ \forall\gamma.$

Consider the function $n:{\rm
Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\times\Sigma_{F}\rightarrow\mathbb{Z}$,
defined as $n(\gamma,\tau)=n_{\gamma\circ\tau}.$ For any $x\in{\rm
Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ we have $n(\gamma x,\tau)=n(\gamma,x\tau).$
The above displayed equation reads

$n(\gamma,\mathfrak{c}\tau)\ =\ n(\mathfrak{c},\gamma\tau).$

Replacing $\tau$ by $\gamma^{-1}\tau$, we also have:

$n(\mathfrak{c},\tau)\ =\ n(\gamma,\mathfrak{c}\gamma^{-1}\tau)\ =\
n(\gamma\mathfrak{c}\gamma^{-1},\tau).$

Replacing $\tau$ by $\mathfrak{c}\tau$ we get

$n(1,\tau)\ =\ n(\gamma\mathfrak{c}\gamma^{-1}\mathfrak{c},\tau).$

Now, take any $x\in{\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$, and we see using
the above relations (for all $\gamma$ and $\tau$) that:

$n(x\gamma\mathfrak{c}\gamma^{-1}\mathfrak{c}x^{-1},\tau)\ =\
n(x\gamma\mathfrak{c}\gamma^{-1}x^{-1}x\mathfrak{c}x^{-1},\tau)\ =\
n((x\gamma)\mathfrak{c}(x\gamma)^{-1},x\mathfrak{c}x^{-1}\tau)\ =\\\
n(\mathfrak{c},x\mathfrak{c}x^{-1}\tau)\ =\
n(x\mathfrak{c}x^{-1},\mathfrak{c}\tau)\ =\
n(x\mathfrak{c}x^{-1}\mathfrak{c},\tau)\ =\ n(1,\tau).$

In other words, $n_{h\tau}=n_{\tau}$ for all $h$ in the normal subgroup
$\mathcal{N}$ of $\mathcal{G}:={\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$
generated by $\\{\gamma\mathfrak{c}\gamma^{-1}\mathfrak{c}:\gamma\in{\rm
Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\\}$. Note that $\mathcal{G}/\mathcal{N}$ is
the largest quotient in which $\mathfrak{c}$ is central. We see that
$\mathcal{G}$ acts transitively on $\Sigma_{F};$ the transitive action of
$\mathcal{G}$ on $\Sigma_{F_{1}}$ is via $\mathcal{G}/\mathcal{N}$ (Lem. 24);
and $\mathcal{N}$ acts transitively on the fiber over
$\tau_{1}\in\Sigma_{F_{1}}$ in the restriction map
$\Sigma_{F}\rightarrow\Sigma_{F_{1}}.$ In particular, if
$\tau,\tau^{\prime}\in\Sigma_{F}$ with
$\tau_{1}=\tau|_{F_{1}}=\tau^{\prime}|_{F_{1}}$, and suppose $h\in\mathcal{N}$
is such that $h\tau=\tau^{\prime}$, then
$n_{\tau^{\prime}}=n_{h\tau}=n_{\tau},$ and we call this common value as
$m_{\tau_{1}}.$ ∎

We are now ready to state the basic fact about the existence of algebraic
Hecke characters for a totally imaginary field in the following

######  Proposition 27.

Let $F$ be a totally imaginary field. Consider an infinity type ${\bf
n}:=(n_{\tau})_{\tau\in\Sigma_{F}},$ $n_{\tau}\in\mathbb{Z},$ satisfying the
purity condition $n_{\gamma\circ\tau}+n_{\gamma\circ\bar{\tau}}={\sf w}$ for
all $\tau\in\Sigma_{F}$ and all $\gamma\in{\rm
Gal}(\bar{\mathbb{Q}}/\mathbb{Q}).$ Then, there exists an algebraic Hecke
character of infinity type ${\bf n}.$

  1. (i)

If $F_{0}=F_{1}$ (this is the case when $F$ has no CM subfield), then there
exists $n\in\mathbb{Z}$ such that $n_{\tau}=n$ for all $\tau\in\Sigma_{F}$.
Any algebraic Hecke character $\chi$ of infinity type ${\bf n}$ is of the form
$\chi=\chi^{\circ}|\\!|\ |\\!|^{n}$ for a Dirichlet character $\chi^{\circ}$
of $F$.

  2. (ii)

If $F_{1}$ is a CM field, then ${\bf n}$ is the base-change of an infinity
type ${\bf m}$ for $F_{1}$ (Prop. 26); any algebraic Hecke character $\chi$ of
$F$ of infinity type ${\bf n}$ is of the form

$\chi\ =\ \chi_{1}\circ N_{F/F_{1}}\otimes\chi^{\circ},$

for some algebraic Hecke character $\chi_{1}$ of $F_{1}$ with infinity type
${\bf m}$ and some Dirichlet character $\chi^{\circ}$ of $F$.

###### Proof.

In both cases from Prop. 26 there exists an infinity type ${\bf m}$ over
$F_{1}$ whose base-change to $F$ is ${\bf n}$. In case (i), since
$F_{1}=F_{0}$ is totally real, all the $m_{\tau_{1}}$ are equal, and hence
$n_{\tau}$ are all equal, to say, $n$. If there is an algebraic Hecke
character $\chi$ of this infinity type then necessarily we have $\chi|\\!|\
|\\!|^{-n}$ is a finite-order character, say, $\chi^{\circ}.$ To construct
$\chi$, take any finite-order Hecke character $\chi^{\circ}$ of $F$, and put
$\chi=\chi^{\circ}|\\!|\ |\\!|^{n}$. In case (ii), from Prop. 25, there exists
an algebraic Hecke character $\chi_{1}$ of $F_{1}$ with infinity type ${\bf
m}.$ Consider its base-change $\chi_{2}:=\chi_{1}\circ N_{F/F_{1}},$ to $F$;
it is clear that this base-change has the same infinity type as $\chi$, if one
knew the existence of $\chi$, and in which case $\chi\chi_{2}^{-1}$ would be a
character of finite-order, say, $\chi^{\circ}.$ Going backwards, construct the
required $\chi$ via $\chi_{1}\circ N_{F/F_{1}}\otimes\chi^{\circ}.$ ∎

I have found it helpful to keep some examples in mind while thinking of a
totally imaginary field $F$, with various possible scenarios for $F_{0}$ and
$F_{1}$. Some of these examples were suggested to me by Haruzo Hida.

######  Example 28.

When $F$ is totally imaginary but not CM, then one has to be careful about
what constitutes a pure infinity type. The following example is instructive:
take $F=\mathbb{Q}(2^{1/3},\omega),$ where $2^{1/3}$ is the real cube root of
$2$ and $\omega=e^{2\pi i/3}.$ Then $\Sigma_{F}={\rm Gal}(F/\mathbb{Q})\simeq
S_{3}$ the permutation group in 3 letters taken to be
$\\{2^{1/3},\,2^{1/3}\omega,\,2^{1/3}\omega^{2}\\}.$ Let $s\in S_{3}$
correspond to $\tau_{s}:F\rightarrow\mathbb{C}.$ Consider the infinity types
${\bf n}=(n_{\tau_{s}})_{s\in S_{3}}$ and ${\bf
n}^{\prime}=(n^{\prime}_{\tau_{s}})_{s\in S_{3}}$:

$\begin{array}[]{|c||c|c|c|c|c|c||}\hline\cr s&e&(12)&(23)&(13)&(123)&(132)\\\
\hline\cr n^{\prime}_{\tau_{s}}&a&b&{\sf w}-a&c&{\sf w}-c&{\sf w}-b\\\
\hline\cr n_{\tau_{s}}&a&{\sf w}-a&{\sf w}-a&{\sf w}-a&a&a\\\
\hline\cr\end{array}$

where $a,b,c,{\sf w}\in\mathbb{Z}$. For the tautological embedding
$F\subset\bar{\mathbb{Q}},$ the set $\Sigma_{F}$ is paired into complex
conjugates as
$\\{(\tau_{e},\tau_{(23)}),(\tau_{(12)},\tau_{(132)}),(\tau_{(13)},\tau_{(123)})\\}.$
Then, $n^{\prime}_{\tau}+n^{\prime}_{\bar{\tau}}={\sf w}.$ However, ${\bf
n}^{\prime}$ is not a pure infinity type in general. All other possible
pairings of $\Sigma_{F}$ into conjugates via automorphisms of
$\bar{\mathbb{Q}}$ ($F$ being Galois this simply boils down to composing these
embeddings $\tau_{s}$ by a fixed one $\tau_{s_{0}},$ and using
$\tau_{s_{0}}\circ\tau_{s}=\tau_{s_{0}s}$) are given by:
$\\{(\tau_{e},\tau_{(12)}),(\tau_{(23)},\tau_{(123)}),(\tau_{(13)},\tau_{(132)})\\},$
and
$\\{(\tau_{e},\tau_{(13)}),(\tau_{(23)},\tau_{(132)}),(\tau_{(12)},\tau_{(123)})\\},$
from which we see that ${\bf n}^{\prime}$ is not a pure infinity type unless
${\sf w}-a,b$ and $c$ are all equal; ${\bf n}$ is pure and has purity weight
${\sf w}.$ Also, ${\bf n}$ is the base-change of the infinity type ${\bf m}$
of $F_{1}=\mathbb{Q}(\omega)$ where $m_{\tau_{e}}=a$ and $m_{\tau_{(12)}}={\sf
w}-a.$

######  Example 29.

Take a CM field $F^{\rm cm}$, and a number field $L$ that is linearly disjoint
from $F^{\rm cm}$, and put $F=L\cdot F^{\rm cm}.$ This is a particularly
pleasant example of a totally imaginary field. If $L$ is totally real, then
$F$ is also of CM type. For concreteness, take $n\geq 3$, then
$F=\mathbb{Q}(2^{1/n},\zeta_{n})$, where $\zeta_{n}=e^{2\pi i/n},$ is a
totally imaginary field of this form; $F^{\rm cm}=\mathbb{Q}(\zeta_{n})$ and
$L=\mathbb{Q}(2^{1/n}).$ This example is slightly more general than the field
considered in the previous example.

######  Example 30.

For a totally imaginary field not of the above form, take $F_{0}=\mathbb{Q}$,
and put $F_{1}=\mathbb{Q}(\mathfrak{i})$ and
$F=\mathbb{Q}(\sqrt{4+\mathfrak{i}},\mathfrak{i}).$ (This example is used in
Sect. 6.2.)

######  Example 31.

For a totally imaginary field with no CM subfield, take $F_{0}$ to be a cubic
totally real field; for example, $F_{0}=\mathbb{Q}(\zeta_{3}+\zeta_{3}^{-1}).$
Choose non-squares $a$ and $b$ in $F_{0}$, with conjugates
$a,a^{\prime},a^{\prime\prime}$ and $b,b^{\prime},b^{\prime\prime}$,
respectively, and chosen so that $a>0,a^{\prime}<0,a^{\prime\prime}<0$ and
$b<0,b^{\prime}<0,b^{\prime\prime}>0.$ Such a choice is possible by weak
approximation. Then the field $F=F_{0}[\sqrt{a},\sqrt{b}]$ is totally
imaginary with no CM subfield. Such fields do not support critical algebraic
Hecke characters since the width of a pure infinity type is $0$ (see Prop.
23).

