# Motives, Periods, and Functoriality

Pierre Deligne & A. Raghuram Institute for Advanced Study, 1 Einstein Drive,
Princeton, NJ 08540, USA<EMAIL_ADDRESS>Dept. of Mathematics, Fordham
University at Lincoln Center, New York, NY 10023, USA<EMAIL_ADDRESS>
###### Abstract.

Given a pure motive $M$ over ${\mathbb{Q}}$ with a multilinear algebraic
structure $\mathsf{s}$ on $M$, and given a representation $V$ of the group
respecting $\mathsf{s}$, we describe a functorial transfer $M^{V}$. We
formulate a criterion that guarantees when the two periods of $M^{V}$ are
equal. This has an implication for the critical values of the $L$-function
attached to $M^{V}.$ The criterion is explicated in a variety of examples such
as: tensor product motives and Rankin–Selberg $L$-functions; orthogonal
motives and the standard $L$-function for even orthogonal groups; twisted
tensor motives and Asai $L$-functions.

###### 2000 Mathematics Subject Classification:

Primary 11F67; Secondary 11G09, 22E55

###### Contents

  1. 1 Introduction
  2. 2 The main theorem on critical motives
    1. 2.1 Betti–de Rham systems of realizations
    2. 2.2 Critical Betti–de Rham realizations
      1. 2.2.2 $c^{+}(M)$ for a critical $M$
      2. 2.2.3 $c^{+}(M)$ for a critical $M$ with coefficients
      3. 2.2.4 Conjecture on critical $L$-values
    3. 2.3 Other determinant of periods
      1. 2.3.1 The invariants $c^{\pm}(M)$
      2. 2.3.2 Behaviour of $c^{\pm}(M)$ under Tate-twists
      3. 2.3.3 Yoshida invariants
    4. 2.4 Two successive critical points
    5. 2.5 Multilinear algebra constructions
      1. 2.5.1 Tannakian formalism giving $V\mapsto M^{V}$
      2. 2.5.2 Explicating the construction of $M^{V}$
      3. 2.5.3 The image of $G(\mathsf{s})_{\rm B}$ under a representation.
    6. 2.6 A criterion for $c^{+}(M)\approx_{E}c^{-}(M)$
    7. 2.7 Multiplicative relations between determinants of periods
      1. 2.7.1
      2. 2.7.3 Grothendieck period conjecture.
  3. 3 Examples
    1. 3.1 Tensor product motives over ${\mathbb{Q}}$
      1. 3.1.2
      2. 3.1.4
      3. 3.1.5 Example: The tensor product of two rank-two motives-I
      4. 3.1.6
    2. 3.2 Artin Motives
    3. 3.3 Systems of Betti–de Rham realizations over a number field $L$
      1. 3.3.1
      2. 3.3.2 Description as module
      3. 3.3.3 A remark on Hodge numbers
      4. 3.3.7 Highest weights
    4. 3.4 Systems of Betti–de Rham realizations over a totally imaginary field
      1. 3.4.1
      2. 3.4.4
      3. 3.4.5 Rankin–Selberg $L$-functions over a totally imaginary field
    5. 3.5 Tensor product motives over a number field.
    6. 3.6 Motives via multiplication induction: Asai motives.
      1. 3.6.1 Definition
      2. 3.6.3 Motivating example-I
      3. 3.6.4 Motivating example-II: Asai motives
    7. 3.7 Symplectic and Orthogonal Motives
      1. 3.7.1 The case when the base field is ${\mathbb{Q}}$
      2. 3.7.3 The case when the base field is totally real
      3. 3.7.4 Example: The tensor product of two rank-two motives-II
      4. 3.7.6 Example: The conjectural motive attached to representations of ${\rm O}(2n)$

##  1\. Introduction

The second author, in independent collaborations with Günter Harder,
Chandrasheel Bhagwat, and Muthu Krishnamurthy, has found a number of cases
where for some reductive group $G$ over a number field $F$, some cuspidal
automorphic representation $\pi$ of $G/F$, and some finite-dimensional
representations $\rho$ of the Langlands dual ${}^{L}G$ of $G$, the
corresponding $L$-function $L(s,\pi,\rho)=:L(s)$ is such that for some values
of $s$, the ratio $L(s)/L(s+1)$ is algebraic:

  * •

Rankin–Selberg $L$-functions for ${\rm GL}_{n}\times{\rm GL}_{m}$ over a
totally real field [HR20];

  * •

Rankin–Selberg $L$-functions for ${\rm GL}_{n}\times{\rm GL}_{m}$ over a
totally imaginary field [R22c], or $L$-functions for ${\rm GL}_{n}$ over a CM
field [R22a];

  * •

Degree-$2n$ $L$-functions for ${\rm GL}_{1}\times{\rm O}(2n)$ over a totally
real field [BR20];

  * •

Degree-$n^{2}$ Asai $L$-functions over $F$ attached to cuspidal
representations of ${\rm GL}_{n}/E$; with $E/F$ a quadratic extension of
totally real fields [KR].