######  Remark 32.

There is a slightly different way to organise one’s thoughts about the
subfields $F_{0}$ and $F_{1}$ of a totally imaginary field $F$, and which at
times is useful to keep in mind. Recall that $\mathfrak{c}$ denote the complex
conjugation on $\mathbb{C}$. Let $F_{1}$ be maximal among all subfields $K$ of
$F$ that have an involution $\iota_{K}:K\rightarrow K$ such that
$\mathfrak{c}\circ\sigma=\sigma\circ\iota_{K}$ for all embeddings
$\sigma:K\rightarrow\bar{\mathbb{Q}}.$ In particular, $F_{1}$ has its
involution $\iota_{F_{1}},$ and let $F_{0}$ denote the subfield of $F_{1}$
that is fixed by $\iota_{F_{1}}.$ There are two cases: (i) $\iota_{F_{1}}$ is
non-trivial; then $F_{1}$ is a totally imaginary quadratic extension of the
totally real $F_{0};$ (ii) $\iota_{F_{1}}$ is trivial; then $F_{1}=F_{0}$ is a
totally real field.

##  5\. Cohomological automorphic representations of ${\rm GL}_{1}$

So far, we have treated Hecke characters as complex valued functions on adèlic
spaces, and one might say that this is from the point of view of automorphic
forms on ${\rm GL}(1).$ We may also treat them as algebro-geometric objects;
see, for example, Deligne [3] or Serre [14]. Of course, these points of view
are inter-related for which the reader is recommended to see Schappacher [13,
Chap. 0] for a lucid summary. In this section we consider Hecke characters
from the point of view often taken in the cohomology of arithmetic groups. We
begin by setting up the context, adopting the treatment in [9, Chap. 2].

###  5.1. Locally symmetric spaces for ${\rm GL}_{1}/F$

Let $G={\rm Res}_{F/\mathbb{Q}}({\rm GL}_{1}/F)$. Then:

$G(\mathbb{Q})={\rm GL}_{1}(F)=F^{\times};$

$G(\mathbb{A})={\rm GL}_{1}(\mathbb{A}_{F})=\mathbb{I}_{F};$

$G(\mathbb{A}_{f})=\mathbb{A}_{F,f}^{\times}$ the group of finite idèles;

$G(\mathbb{R})={\rm
GL}_{1}(F\otimes\mathbb{R})=F_{\infty}^{\times}=\prod_{v\in
S_{r}}\mathbb{R}^{\times}\times\prod_{w\in S_{c}}\mathbb{C}^{\times};$

$G(\mathbb{R})^{\circ}=F_{\infty+}^{\times}$, the connected component of the
identity in $G(\mathbb{R});$

$K_{\infty}=\prod_{v\in S_{r}}\\{\pm 1\\}\times\prod_{w\in S_{c}}S^{1},$ the
maximal compact subgroup of $G(\mathbb{R});$

$K_{\infty}^{\circ}\simeq\prod_{w\in S_{c}}S^{1}$, the connected component of
the identity in $K_{\infty};$

$A={\rm GL}_{1}/\mathbb{Q}\hookrightarrow G/\mathbb{Q}$ is the canonical
inclusion in ${\rm Hom}({\rm GL}_{1}/\mathbb{Q},{\rm Res}_{F/\mathbb{Q}}({\rm
GL}_{1}/F));$

$A(\mathbb{R})$ is the diagonal copy of $\mathbb{R}^{\times}$ inside
$G(\mathbb{R})=\prod_{v\in S_{r}}\mathbb{R}^{\times}\times\prod_{w\in
S_{c}}\mathbb{C}^{\times};$

$A(\mathbb{R})^{\circ}=\mathbb{R}_{+}^{\times}$, the connected component of
the identity in $A(\mathbb{R});$

$\mathfrak{m}=\prod_{\mathfrak{p}}\mathfrak{p}^{m_{\mathfrak{p}}}$, an
integral ideal of $F$, which gives us a level structure, by which we mean:

$K_{\mathfrak{m}}\subset G(\mathbb{A}_{f}),$ an open compact subgroup:
$K_{\mathfrak{m}}:=\mathbb{U}_{F,f}(\mathfrak{m})=\prod_{\mathfrak{p}|\mathfrak{m}}(1+\mathfrak{p}^{m_{\mathfrak{p}}})\times\prod_{\mathfrak{p}\not\,|\mathfrak{m}}U_{\mathfrak{p}}.$

There are various quotients of $\mathbb{I}_{F}$ that one may consider
depending on how much to divide by inside $G(\mathbb{R})$. Here are five
possible situations:

(33a) $\displaystyle S^{G}_{\mathfrak{m}}$ $\displaystyle\ :=\
G(\mathbb{Q})\backslash G(\mathbb{A})/K_{\mathfrak{m}}.$ (33b) $\displaystyle
X^{G}_{\mathfrak{m}}$ $\displaystyle\ :=\ G(\mathbb{Q})\backslash
G(\mathbb{A})/A(\mathbb{R})^{\circ}K_{\mathfrak{m}}.$ (33c) $\displaystyle
Y^{G}_{\mathfrak{m}}$ $\displaystyle\ :=\ G(\mathbb{Q})\backslash
G(\mathbb{A})/K_{\infty}^{\circ}K_{\mathfrak{m}}.$ (33d) $\displaystyle
T^{G}_{\mathfrak{m}}$ $\displaystyle\ :=\ G(\mathbb{Q})\backslash
G(\mathbb{A})/A(\mathbb{R})^{\circ}K_{\infty}^{\circ}K_{\mathfrak{m}}.$ (33e)
$\displaystyle C^{G}_{\mathfrak{m}}$ $\displaystyle\ :=\
G(\mathbb{Q})\backslash G(\mathbb{A})/G(\mathbb{R})^{\circ}K_{\mathfrak{m}}.$

Of course, $G$ being abelian, the double-coset-spaces are all quotients of
$\mathbb{I}_{F}$; for example

$C^{G}_{\mathfrak{m}}\ =\
G(\mathbb{A})/(G(\mathbb{Q})G(\mathbb{R})^{\circ}K_{\mathfrak{m}})\ =\
\mathbb{I}_{F}/(F^{\times}\mathbb{U}_{F}(\mathfrak{m}))\ \approx\
\mathcal{C}l^{+}_{F}(\mathfrak{m}),$

the narrow ray class group modulo $\mathfrak{m}.$ Writing as a double-coset
space is simply to suggest how things would generalize if $G$ is taken to be a
general reductive group over $\mathbb{Q}$. There are the canonical projection
maps:

(34)
$\textstyle{S^{G}_{\mathfrak{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{G}_{\mathfrak{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y^{G}_{\mathfrak{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T^{G}_{\mathfrak{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{G}_{\mathfrak{m}}\approx\mathcal{C}l^{+}_{F}(\mathfrak{m})}$

The bottom most is a finite-abelian group being a narrow ray class group. To
get a feel for the space $S^{G}_{\mathfrak{m}}$ at the top, think of

$G(\mathbb{A})/K_{\mathfrak{m}}\ =\
G(\mathbb{R})\times(G(\mathbb{A}_{f})/K_{\mathfrak{m}}),$

as the product of $G(\mathbb{R})$ with the totally disconnected space
$G(\mathbb{A}_{f})/K_{\mathfrak{m}}$; on which the discrete subgroup
$G(\mathbb{Q})$–it is discrete in $G(\mathbb{A})$–acts diagonally, and taking
the quotient by that action we have
$G(\mathbb{Q})\backslash(G(\mathbb{R})\times(G(\mathbb{A}_{f})/K_{\mathfrak{m}})).$
Note that $G(\mathbb{Q})\backslash G(\mathbb{A}_{f})/K_{\mathfrak{m}}$ is a
finite set:

$G(\mathbb{Q})\backslash G(\mathbb{A}_{f})/K_{\mathfrak{m}}\ =\
\mathbb{A}_{F,f}^{\times}/(F^{\times}\mathbb{U}_{F,f}(\mathfrak{m}))\
\hookrightarrow\
\mathbb{I}_{F}/(F^{\times}(F_{\infty+}^{\times}\mathbb{U}_{F,f}(\mathfrak{m})))\
=\ \mathcal{C}l_{F}^{+}(\mathfrak{m}).$

There exists a finite set $\\{x_{1,f},\dots,x_{k,f}\\}$ of finite idèles such
that
$\mathbb{A}_{F,f}^{\times}=\cup_{i=1}^{k}x_{i,f}F^{\times}\mathbb{U}_{F,f}(\mathfrak{m}).$
Using these representatives a bijection

$S^{G}_{\mathfrak{m}}\ \simeq\
\coprod_{i=1}^{k}(F_{\infty}^{\times}/U_{F}^{1}(\mathfrak{m}))x_{i,f}$

can described as follows: for
$x=(x_{\infty},x_{f})\in\mathbb{I}_{F}=F_{\infty}^{\times}\times\mathbb{A}_{F,f}^{\times}$,
there is a unique $i$ such that $x_{f}=x_{i,f}\,\gamma\,u_{f}$ for some
$\gamma\in F^{\times}$ and $u_{f}\in\mathbb{U}_{F,f}(\mathfrak{m})$; the
bijection is given by:

$S^{G}_{\mathfrak{m}}\ni x(F^{\times}\mathbb{U}_{F,f}(\mathfrak{m}))\ \mapsto\
(\gamma^{-1}x_{\infty}U_{F}^{1}(\mathfrak{m}),x_{i,f})\in(F_{\infty}^{\times}/U_{F}^{1}(\mathfrak{m}))x_{i,f}.$

Let us describe the connected components of $S^{G}_{\mathfrak{m}}$, which are
all copies of

$\frac{F^{\times}_{\infty+}}{U_{F}^{1}(\mathfrak{m})}\ \simeq\
\frac{(\mathbb{R}^{\times}_{+})^{r_{1}}\times(\mathbb{C}^{\times})^{r_{2}}}{U_{F}^{1}(\mathfrak{m})}.$

Let ${}^{0}F^{\times}_{\infty+}$ denote the kernel of norm at infinity $|\\!|\
|\\!|_{\infty}:F^{\times}_{\infty+}\rightarrow\mathbb{R}^{\times}_{+}.$ Then

$F^{\times}_{\infty+}\ =\ {}^{0}F^{\times}_{\infty+}\times
A(\mathbb{R})^{\circ}.$

By the product formula, $F^{\times}_{\infty+}$ contains
$U_{F}^{1}(\mathfrak{m})$. Note that if $u\in U_{F}^{1}(\mathfrak{m})\cap
A(\mathbb{R})^{\circ}$ then $\sigma(u)=\sigma^{\prime}(u)$ for all
$\sigma,\sigma^{\prime}\in\Sigma_{F}$, hence $u\in\mathbb{Q}^{\times}$; whence
$u\in\\{\pm 1\\}.$ Assume the mild condition that
$U_{F}^{1}(\mathfrak{m})\cap\\{\pm 1\\}=\\{1\\}.$ Hence we get

$\frac{F^{\times}_{\infty+}}{U_{F}^{1}(\mathfrak{m})}\ =\
A(\mathbb{R})^{\circ}\times\frac{{}^{0}F^{\times}_{\infty+}}{U_{F}^{1}(\mathfrak{m})}.$

Next, under the identification
$F^{\times}_{\infty+}\simeq(\mathbb{R}^{\times}_{+})^{r_{1}}\times(\mathbb{C}^{\times})^{r_{2}}$,
after writing $\mathbb{C}^{\times}=\mathbb{R}^{\times}_{+}\times S^{1}$, and
noting that $K(\mathbb{R})^{\circ}=(S^{1})^{r_{2}}$ which is the product of
the $S^{1}$’s coming from the complex places, we have:

${}^{0}F^{\times}_{\infty+}\ \simeq\
K(\mathbb{R})^{\circ}\times{}^{0}(\mathbb{R}^{\times}_{+})^{r_{1}+r_{2}},$

where, ${}^{0}(\mathbb{R}^{\times}_{+})^{r_{1}+r_{2}}$ consists of tuples
$(t_{v})_{v\in S_{\infty}}$ such that $\sum_{v}{\rm log}|t_{v}|_{v}=0.$ Assume
furthermore that $\mathfrak{m}$ is deep enough so that
$U_{F}^{1}(\mathfrak{m})\cap K(\mathbb{R})^{\circ}$, which a priori is finite
being discrete and compact in $F^{\times}_{\infty+}$, is in fact trivial,
i.e., it amounts to assuming that $U_{F}^{1}(\mathfrak{m})$ has no roots of
unity. Then, by the proof of Dirichlet’s unit theorem using the Minkowski map
(see the notations in the proof of Lem. 17) one has:

$\frac{{}^{0}(\mathbb{R}^{\times}_{+})^{r_{1}+r_{2}}}{U_{F}^{1}(\mathfrak{m})}\approx\frac{\mathcal{H}\approx\mathbb{R}^{r_{1}+r_{2}-1}}{\Gamma_{F}^{1}(\mathfrak{m})}\approx(S^{1})^{r_{1}+r_{2}-1}.$

To summarise, any connected component of $S^{G}_{\mathfrak{m}}$, for
$\mathfrak{m}$ deep enough, is homeomorphic to

$A(\mathbb{R})^{\circ}\times
K(\mathbb{R})^{\circ}\times(S^{1})^{r_{1}+r_{2}-1},$

which in turn is isomorphic to
$\mathbb{R}^{\times}_{+}\times(S^{1})^{r_{1}+2r_{2}-1}.$ Note that

${\rm dim}(S^{G}_{\mathfrak{m}})=r_{1}+2r_{2}=d_{F}=[F:\mathbb{Q}].$

Similarly, the other spaces in (33) can be described. We can grant ourselves
the license to call any of these as a ‘locally symmetric space’ for ${\rm
GL}_{1}$ over $F$ with level $\mathfrak{m}.$

###  5.2. Sheaves on locally symmetric spaces

Take a field $E,$ which is assumed to be a finite Galois extension of
$\mathbb{Q}$ that takes a copy of $F$; hence ${\rm Hom}(F,E)={\rm
Hom}(F,\bar{E}).$ This field $E$ may be called the field of coefficients.
Consider an infinity type ${\bf n}=(n_{\tau})_{\tau:F\rightarrow E}$ with
$n_{\tau}\in\mathbb{Z};$ one may also denote this as ${\bf
n}\in\mathbb{Z}[{\rm Hom}(F,E)]$. This is like an infinity type from before,
except that it is now parametrised over ${\rm Hom}(F,E)$ instead of ${\rm
Hom}(F,\bar{\mathbb{Q}})$ as before. An embedding
$\iota:E\rightarrow\bar{\mathbb{Q}}$ gives an identification $\iota_{*}:{\rm
Hom}(F,E)\rightarrow{\rm Hom}(F,\bar{\mathbb{Q}})$ as
$\tau\mapsto\iota_{*}\tau=\iota\circ\tau.$ Such an ${\bf n}$ gives us an
algebraic representation $\vartheta_{\bf n}$ on a one-dimensional $E$-vector
space $\mathcal{M}_{{\bf n},E}$ denoted:

$\vartheta_{\bf n}:G\times E\ \longrightarrow\ {\rm GL}_{1}(\mathcal{M}_{{\bf
n},E}).$

Since $G\times E={\rm Res}_{F/\mathbb{Q}}({\rm GL}_{1}/F)\times
E=\prod_{\tau:F\rightarrow E}{\rm GL}_{1}\times_{F,\tau}E,$ and on the
component indexed by $\tau$, the representation is
$x\mapsto\vartheta_{n^{\tau}}(x):=x^{n_{\tau}}$ for $x\in{\rm GL}_{1}(E).$ For
$a\in G(\mathbb{Q})=F^{\times}$, we have

$\vartheta_{\bf n}(a)\ =\ \prod_{\tau:F\rightarrow
E}\vartheta_{n_{\tau}}(\tau(a))\ =\ \prod_{\tau:F\rightarrow
E}\tau(a)^{n_{\tau}}.$

Given an infinity type ${\bf n}\in\mathbb{Z}[{\rm Hom}(F,E)]$, and the
associated representation $(\vartheta_{\bf n},\mathcal{M}_{{\bf n},E})$, there
is a sheaf $\widetilde{\mathcal{M}}_{{\bf n},E}$ of $E$-vector spaces on
$S^{G}_{\mathfrak{m}}$ constructed as follows. Consider the canonical
projection map of dividing by $F^{\times}$:

$\pi\ :\ \mathbb{I}_{F}/\mathbb{U}_{F,f}(\mathfrak{m})\ \longrightarrow\
F^{\times}\backslash\mathbb{I}_{F}/\mathbb{U}_{F,f}(\mathfrak{m})=S^{G}_{\mathfrak{m}}.$

For any open subset $V\subset S^{G}_{\mathfrak{m}}$, define the sections of
$\widetilde{\mathcal{M}}_{{\bf n},E}$ over $V$ as the $E$-vector space:

(35) $\widetilde{\mathcal{M}}_{{\bf n},E}(V)\ =\
\\{s:\pi^{-1}(V)\rightarrow\mathcal{M}_{{\bf n},E}\ :\ s(ax)=\vartheta_{\bf
n}(a)s(x),\ \ \forall a\in F^{\times},\ \forall x\in\pi^{-1}(V)\\}.$

Checking the sheaf axioms is a routine exercise. The purity lemma (Lem. 17)
can be recast as the following lemma:

######  Lemma 36 (Purity).

Given ${\bf n}\in\mathbb{Z}[{\rm Hom}(F,E)]$, the sheaf
$\widetilde{\mathcal{M}}_{{\bf n},E}$ of $E$-vector spaces on
$S^{G}_{\mathfrak{m}}$ is non-zero if and only if: there exists ${\sf
w}\in\mathbb{Z}$ such that

  1. (i)

if $S_{r}\neq\emptyset$ then $n_{\tau}={\sf w}$ for all $\tau\in{\rm
Hom}(F,E),$ and

  2. (ii)

if $S_{r}=\emptyset$ then
$n_{\iota\circ\tau}+n_{\overline{\iota\circ\tau}}={\sf w}$ for all
$\tau\in{\rm Hom}(F,E)$ and for all $\iota\in{\rm Hom}(E,\bar{\mathbb{Q}}).$

###### Proof.

A sheaf is nonzero if and only if it admits a nonzero stalk. A stalk over a
point $y\in V$, consists of all $s:\pi^{-1}(y)\rightarrow\mathcal{M}_{{\bf
n},E}$, such that $s(ax)=\vartheta_{\bf n}(a)s(x),$ for all $a\in F^{\times}$
and all $x\in\pi^{-1}(y).$ This is possible if and only if $\vartheta_{\bf
n}(a)=1$ for all $a\in
F^{\times}\cap\mathbb{U}_{F,f}(\mathfrak{m})=U_{F}^{1}(\mathfrak{m})$. After
passing to $\bar{\mathbb{Q}}$ via an $\iota$, the rest is basically the same
as proof of Lem. 17. ∎

Analogously, one may construct sheaves on the other spaces in (33); one may
pull-back or push-forward the so constructed sheaves, and explicate the mutual
relations between them.

###  5.3. Cohomology of ${\rm GL}_{1}$ and algebraic Hecke characters of $F$
with coefficients in $E$

Given an integral ideal $\mathfrak{m}$, and an infinity type ${\bf
n}\in\mathbb{Z}[{\rm Hom}(F,E)]$ satisfying the purity conditions of Lem. 36,
a fundamental object of interest is

$H^{\bullet}(S^{G}_{\mathfrak{m}},\,\widetilde{\mathcal{M}}_{{\bf n},E}),$

the sheaf-cohomology of a locally symmetric space for ${\rm GL}_{1}/F$ with
coefficients in the sheaf $\widetilde{\mathcal{M}}_{{\bf n},E}.$ One may also
consider $H^{\bullet}_{c}(S^{G}_{\mathfrak{m}},\,\widetilde{\mathcal{M}}_{{\bf
n},E}),$ which is the cohomology with compact supports. It will take us way
too much of an effort without substantial dividends for the purposes of this
article to develop this systematically. Any reader who might feel this is
brewing heavy weather is recommended to look into Harder [6, § 2.5] on the
cohomology of tori, where the torus at hand is the diagonal torus in ${\rm
GL}_{2}$, which, up to the easily controlled centre, is basically the above
kind of cohomology group. This context naturally arises in the theory of
Eisenstein cohomology where the cohomology of the Levi subgroups (in
particular, tori) play a crucial role. The reader is referred to Harder’s
forthcoming book [7] where this point of view is treated in great detail.

Let us suffice here to note that an algebraic Hecke character $\chi$ of $F$
modulo $\mathfrak{m}$, taking values in the algebraic closure
$\bar{\mathbb{Q}}$ of $\mathbb{Q}$ inside $\mathbb{C}$, and with infinity type
${\bf n}$, contributes to
$H^{0}(S^{G}_{\mathfrak{m}},\,\widetilde{\mathcal{M}}_{{\bf
n},\bar{\mathbb{Q}}})$, and by duality also to the top-degree cohomology with
compact supports:
$H^{d_{F}}_{c}(S^{G}_{\mathfrak{m}},\,\widetilde{\mathcal{M}}_{{\bf
n},\bar{\mathbb{Q}}}).$ For arithmetic applications, it is better to work with
an arbitrary coefficient field $E$, from which one can then map into
$\bar{\mathbb{Q}}$ or $\mathbb{C}$; this necessitates clarifying the meaning
of an algebraic Hecke character of $F$ with coefficients in $E$.

The definition below follows Deligne [3, §5]; see also Schappacher [13, Chap.
0]. Whereas in [3] and [13] $E$ is any number field with no relation to $F$,
we will continue with our restriction that $E$ is a finite Galois extension of
$\mathbb{Q}$ that contains a copy of $F.$ Given an integral ideal
$\mathfrak{m}$ of $F$, and an infinity type ${\bf n}\in\mathbb{Z}[{\rm
Hom}(F,E)],$ define an algebraic Hecke character of $F$ with values in $E$ of
modulus $\mathfrak{m}$ and infinity type ${\bf n}$ as a homomorphism

$\chi:\ \mathbb{J}_{F}(\mathfrak{m})\ \rightarrow\ E^{\times}$

such that for any principal ideal $(a)\in\mathbb{P}^{+}_{F}(\mathfrak{m})$
(recall: $a\gg 0$ and $a\equiv 1\pmod{\mathfrak{m}}$) one has

$\chi((a))\ =\ \prod_{\tau\in{\rm Hom}(F,E)}\tau(a)^{n_{\tau}}.$

Note the similarity to Hecke’s definition of a Größencharakter mod
$\mathfrak{m}$, but with values in $E^{\times}$ instead of
$\mathbb{C}^{\times}$, and while asking for the character at infinity to be of
algebraic type. To see the relation with our previous treatment of algebraic
Hecke characters, following [13, §0.5], ‘localise’ over any place of $E$. We
will work with archimedean places, but note that as in loc. cit. we may also
go through the discussion below for any finite place which would give us
$\ell$-adic versions of algebraic Hecke characters.

Let $\chi$ be an algebraic Hecke character of $F$ with values in $E$ of
modulus
$\mathfrak{m}=\prod_{\mathfrak{p}|\mathfrak{m}}\mathfrak{p}^{m_{\mathfrak{p}}}$
and infinity type ${\bf n}.$ Then there is unique continuous homomorphism

(37) $\chi_{\mathbb{A}}:\mathbb{I}_{F}\ \rightarrow\ E^{\times},$

with the discrete topology on $E^{\times}$, such that

  1. (1)

$\chi_{\mathbb{A}}^{-1}(1)=\prod_{v\in
S_{\infty}}F_{v}^{\times}\prod_{\mathfrak{p}|\mathfrak{m}}(1+\mathfrak{p}^{m_{\mathfrak{p}}}\mathcal{O}_{p})\prod_{\mathfrak{p}\not\,|\,\mathfrak{m}}\mathcal{O}_{\mathfrak{p}}^{\times},$
which is open in $\mathbb{I}_{F}.$

  2. (2)

for $\mathfrak{p}\\!\not|\,\mathfrak{m}$, the local component
$\chi_{\mathfrak{p}}$ of $\chi_{\mathbb{A}}$ satisfies:
$\chi_{\mathfrak{p}}(\varpi_{\mathfrak{p}})=\chi(\mathfrak{p}).$

  3. (3)

$\chi_{\mathbb{A}}|_{F^{\times}}=\vartheta_{\bf n};$ (recall: $\vartheta_{\bf
n}(a)=\prod_{\tau:F\rightarrow E}\tau(a)^{n_{\tau}}$).

By weak approximation, $\mathbb{I}_{F}=F^{\times}\cdot(\prod_{v\in
S_{\infty}}F_{v}^{\times}\prod_{\mathfrak{p}|\mathfrak{m}}(1+\mathfrak{p}^{m_{\mathfrak{p}}}\mathcal{O}_{p})\prod_{\mathfrak{p}\not\,|\,\mathfrak{m}}F_{\mathfrak{p}}^{\times})$,
giving us the uniqueness of $\chi_{\mathbb{A}}.$ Let us note en passant that
such a $\chi_{\mathbb{A}}$ is an element of
$H^{0}(S^{G}_{\mathfrak{m}},\widetilde{\mathcal{M}}_{{\bf n},E}).$ Fix an
embedding $\iota:E\hookrightarrow\mathbb{C}$, and suppose it corresponds to
the archimedean place $v_{\iota}$ of $E$. The character $\vartheta_{\bf n}$ is
an element of ${\rm Hom}_{\mathbb{Q}-{\rm alg}}({\rm Res}_{F/\mathbb{Q}}({\rm
GL}_{1}/F),{\rm Res}_{E/\mathbb{Q}}({\rm GL}_{1}/E))$, and taking the
$\mathbb{A}$-points, one gets a continuous homomorphism $\vartheta_{{\bf
n},\mathbb{A}}:\mathbb{I}_{F}\rightarrow\mathbb{I}_{E}.$ Composing with the
projection $\mathbb{I}_{E}^{\times}\rightarrow
E_{v_{\iota}}^{\times}\simeq\mathbb{C}^{\times}$ gives:

(38) $\vartheta_{{\bf n},\iota}:\mathbb{I}_{F}\rightarrow\mathbb{C}^{\times}.$

Now, define a homomorphism $\mathbb{I}_{F}\rightarrow\mathbb{C}^{\times}$ as:

(39) ${}^{\iota}\chi\ :=\ (\iota\circ\chi_{\mathbb{A}})\cdot\vartheta_{{\bf
n},\iota}^{-1},$

Note that $\iota$ induces a bijection $\iota_{*}:{\rm Hom}(F,E)\rightarrow{\rm
Hom}(F,\mathbb{C})$ as $\iota_{*}(\tau)=\iota\circ\tau.$ The standing
hypothesis on $E$ being Galois and containing a copy of $F$ is used here. In
particular, there is a mapping of infinity types: $\mathbb{Z}[{\rm
Hom}(F,E)]\rightarrow\mathbb{Z}[{\rm Hom}(F,\mathbb{C})]$ as ${\bf
n}\mapsto{}^{\iota}{\bf n};$ for $\eta\in{\rm Hom}(E,\mathbb{C})$, the
$\eta$-component of ${}^{\iota}{\bf n}$ is given by: ${}^{\iota}{\bf
n}_{\eta}=n_{\iota_{*}^{-1}\eta}.$ Hence, $\iota\circ\vartheta_{\bf
n}=\vartheta_{{}^{\iota}{\bf n}}$, as homomorphisms from $F^{\times}$ to
$\mathbb{C}^{\times}$ since

$(\iota\circ\vartheta_{\bf n})(a)\ =\ \iota(\prod_{\tau\in{\rm
Hom}(F,E)}\tau(a)^{n_{\tau}})\ =\ \prod_{\tau\in{\rm
Hom}(F,E)}(\iota_{*}\tau(a))^{n_{\tau}}\ =\ \prod_{\eta\in{\rm
Hom}(F,\mathbb{C})}(\eta(a))^{n_{\iota_{*}^{-1}\eta}}.$

The properties of ${}^{\iota}\chi$ are summarised in the following proposition
the proof of which is now a routine check.

######  Proposition 40.

Let $\chi$ be an algebraic Hecke character of $F$ with values in $E$ of
modulus $\mathfrak{m}$ and infinity type ${\bf n}.$ For any embedding
$\iota:E\rightarrow\mathbb{C}$, the homomorphism ${}^{\iota}\chi$ satisfies
the following:

  1. (1)

${}^{\iota}\chi:\mathbb{I}_{F}/F^{\times}\rightarrow\mathbb{C}^{\times}$ is a
continuous homomorphism;

  2. (2)

For all $\mathfrak{p}\notin S_{\infty}$,
${}^{\iota}\chi_{\mathfrak{p}}=\iota\circ\chi_{\mathfrak{p}}$; recall that
$\chi_{\mathfrak{p}}$ is the local character of $\chi_{\mathbb{A}};$

  3. (3)

${}^{\iota}\chi_{\infty}=\vartheta_{{\bf
n},\iota}^{-1}|_{F_{\infty}^{\times}}$ is completely determined by its values
on $F^{\times}$ embedded diagonally in $F_{\infty}^{\times},$ on which we have
$\vartheta_{{\bf n},\iota}^{-1}|_{F^{\times}}=\iota\circ\vartheta_{\bf
n}^{-1}=\vartheta_{-{}^{\iota}{\bf n}};$

i.e, ${}^{\iota}\chi$ is a Hecke character of $F$ of modulus $\mathfrak{m}$
and infinity type $-{}^{\iota}{\bf n}.$

Furthermore, for $k\in\mathbb{Z}$, the Tate-twist
${}^{\iota}\chi(k):={}^{\iota}\chi\otimes|\\!|\ |\\!|^{k}$, has infinity type
$-{}^{\iota}{\bf n}+k.$

We may say that ${}^{\iota}\chi$ is an automorphic representation of ${\rm
GL}_{1}/F$ of cohomological type. We are interested in the special values of
the Hecke $L$-function attached to ${}^{\iota}\chi$.

##  6\. Ratios of critical values of Hecke $L$-functions

Let $\chi$ be an algebraic Hecke character of $F$ with values in $E$ of
modulus $\mathfrak{m}$ and infinity type ${\bf n}=\sum_{\tau:F\rightarrow
E}n_{\tau}\tau.$ For any embedding $\iota:E\rightarrow\mathbb{C}$, consider
the algebraic Hecke character ${}^{\iota}\chi$ as in Prop. 40. The rest of
this article concerns the special values of the $\mathbb{C}$-valued
$L$-function:

$L(s,\iota,\chi)\ :=\ L(s,{}^{\iota}\chi),$

the right hand side is the usual Hecke $L$-function; see 4.2.1. It is
sometimes convenient to consider the $E\otimes\mathbb{C}$-valued $L$-function:

$\mathbb{L}(s,\chi)\ :=\
\\{L(s,\iota,\chi)\\}_{\iota:E\rightarrow\mathbb{C}},$

where $E\otimes\mathbb{C}$ is identified with
$\prod_{\iota:E\rightarrow\mathbb{C}}\mathbb{C}.$ From Prop. 23 and Lem. 49
below it follows that the critical set of $L(s,{}^{\iota}\chi)$ is independent
of $\iota$; one may therefore allude to the critical set for
$\mathbb{L}(s,\chi).$ Furthermore, we are interested in the rationality
results for the ratios of successive critical $L$-values that arise naturally
in the theory of Eisenstein cohomology as in Harder [6]. Henceforth, we assume
that $F$ is a totally imaginary number field, in which case it follows from
Prop. 23 that if the width $\ell$ of ${}^{\iota}\chi$ satisfies $\ell\geq 2$,
then we are guaranteed the existence of two consecutive integers that are
critical for $\mathbb{L}(s,\chi).$

###  6.1. A result of Harder on ratios of critical $L$-values

The relative discriminant $\delta_{F/K}$ of any extension $F/K$ of number
fields is defined as: suppose ${\rm
Hom}(F,\bar{K})=\\{\sigma_{1},\dots,\sigma_{r}\\}$, and
$\\{\omega_{1},\dots,\omega_{r}\\}$ is a $K$-basis of $F$, then define
$\delta_{F/K}\ :=\ \det([\sigma_{i}(\omega_{j})])^{2}$; taking the square
makes it invariant under ${\rm Gal}(\bar{K}/K)$, and hence $\delta_{F/K}$ is
in $K^{\times}$, and as an element of $K^{\times}/K^{\times}{}^{2}$ it is
independent of the choice of the basis $\\{\omega_{j}\\}$. The absolute
discriminant of $F$ is $\delta_{F/\mathbb{Q}}.$ Harder proves in [6, Cor.
(4.2.2)] that if $m$ and $m+1$ are critical for $\mathbb{L}(s,\chi),$ then the
number

(41) $C({}^{\iota}\chi)\ :=\
|\delta_{F/\mathbb{Q}}|^{1/2}\frac{L(m,{}^{\iota}\chi)}{L(m+1,{}^{\iota}\chi)}$

is in $\bar{\mathbb{Q}}$, and for any $\varsigma\in{\rm
Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$, one has

(42) $\varsigma(C({}^{\iota}\chi))\ =\ C({}^{\varsigma\circ\iota}\chi).$

Let us note that we work with the completed $L$-function, and not with just
the finite part of the $L$-function as in loc. cit. In particular, the power
of $\bm{\uppi}$ in [6, Cor. (4.2.2)] is subsumed by the ratio of archimedean
local-factors. Also, the result is stated therein for $m=-1$; for a general
critical integer $m$ (with $m+1$ also critical) follows by taking Tate-twists
of $\chi.$

Note that (41) is asserting that the ratio of $L$-values therein is algebraic,
since $|\delta_{F/\mathbb{Q}}|^{1/2}$ is algebraic; but if we only look at the
ratios of $L$-values then the reciprocity law becomes obscured. At any rate,
the algebraicity result as stated in (41) together with the reciprocity law in
(42) gives that:

(43)
$|\delta_{F/\mathbb{Q}}|^{1/2}\frac{L(m,{}^{\iota}\chi)}{L(m+1,{}^{\iota}\chi)}\
\in\iota(E).$

We contend that whereas just the algebraicity result in (41) is correct, but
restated as the stronger statement in (43) via the reciprocity law it is not
correct as it stands by producing an example when $F$ is a totally imaginary
field that is not of CM type. The reason turns out to be, and this is the
raison d’être for writing this article, that the reciprocity law in (42) needs
to be modified by a sign; this sign is rather complicated, and turns out to be
trivial when $F$ is a CM field, and can be nontrivial for a general totally
imaginary field. We will first discuss this example in Sect. 6.2 that says
that (43) is not stable under base-change making it incorrect as it stands in
general. In Sect. 6.3, we will see how to rectify the reciprocity law in (42)
using, interestingly enough, some other passages in [6] itself.

###  6.2. An example: stability under base-change

Consider the fields in Example 30: $F_{0}=\mathbb{Q}$, and put
$F_{1}=\mathbb{Q}(\mathfrak{i})$ and
$F=\mathbb{Q}(\mathfrak{i},\sqrt{4+\mathfrak{i}}).$ To emphasize, this $F$ is
totally imaginary but not CM, and $F_{1}$ is a CM subfield. Changing notations
mildly to make it more suggestive we denote:

$\textstyle{F=\mathbb{Q}(\mathfrak{i},\sqrt{4+\mathfrak{i}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{\rm
cm}=\mathbb{Q}(\mathfrak{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{+}=\mathbb{Q}}$

Let us compute $\delta_{F/\mathbb{Q}}.$ First of all,

$\delta_{F/F^{\rm cm}}\ =\
\delta_{\mathbb{Q}(\mathfrak{i},\sqrt{4+\mathfrak{i}})/\mathbb{Q}(\mathfrak{i})}\
=\ 4(4+\mathfrak{i}),$

computed with respect to the basis $\\{1,\sqrt{4+\mathfrak{i}}\\}$; hence:

$N_{F^{\rm cm}/\mathbb{Q}}(\delta_{F/F^{\rm cm}})\ =\
N_{\mathbb{Q}(\mathfrak{i})/\mathbb{Q}}(4(4+\mathfrak{i}))\ =\ 16\cdot 17.$

Also, using the basis $\\{1,\mathfrak{i}\\}$ for $\mathbb{Q}(\mathfrak{i})$,
we have:

$\delta_{F^{\rm
cm}/\mathbb{Q}}=\delta_{\mathbb{Q}(\mathfrak{i})/\mathbb{Q}}=-4.$

Using the well-known formula for the discriminant of a tower of fields, we
get:

$\delta_{F/\mathbb{Q}}\ =\ \delta_{F^{\rm cm}/\mathbb{Q}}^{[F:F^{\rm
cm}]}\cdot N_{F^{\rm cm}/\mathbb{Q}}(\delta_{F/F^{\rm cm}})\ =\ (-4)^{2}\cdot
16\cdot 17\ =\ 2^{8}\cdot 17.$

Let $\psi$ be an algebraic Hecke character of $F^{\rm
cm}=\mathbb{Q}(\mathfrak{i})$; as a continuous homomorphism
$\psi:\mathbb{A}_{\mathbb{Q}(\mathfrak{i})}^{\times}/\mathbb{Q}(\mathfrak{i})^{\times}\rightarrow\mathbb{C}^{\times}$,
with the character at infinity,
$\psi_{\infty}:\mathbb{C}^{\times}\rightarrow\mathbb{C}^{\times},$ being of
the form $\psi_{\infty}(z)=z^{a}\bar{z}^{b},$ for $a,b\in\mathbb{Z}$. We may
take (without loss of much generality) that $a\geq b$. The critical set for
$L(s,\psi)$ is given by:

${\rm Crit}(L(s,\psi))\ =\ \\{1-a,\,2-a,\,\dots,\,-b\\}.$

Let us further assume that $a-b\geq 2$ so that there are at least two critical
points; let $m,m+1\in{\rm Crit}(L(s,\psi)).$ Then, (43) will say that

$|\delta_{F^{\rm cm}/\mathbb{Q}}|^{1/2}\ \frac{L(m,\psi)}{L(m+1,\psi)}\
\in\mathbb{Q}(\psi),$

where $\mathbb{Q}(\psi)$ is the number field generated by the values of the
finite part $\psi_{f}$ of $\psi.$ (We could have set up the context so that
$\psi$ takes values in a coefficient field $E$, and then used an
$\iota:E\rightarrow\mathbb{C}$ and to assert that the left hand side, for
$L(s,\iota,\psi)$ in place of $L(s,\psi),$ lies in $\iota(E).$) Since
$|\delta_{F^{\rm cm}/\mathbb{Q}}|^{1/2}=|-4|^{1/2}\in\mathbb{Q}^{\times}$ we
have:

(44) $\frac{L(m,\psi)}{L(m+1,\psi)}\ \in\ \mathbb{Q}(\psi).$

Suppose $\omega$ is a quadratic Dirichlet character of $F^{\rm cm}$, then
$\omega_{\infty}$ is trivial, and (44) applied to $\psi\omega$ will give:

(45) $\frac{L(m,\psi\omega)}{L(m+1,\psi\omega)}\ \in\ \mathbb{Q}(\psi\omega)\
=\ \mathbb{Q}(\psi).$

Now let $\chi$ be the base-change of $\psi$ to give a Hecke character of $F$:

$\chi\ :=\ \psi\circ N_{F/F^{\rm cm}}.$

Note that $F_{\infty}=\mathbb{C}\times\mathbb{C}$, $F^{\rm
cm}_{\infty}=\mathbb{C}$, and $N_{F_{\infty}/F^{\rm
cm}_{\infty}}(z_{1},z_{2})=z_{1}z_{2}$. Hence:
$\chi_{\infty}(z_{1},z_{2})=(z_{1}z_{2})^{a}\overline{(z_{1}z_{2})}^{b}=z_{1}^{a}z_{2}^{a}\bar{z_{1}}^{b}\bar{z_{2}}^{b},$
which is to say $\chi_{\infty}=\psi_{\infty}\otimes\psi_{\infty}$, whence
$L_{\infty}(s,\chi)=L_{\infty}(s,\psi)^{2}$, from which we deduce:

${\rm Crit}(L(s,\chi))\ =\ {\rm Crit}(L(s,\psi)).$

Also, a basic property of base-change is that:

(46) $L(s,\chi)\ =\ L(s,\psi)L(s,\psi\omega),$

where $\omega=\omega_{F/F^{\rm cm}}$ is the quadratic character of $F^{\rm
cm}$ attached by class field theory to the quadratic extension $F/F^{\rm cm}.$
From (46), (44), and (45) we deduce:

(47) $\frac{L(m,\chi)}{L(m+1,\chi)}\ \in\ \mathbb{Q}(\psi)=\mathbb{Q}(\chi).$

But, on the other hand, one may directly apply (43) to $L(s,\chi)$ to get:

$|\delta_{F/\mathbb{Q}}|^{1/2}\ \frac{L(m,\chi)}{L(m+1,\chi)}\
\in\mathbb{Q}(\chi).$

For the discriminant term: $|\delta_{F/\mathbb{Q}}|^{1/2}=(2^{8}\cdot
17)^{1/2}=\sqrt{17}\pmod{\mathbb{Q}^{\times}}.$ Hence:

(48) $\sqrt{17}\,\frac{L(m,\chi)}{L(m+1,\chi)}\ \in\ \mathbb{Q}(\chi).$

Of course, (47) and (48) are contradictory, because in general
$\sqrt{17}\notin\mathbb{Q}(\chi).$ We may say that that the formal statement
in (43) is not stable under base-change. The reader can readily appreciate
that the above example can be generalized.

###  6.3. Variations on a result of Harder

In this subsection we state the main theorem of this article; see Theorem 56.
Before doing so, we need to introduce notions and notations that lead to a
delicate signature that appears in that theorem. For the rest of this
subsection we will let $F$ be a totally imaginary number field, and $E$ a
number field that is Galois over $\mathbb{Q}$ and contains a copy of $F$. Let
${\bf n}=\sum_{\tau}n_{\tau}\tau\in\mathbb{Z}[{\rm Hom}(F,E)]$ be an infinity
type satisfying the purity condition of Lem. 36, with purity weight ${\sf w}.$
Suppose $\chi$ is an algebraic Hecke character of $F$ with values in $E$ of
infinity type ${\bf n}.$ Take any embedding $\iota:E\rightarrow\mathbb{C},$
and let ${}^{\iota}\chi$ be the character as in Prop. 40.

Recall from Prop. 23 that the width of ${\bf n},$ for a given $\iota,$ is
defined as:

$\ell({}^{\iota}\chi)\ =\ \ell({}^{\iota}{\bf n})\ :=\ \min_{\tau:F\rightarrow
E}\\{|n_{\iota\circ\tau}-n_{\overline{\iota\circ\tau}}|\\}.$

The Hecke $L$-function $L(s,{}^{\iota}\chi)$ has $\ell({}^{\iota}\chi)$ many
critical points. Note the following simple but important lemma:

######  Lemma 49.

$\ell({}^{\iota}{\bf n})$ is independent of $\iota$, and depends only on the
pure infinity type ${\bf n}.$

###### Proof.

If $F$ has no CM subfield ($F_{1}=F_{0}$ is totally real), then from Prop. 27
is follows that $\ell({}^{\iota}{\bf n})=0,$ which proves the lemma; but this
case is not interesting because we would like to have critical points. Assume
henceforth that $F$ is totally imaginary field that contains a (maximal) CM
subfield $F_{1},$ in which case ${\bf n}$ is the base-change from a pure
infinity type ${\bf m}$ over $F_{1}$. It is clear that $\ell({}^{\iota}{\bf
n})=\ell({}^{\iota}{\bf m}).$ Suppose ${\rm
Hom}(F_{1},E)=\\{\tau_{1},\tau_{1}^{\prime},\dots,\tau_{k},\tau_{k}^{\prime}\\}$,
where $2k=[F_{1}:\mathbb{Q}]$, and $\tau_{j}$ and $\tau_{j}^{\prime}$ have the
same restriction to the maximal totally real subfield $F_{0}$ of $F_{1}$. Then
it follows from (the proof of) Lem. 24 that: $\ell({}^{\iota}{\bf m})\ =\
\min_{j}\\{|m_{\tau_{j}}-m_{\tau_{j}^{\prime}}|\\}.$ ∎

The above lemma justifies the notation $\ell=\ell({\bf n}).$

####  6.3.1. A combinatorial lemma

The following is a special case of a general ‘combinatorial lemma’ that is at
the heart of what makes the theory of Eisenstein cohomology work so well for
studying the special values to automorphic $L$-functions.

######  Lemma 50.

The following three conditions are equivalent:

  1. (1)

$s=-1$ and $s=0$ are critical for $L(s,{}^{\iota}\chi).$

  2. (2)

$-\ell\leq{\sf w}\leq-4+\ell.$

  3. (3)

For each $\tau:F\rightarrow E$ and $\iota:E\rightarrow\mathbb{C}$, there
exists $w_{\iota\circ\tau}$ in the Weyl group of ${\rm GL}(2)$ such that

    1. (a)

$l(w_{\iota\circ\tau})+l(w_{\overline{\iota\circ\tau}})=1,$ and

    2. (b)

$w_{\iota\circ\tau}\cdot\left(\begin{smallmatrix}n_{\iota\circ\tau}&\\\
&0\end{smallmatrix}\right)$ is dominant.

The reader is referred to [12, Sect. 3.2] for a proof; to compare Statement
(2) above to the corresponding statement in [12, Lem. 3.16], note that $N$,
$\ell(\mu,\mu^{\prime})$, and $a(\mu,\mu^{\prime})$ of [12] take the values
$2$, $\ell$, and ${\sf w}/2,$ respectively in this article; Statement (3) and
its consequences need to be elaborated upon. Given $n_{1},n_{2}\in\mathbb{Z}$,
by $\left(\begin{smallmatrix}n_{1}&\\\ &n_{2}\end{smallmatrix}\right)$ one
means the algebraic character of the diagonal torus of ${\rm GL}(2)$ given by
${\rm diag}(t_{1},t_{2})\mapsto t_{1}^{n_{1}}t_{2}^{n_{2}}.$ Such a character
is said to be dominant if $n_{1}\geq n_{2}$ (where the choice of the Borel
subgroup $B$ is taken to be the upper-triangular matrices in ${\rm GL}(2)$).
The Weyl group $W$ of ${\rm GL}(2)$, which has order $2$, acts on this torus
and hence on its characters. The twisted action of the nontrival element
$w_{0}\in W$ on such a character is explicitly given by:
$w_{0}\cdot\left(\begin{smallmatrix}n_{1}&\\\
&n_{2}\end{smallmatrix}\right)=\left(\begin{smallmatrix}n_{2}-1&\\\
&n_{1}+1\end{smallmatrix}\right).$ Of course the length $l(w)=1$ if $w=w_{0}$,
and $l(w)=0$ if $w$ is the trivial element. Now the statement in (3) makes
sense. Furthermore, in each pair
$\\{\iota\circ\tau,\,\overline{\iota\circ\tau}\\}$ of conjugate embeddings
$\Sigma_{F}$, by (a), exactly one of the Weyl group elements is nontrivial and
so having length $1$, and the other is trivial; this defines a CM-type for $F$
(recall that by a CM-type $\Phi$ for $F$ one means $\Phi\subset\Sigma_{F}$
such that $\Sigma_{F}=\Phi\amalg\bar{\Phi}$):

(51) $\Phi({\bf n},\iota)\ :=\ \left\\{\eta\in\Sigma_{F}\ :\
l(w_{\eta})=1\right\\},$

and a bijection $\beta_{{\bf n},\iota}:\Phi({\bf n},\iota)\rightarrow
S_{\infty}$ with the set of archimedean places of $F.$ Statement (3) also
implies:

(52) $\eta\in\Phi({\bf n},\iota)\implies n_{\eta}\leq-2,\quad{\rm
and}\quad\eta\notin\Phi({\bf n},\iota)\implies n_{\eta}\geq 0.$

For each $\eta\in\Sigma_{F}$ we have ($n_{\eta}\leq-2$ and $n_{\bar{\eta}}\geq
0$) or ($n_{\eta}\geq 0$ and $n_{\bar{\eta}}\leq-2$).

Let us digress for a moment: what condition (3) really says is that the
automorphic representation ${}^{\rm a}{\rm Ind}_{B(\mathbb{A}_{F})}^{{\rm
GL}_{2}(\mathbb{A}_{F})}({}^{\iota}\chi,1\\!\\!1),$ that is algebraically and
parabolically induced from the Borel subgroup, contributes to the cohomology
in degree $[F:\mathbb{Q}]/2$ of the Borel–Serre boundary $\partial_{B}S^{G}$
of a locally symmetric space $S^{G}$ for $G={\rm GL}(2)/F$. This is the
starting point of using Eisenstein cohomology for ${\rm GL}(2)/F$ to deduce
results on the special values of $L$-functions attached to algebraic Hecke
characters that was pioneered by Harder in [6]. In that paper, Harder proves,
under the conditions of Lem. 50, a rationality result for the ratio of
critical values $L(-1,{}^{\iota}\chi)/L(0,{}^{\iota}\chi),$ from which an
analogous rationality result follows for the ratios of other successive pair
of critical values by allowing ourselves Tate-twists of $\chi$; Statement (2)
bounds the possible twists we can make, and it is a striking aspect of the
lemma that one so obtains all successive pairs of critical values, no more and
no less! Let me note that in [6] such a lemma and its consequences are
implicit; the explicit version of the above lemma and its philosophical
content came out in my work with Harder [9]; that was further developed in my
sequel [12]–especially, see paragraph 5.3.2.2 in [12].

####  6.3.2. A tale of two signatures

Fix an ordering on the set $S_{\infty}$ of archimedean places of $F$; say
$S_{\infty}=\\{w_{1},\dots,w_{r}\\},$ where $r=d_{F}/2=[F:\mathbb{Q}]/2.$ For
each $\varsigma\in{\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ define a permutation
$\pi_{{\bf n},\iota}(\varsigma)$ of $S_{\infty}$ by the diagram:

$\textstyle{\Phi({\bf
n},\iota)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{{\bf
n},\iota}}$$\scriptstyle{\varsigma\circ-}$$\textstyle{S_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{{\bf
n},\iota}(\varsigma)}$$\textstyle{\Phi({\bf
n},\varsigma\circ\iota)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{{\bf
n},\varsigma\circ\iota}}$$\textstyle{S_{\infty},}$

where the left vertical arrow needs an explanation: the $\iota$-independence
of (1) and/or (2) of Lem. 50 translates to $\iota$-independence of (3) also,
whence if $\iota\circ\tau\in\Phi({\bf n},\iota)$, then
$\varsigma\circ(\iota\circ\tau)\in\Phi({\bf n},\varsigma\circ\iota).$ Put in
other words, in general, a Galois conjugate of a CM-type of a totally
imaginary field need not be a CM-type; however, since ${\bf n}$ is a base-
change of an infinity-type ${\bf m}$ from $F_{1}$ (the key ingredient that
went into $\iota$-independence) we see that the CM-type $\Phi({\bf n},\iota)$
for $F$ is deduced from the CM-type $\Phi({\bf m},\iota)$ for $F_{1}$ under
the canonical map $\Sigma_{F}\rightarrow\Sigma_{F_{1}};$ and for a CM-field
such as $F_{1}$, a Galois conjugate of a CM-type is a CM-type; hence we deduce
that $(\varsigma\circ-)(\Phi({\bf n},\iota))=\Phi({\bf
n},\varsigma\circ\iota)$ is a CM-type. The sign of the permutation $\pi_{{\bf
n},\iota}(\varsigma)$ defines the signature:

(53) $\varepsilon_{{\bf n},\iota}(\varsigma)\ :=\ {\rm sgn}(\pi_{{\bf
n},\iota}(\varsigma)).$

Let us digress for another moment: the key idea in the theory of Eisenstein
cohomology is to see the map induced in cohomology by the standard
intertwining operator:

$T_{\rm st}(s)|_{s=-1}\ :\ {}^{\rm a}{\rm Ind}_{B(\mathbb{A}_{F})}^{{\rm
GL}_{2}(\mathbb{A}_{F})}({}^{\iota}\chi,1\\!\\!1)\ \longrightarrow\ {}^{\rm
a}{\rm Ind}_{B(\mathbb{A}_{F})}^{{\rm
GL}_{2}(\mathbb{A}_{F})}(1\\!\\!1(1),{}^{\iota}\chi(-1))$

is realised in terms of the restriction map from total cohomology of $S^{G}$
to the cohomology of the isotypic component cut out by these induced
representations in the cohomology of the Borel–Serre boundary
$\partial_{B}S^{G}$. (See [9, pp. 82–83].) The infinity type of the inducing
representation ${}^{\iota}\chi\otimes 1\\!\\!1$ in the domain of $T_{\rm
st}(s)|_{s=-1}$ corresponds to the weight ${}^{\iota}\underline{\bf
n}:=\left(\begin{smallmatrix}{}^{\iota}{\bf n}&\\\
&0\end{smallmatrix}\right)$; similarly the infinity type of the inducing
representation $1\\!\\!1(1)\otimes{}^{\iota}\chi(-1)$ for the codmain is the
weight ${}^{\iota}\tilde{\underline{\bf n}}:=\left(\begin{smallmatrix}-1&\\\
&{}^{\iota}{\bf n}+1\end{smallmatrix}\right)$.

Complementary to $\Phi({\bf n},\iota)$, we define the CM-type

(54) $\Phi(\tilde{\bf n},\iota)\ :=\ \left\\{\eta\in\Sigma_{F}\ :\
l(w_{\eta})=0\right\\}\ =\ \left\\{\eta\in\Sigma_{F}\ :n_{\eta}\geq
0\right\\}.$

Note that $\Phi({\bf n},\iota)=\overline{\Phi}(\tilde{\bf n},\iota),$ and the
bijection $\beta_{\tilde{\bf n},\iota}:\Phi(\tilde{\bf n},\iota)\rightarrow
S_{\infty}$ is the complex-conjugate of $\beta_{{\bf n},\iota}.$ For each
$\varsigma\in{\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ define a permutation
$\pi_{\tilde{\bf n},\iota}(\varsigma)$ of $S_{\infty}$ by the diagram:

$\textstyle{\Phi(\tilde{\bf
n},\iota)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{\tilde{\bf
n},\iota}}$$\scriptstyle{\varsigma\circ-}$$\textstyle{S_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\tilde{\bf
n},\iota}(\varsigma)}$$\textstyle{\Phi(\tilde{\bf
n},\varsigma\circ\iota)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{\tilde{\bf
n},\varsigma\circ\iota}}$$\textstyle{S_{\infty},}$

the sign of which defines the signature

(55) $\varepsilon_{\tilde{\bf n},\iota}(\varsigma)\ :=\ {\rm
sgn}(\pi_{\tilde{\bf n},\iota}(\varsigma)).$

####  6.3.3. The main theorem on $L$-values

We are now ready to state the main theorem which is a variation of a result
due to Harder:

######  Theorem 56.

Let $F$ be a totally imaginary number field, and $E$ a number field that is
Galois over $\mathbb{Q}$ and containing a copy of $F$. Let ${\bf
n}=\sum_{\tau}n_{\tau}\tau\in\mathbb{Z}[{\rm Hom}(F,E)]$ be an infinity type
satisfying the purity condition of Lem. 36, with purity weight ${\sf w}.$
Suppose $\chi$ is an algebraic Hecke character of $F$ with values in $E$ of
infinity type ${\bf n}.$ Take any embedding $\iota:E\rightarrow\mathbb{C},$
and let ${}^{\iota}\chi$ be the character as in Prop. 40. Assume the condition
on ${\bf n}$ from Lem. 50:

$-\ell\leq{\sf w}\leq-4+\ell.$

Hence $\ell\geq 2$ and ${\bf n}$ is the base-change to $F$ from an infinity-
type of a maximal CM-subfield of $F$. Suppose $m,m+1\in{\rm
Crit}(\mathbb{L}(s,\chi));$ see Prop. 23. Then:

(57)
$|\delta_{F/\mathbb{Q}}|^{1/2}\frac{L(m,{}^{\iota}\chi)}{L(m+1,{}^{\iota}\chi)}\
\in\iota(E),$

and, furthermore, for every $\varsigma\in{\rm
Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ we have the reciprocity law:

(58)
$\varsigma\left(|\delta_{F/\mathbb{Q}}|^{1/2}\frac{L(m,{}^{\iota}\chi)}{L(m+1,{}^{\iota}\chi)}\right)\
=\ \varepsilon_{{\bf n},\iota}(\varsigma)\cdot\varepsilon_{\tilde{\bf
n},\iota}(\varsigma)\cdot|\delta_{F/\mathbb{Q}}|^{1/2}\frac{L(m,{}^{\varsigma\circ\iota}\chi)}{L(m+1,{}^{\varsigma\circ\iota}\chi)}.$

####  6.3.4. Remarks on the proof

The above theorem is a special case of a more general theorem on
Rankin–Selberg $L$-functions for ${\rm GL}(n)\times{\rm GL}(n^{\prime})$ over
a totally imaginary field; see [12, Thm. 5.16, (1), (2)]. When $F$ is a CM
field, one has
$\varsigma\circ\mathfrak{c}\circ\eta=\mathfrak{c}\circ\varsigma\circ\eta$ (see
Lem. 24), hence, $\pi_{{\bf n},\iota}(\varsigma)=\pi_{\tilde{\bf
n},\iota}(\varsigma)$ an equality of the permutations of $S_{\infty}$; whence,
$\varepsilon_{{\bf n},\iota}(\varsigma)=\varepsilon_{\tilde{\bf
n},\iota}(\varsigma).$ In particular, the reciprocity law in (58) is the same
as (42). As will be seen below this is no longer the case for a general total
imaginary field; even though one has an equality of sets
$\mathfrak{c}(\Phi({\bf
n},\varsigma\circ\iota))=\varsigma(\mathfrak{c}(\Phi({\bf n},\iota))),$ the
permutations $\pi_{{\bf n},\iota}(\varsigma)$ and $\pi_{\tilde{\bf
n},\iota}(\varsigma)$ can be distinct, and the signature $\varepsilon_{{\bf
n},\iota}(\varsigma)\cdot\varepsilon_{\tilde{\bf n},\iota}(\varsigma)$ can be
nontrivial. (This is in essence the explanation of the counterexample in Sect.
6.2.) Even though Thm. 56 and especially (58), for a general totally imaginary
field, are not stated like this in Harder [6], its proof follows by putting
together the proof of [6, Cor. 4.2.2] amplified by the discussion in [6, Sect.
2.4] of a rational system of generators for the unipotent cohomology of the
coefficient system on ${\rm GL}(2).$ The signature $\varepsilon_{{\bf
n},\iota}(\varsigma)$ is described in [6, (2.4.1)].

###  6.4. Compatibility with Deligne’s Conjecture

In this subsection we recall Deligne’s celebrated conjecture on the special
values of motivic $L$-functions, as applied to ratios of successive critical
values, and also recall a result of Deligne on the ratios of motivic periods
that gives a motivic explanation to the ratio of successive critical
$L$-values for a Hecke $L$-function.

####  6.4.1. Statement of Deligne’s Conjecture

We will freely use the notation of [4]; a motive $M$ over $\mathbb{Q}$ with
coefficients in a field $E$ will be thought in terms of its Betti, de Rham,
and $\ell$-adic realizations. For a critical motive $M$ we have its periods
$c^{\pm}(M)\in(E\otimes\mathbb{C})^{\times}$ as in loc. cit., that are well-
defined in $(E\otimes\mathbb{C})^{\times}/E^{\times}.$ We begin with a
relation between the two periods over a totally imaginary base field $F$;
recall that $F_{0}$ is the largest totally real subfield of $F$; then $F_{0}$
admits at most one totally imaginary quadratic extension contained inside $F$;
if $F$ does admit such a CM subfield, we denote it as $F_{1}$; and if there is
no such extension inside $F$, then we put $F_{1}=F_{0}.$ If $F_{1}$ is indeed
a CM field, and suppose $F_{1}=F_{0}(\sqrt{D})$ for a totally negative $D\in
F_{0}$, then define

$\Delta_{F_{1}}\ :=\ \sqrt{N_{F_{0}/\mathbb{Q}}(D)},\quad\Delta_{F}\ :=\
\Delta_{F_{1}}^{[F:F_{1}]}.$

If $F_{1}=F_{0}$ is totally real, then define

$\Delta_{F_{1}}\ :=\ 1,\quad\Delta_{F}\ :=\ \Delta_{F_{1}}^{[F:F_{1}]}=1.$

The following proposition is proved in [5]:

######  Proposition 59.

Let $M_{0}$ be a pure motive of rank $1$ over a totally imaginary number field
$F$ with coefficients in a number field $E$. Put $M={\rm
Res}_{F/\mathbb{Q}}(M_{0}),$ and suppose that $M$ has no middle Hodge type.
Let $c^{\pm}(M)$ be the periods defined in [4]. Then

$\frac{c^{+}(M)}{c^{-}(M)}\ =\ 1\otimes\Delta_{F},\quad\mbox{in
$(E\otimes\mathbb{C})^{\times}/E^{\times}$.}$

Let us also note that $1\otimes\Delta_{F}$ is $\pm 1$ in each component of
$(E\otimes\mathbb{C})^{\times}/E^{\times}$, since its square is trivial. Based
on Prop. 59, Deligne’s conjecture [4] for the ratios of successive critical
values of the completed $L$-function of a rank-one motive may be stated as:

######  Conjecture 60 (Deligne).

Let $M_{0}$ be a pure motive of rank $1$ over a totally imaginary $F$ with
coefficients in $E$. Put $M={\rm Res}_{F/\mathbb{Q}}(M_{0}),$ and suppose that
$M$ has no middle Hodge type. For $\iota:E\rightarrow\mathbb{C}$ let
$L(s,\iota,M)$ denote the completed $L$-function attached to $(M,\iota).$ Put
$L(s,M)=\\{L(s,\iota,M)\\}_{\iota:E\rightarrow\mathbb{C}}$ for the array of
$L$-functions taking values in $E\otimes\mathbb{C}.$ Suppose $m$ and $m+1$ are
critical integers for $L(s,M)$, and assuming that $L(m+1,M)\neq 0$, we have

$\frac{L(m,M)}{L(m+1,M)}\ =\ 1\otimes\mathfrak{i}^{d_{F}/2}\Delta_{F},$

where the equality is in $(E\otimes\mathbb{C})^{\times}/E^{\times}.$

A word of explanation is in order, since, in [4], Deligne formulated his
conjecture for critical values of $L_{f}(s,M)$–the finite-part of the
$L$-function attached to $M;$ which takes the shape:

$\frac{L_{f}(m,M)}{L_{f}(m+1,M)}\ =\
(1\otimes(2\bm{\uppi}\mathfrak{i})^{-d^{\pm}(M)})\,\frac{c^{\pm}(M)}{c^{\mp}(M)},\quad\mbox{in
$E\otimes\mathbb{C}$}.$

The assumption on $M$ that there is middle Hodge type gives
$d^{\pm}(M)=d(M)/2=d_{F}/2.$ Knowing the $L$-factor at infinity we have:

$L_{\infty}(m,M)/L_{\infty}(m+1,M)=1\otimes(2\bm{\uppi})^{d_{F}/2}.$

For the completed $L$-function, we get:

(61) $\frac{L(m,M)}{L(m+1,M)}\ =\
(1\otimes\mathfrak{i}^{d_{F}/2})\,\frac{c^{\pm}(M)}{c^{\mp}(M)}.$

It is clear now that (61) and Prop. 59 give Conj. 60.

Using the conjectural correspondence between an algebraic Hecke character of
$F$ with values in $E$, and a rank-one motive over $F$ with coefficients in
$E$, Conj. 60 may be restated as the following:

######  Conjecture 62 (Deligne’s Conjecture for ratios of critical values of
Hecke $L$-functions).

Let the notations and hypothesis be as in Thm. 56. Then:

(63)
$\mathfrak{i}^{d_{F}/2}\Delta_{F}\,\frac{L(m,{}^{\iota}\chi)}{L(m+1,{}^{\iota}\chi)}\
\in\iota(E),$

and, furthermore, for every $\varsigma\in{\rm
Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ we have the reciprocity law:

(64)
$\varsigma\left(\mathfrak{i}^{d_{F}/2}\Delta_{F}\,\frac{L(m,{}^{\iota}\chi)}{L(m+1,{}^{\iota}\chi)}\right)\
=\
\mathfrak{i}^{d_{F}/2}\Delta_{F}\,\frac{L(m,{}^{\varsigma\circ\iota}\chi)}{L(m+1,{}^{\varsigma\circ\iota}\chi)}.$

####  6.4.2. Thm. 56 $\implies$ Conj. 62

This follows from the following:

######  Proposition 65.

Let the notations and hypothesis be as in Thm. 56. Then:

$\frac{\varsigma(\mathfrak{i}^{d_{F}/2}\Delta_{F})}{\mathfrak{i}^{d_{F}/2}\Delta_{F}}\
=\
\frac{\varsigma(|\delta_{F/\mathbb{Q}}|^{1/2})}{|\delta_{F/\mathbb{Q}}|^{1/2})}\cdot\varepsilon_{{\bf
n},\iota}(\varsigma)\cdot\varepsilon_{\tilde{\bf n},\iota}(\varsigma).$

###### Proof.

When $F$ is a CM-field, we have $\varepsilon_{{\bf
n},\iota}(\varsigma)\cdot\varepsilon_{\tilde{\bf n},\iota}(\varsigma)=1$ from
6.3.4; the proof will follow from the following lemma:

######  Lemma 66.

Let $F$ be a CM field. Then, as elements of
$\mathbb{C}^{\times}/\mathbb{Q}^{\times}$, we have:

$|\delta_{F/\mathbb{Q}}|^{1/2}\ =\ \mathfrak{i}^{d_{F}/2}\cdot\Delta_{F}.$

###### Proof of Lem. 66.

Let $F_{0}$ be the maximal totally real subfield of $F;$ suppose
$F=F_{0}(\sqrt{D})$, for a totally negative $D\in F_{0};$ then
$\Delta_{F}=\sqrt{N_{F_{0}/\mathbb{Q}}(D)}.$ For the absolute discriminant of
$F$, using the well-known formula for the discriminant of a tower of fields,
we have:

$\delta_{F/\mathbb{Q}}\ =\ \delta_{F_{0}/\mathbb{Q}}^{[F:F_{0}]}\cdot
N_{F_{0}/\mathbb{Q}}(\delta_{F/F_{0}})\ =\ \delta_{F_{0}/\mathbb{Q}}^{2}\cdot
N_{F_{0}/\mathbb{Q}}(\delta_{F/F_{0}}).$

Using the basis $\\{1,\sqrt{D}\\}$ we get the relative discriminant of
$F/F_{0}$ as $\delta_{F/F_{0}}=4D.$ Hence, taking square-root of the absolute
value of $\delta_{F/\mathbb{Q}}$ we get:

$\sqrt{|\delta_{F/\mathbb{Q}}|}\ =\
\sqrt{|N_{F_{0}/\mathbb{Q}}(D)|}\pmod{\mathbb{Q}^{\times}}.$

Since $D$ is totally negative, the sign of $N_{F_{0}/\mathbb{Q}}(D)$ is the
same as $(-1)^{[F_{0}:\mathbb{Q}]}=(-1)^{d_{F}/2},$ i.e.,
$|N_{F_{0}/\mathbb{Q}}(D)|=(-1)^{d_{F}/2}N_{F_{0}/\mathbb{Q}}(D).$ Hence:

$|\delta_{F/\mathbb{Q}}|^{1/2}\ =\
\sqrt{(-1)^{d_{F}/2}N_{F_{0}/\mathbb{Q}}(D)}\ =\
\mathfrak{i}^{d_{F}/2}\sqrt{N_{F_{0}/\mathbb{Q}}(D)}\ =\
\mathfrak{i}^{d_{F}/2}\Delta_{F}\pmod{\mathbb{Q}^{\times}}.$

∎

Suppose now that $F$ is a totally imaginary field; the hypothesis of Thm. 56
guarantees that $F$ contains a CM subfield $F_{1}$ and its maximal totally
real subfield $F_{0}.$ Lem. 66 generalizes to the following lemma:

######  Lemma 67.

Let $F$ be a totally imaginary field, and $F_{1}$ its maximal CM subfield.
Then, as elements of $\mathbb{C}^{\times}/\mathbb{Q}^{\times}$, we have:

$|\delta_{F/\mathbb{Q}}|^{1/2}\ =\
\mathfrak{i}^{d_{F}/2}\cdot\Delta_{F}\cdot\left(N_{F_{1}/\mathbb{Q}}(\delta_{F/F_{1}})\right)^{1/2}.$

###### Proof of Lem. 67.

Begin by noting that $\delta_{F/\mathbb{Q}}\ =\
\delta_{F_{0}/\mathbb{Q}}^{[F:F_{0}]}\cdot
N_{F_{0}/\mathbb{Q}}(\delta_{F/F_{0}}),$ and since $[F:F_{0}]=2[F:F_{1}]$ is
even, we get

$|\delta_{F/\mathbb{Q}}|^{1/2}\ =\
|N_{F_{0}/\mathbb{Q}}(\delta_{F/F_{0}})|^{1/2}\pmod{\mathbb{Q}^{\times}}.$

Next, since $\delta_{F/F_{0}}\ =\ \delta_{F_{1}/F_{0}}^{[F:F_{1}]}\cdot
N_{F_{1}/F_{0}}(\delta_{F/F_{1}});$ using the $F_{0}$-basis $\\{1,\sqrt{D}\\}$
for $F_{1}$, one has $\delta_{F_{1}/F_{0}}=4D;$ therefore,

$N_{F_{0}/\mathbb{Q}}(\delta_{F/F_{0}})\ =\
N_{F_{0}/\mathbb{Q}}(4D)^{[F:F_{1}]}\cdot
N_{F_{0}/\mathbb{Q}}(N_{F_{1}/F_{0}}(\delta_{F/F_{1}}))\\\ =\
N_{F_{0}/\mathbb{Q}}(D)^{[F:F_{1}]}\cdot
N_{F_{1}/\mathbb{Q}}(\delta_{F/F_{1}})\pmod{\mathbb{Q}^{\times}{}^{2}}.$

Since $F_{1}/\mathbb{Q}$ is a CM-extension,
$N_{F_{1}/\mathbb{Q}}(\delta_{F/F_{1}})>0$; hence, we get

$|\delta_{F/\mathbb{Q}}|^{1/2}\ =\
|N_{F_{0}/\mathbb{Q}}(\delta_{F/F_{0}})|^{1/2}\ =\
|N_{F_{0}/\mathbb{Q}}(D)|^{[F:F_{1}]/2}\cdot\left(N_{F_{1}/\mathbb{Q}}(\delta_{F/F_{1}})\right)^{1/2}\pmod{\mathbb{Q}^{\times}}.$

Since $D\ll 0$ in $F_{0}$, we see that
$(-1)^{[F_{0}:\mathbb{Q}]}N_{F_{0}/\mathbb{Q}}(D)>0$. Hence,

$|N_{F_{0}/\mathbb{Q}}(D)|^{[F:F_{1}]/2}\ =\
((-1)^{[F_{0}:\mathbb{Q}]}N_{F_{0}/\mathbb{Q}}(D))^{[F:F_{1}]/2}\ =\
(\mathfrak{i}^{[F_{0}:\mathbb{Q}]}\Delta_{F_{1}})^{[F:F_{1}]}\\\ =\
\mathfrak{i}^{[F_{0}:\mathbb{Q}][F:F_{1}]}\Delta_{F}\ =\
\mathfrak{i}^{d_{F}/2}\Delta_{F}.$

∎

######  Lemma 68.

With notations as in Lem. 67 and Thm. 56, we have the equality of signatures:

$\varepsilon_{{\bf n},\iota}(\varsigma)\cdot\varepsilon_{\tilde{\bf
n},\iota}(\varsigma)\ =\
\frac{\varsigma\left(N_{F_{1}/\mathbb{Q}}(\delta_{F/F_{1}})^{1/2}\right)}{N_{F_{1}/\mathbb{Q}}(\delta_{F/F_{1}})^{1/2}}.$

###### Proof of Lem. 68.

For the left hand side, recall that $\varepsilon_{{\bf n},\iota}(\varsigma)$
(resp., $\varepsilon_{\tilde{\bf n},\iota}(\varsigma)$) is the sign of the
permutation $\pi_{{\bf n},\iota}(\varsigma)$ (resp., $\pi_{\tilde{\bf
n},\iota}(\varsigma)$) induced by the map $\varsigma\circ-:\Phi_{{\bf
n},\iota}\rightarrow\Phi_{{\bf n},\varsigma\circ\iota}$ (resp.,
$\Phi_{\tilde{\bf n},\iota}\rightarrow\Phi_{\tilde{\bf
n},\varsigma\circ\iota}$). For brevity, let $\Phi=\Phi_{{\bf n},\iota}$, then
$\bar{\Phi}=\Phi_{\tilde{\bf n},\iota}$. Recall from Prop. 26 that ${\bf n}$
is the base-change of ${\bf m}$, hence the CM-type $\Psi=\Phi_{{\bf m},\iota}$
for the CM-field $F_{1}$ has the property that $\Phi\rightarrow\Psi$ under the
restriction map; also $\bar{\Phi}\rightarrow\bar{\Psi}.$ One has the following
relation with respect to conjugating by a Galois element $\varsigma$:

$\textstyle{\Phi\cup\bar{\Phi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varsigma\circ-}$$\textstyle{\varsigma(\Phi)\cup\varsigma(\bar{\Phi})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Psi\cup\bar{\Psi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varsigma\circ-}$$\textstyle{\varsigma(\Psi)\cup\varsigma(\bar{\Psi})}$

Fixing an ordering on $S_{\infty}(F_{1})$, the bottom row involves two
permutations corresponding to $\Psi\rightarrow\varsigma(\Psi)$ and
$\bar{\Psi}\rightarrow\varsigma(\bar{\Psi})$, but from Lem. 24 it follows that
they are equal, hence $\varepsilon_{{\bf
m},\iota}(\varsigma)=\varepsilon_{\tilde{\bf m},\iota}(\varsigma).$ However,
the two permutations and their signatures corresponding to
$\Phi\rightarrow\varsigma(\Phi)$ and
$\bar{\Phi}\rightarrow\varsigma(\bar{\Phi})$ need not be equal.

For the right hand side: let
$\Sigma_{F_{1}}=\\{\theta_{1},\dots,\theta_{d_{1}}\\};$
$d_{1}=[F_{1}:\mathbb{Q}];$ then for any $x\in F_{1}^{\times}$ one has
$N_{F_{1}/\mathbb{Q}}=\prod_{j=1}^{d_{1}}\theta_{j}(x)>0.$ Also, suppose
$\\{\rho_{1},\dots,\rho_{k}\\}$ denotes all embeddings of $F$ into