In all cases, building on previous work of Harder [H87], [H10], rationality
results follow from a study of rank-one Eisenstein cohomology on an ambient
reductive group having $G$ as a Levi quotient, giving a cohomological
interpretation to some aspects of the Langlands–Shahidi method [Sha10].

In all cases, there is conjecturally a motive $M$ such that, possibly after a
shift $s^{\prime}=s+a$, $L(s)$ is the $L$-function attached to $M$:

$L(s)\ =\ L(M,s^{\prime})$

and the values $s$ and $s+1$ considered for $L(s)$ correspond to values
$s^{\prime}$ and $s^{\prime}+1$ which are critical for $M$.

In [D79], the first author proposed a conjecture on the critical values
$L(M,s)$ of motivic $L$-functions. The $L$-functions considered were without
the $L$-factors at infinity. For a motive $M$ with coefficients in a number
field $E$, the $L$-function takes values in $E\otimes{\mathbb{C}}$, and for
$s$ critical for $M$, the conjecture determined $L(M,s)$ up to multiplication
by an element of $E$. It was given by some determinant of periods
$c^{+}(M(s))$ of the motive $M(s)$. Here, $s$ is always an integer, and $M(s)$
is a Tate-twist of $M$. There is another determinant of periods $c^{-}(M(s))$,
such that $c^{+}(M(s+1))=c^{-}(M(s))$ up to multiplication by some power of
$2\pi i$. Reintroducing the $L$-factors at infinity also introduces powers of
$\pi$ and $i.$

The aim of the present work is to prove rationality results for some ratios
$c^{+}(M)/c^{-}(M)$ and to explain how they a posteriori predict some of the
results of the first author mentioned above. The interest is two-fold. In one
direction, it gives partial confirmations of [D79, Conj. 2.8]. In the other
direction, it has led to corrections to the results on special values of
automorphic $L$-functions in the literature (see the commentary in [R22b,
Sect. 6]), and in the results of the first author (especially, see [R22c,
Sect. 5.4] and 3.4.5 below). It also suggested new cases to be investigated.

In Sect. 2, after reviewing the basics of a critical motive and its periods,
we describe the realizations of a motive $M^{V}$ attached to an algebraic
representation $V$ of the group of automorphisms of a realization of a motive
$M$ that respects some given multilinear algebraic structure $\mathsf{s}$ on
$M.$ The assignment $M\mapsto M^{V}$ is to be construed as the motivic
analogue of Langlands transfer. We then state and prove the main result on
period relations; see Thm. 2.6.1; the proof is inspired by the methods of
Yoshida [Y01]. In Sect. 3 we discuss a variety of concrete examples relevant
to the above mentioned automorphic $L$-functions; especially, see Prop. 3.1.1,
Thm. 3.4.2, Prop. 3.5.1, Thm. 3.6.2, and Thm. 3.7.2.

##  2\. The main theorem on critical motives

The $L$-function of a motive over a number field $F$ coincides with the
$L$-function of the motive over ${\mathbb{Q}}$ deduced from it by restriction
of scalars. Because of this, we will often consider only (pure) motives over
${\mathbb{Q}}.$ They will be thought in terms of corresponding system of
realizations in Betti, de Rham, and $\ell$-adic cohomologies, and the
comparison isomorphisms. We will also consider motives with coefficients in a
number field $E$, that is given with $E\to\mathrm{End}(M).$

Let $M$ be a motive with coefficients in $E$. We will review what it means for
$M$ to be critical, and we will discuss the definition of the periods
$c^{\pm}(M)\in(E\otimes{\mathbb{C}})^{*}/E^{*}$ attached by the first author
([D79]) to critical motives. We will exhibit cases where $c^{+}(M)=c^{-}(M).$
The inspiration for the proof comes from Yoshida [Y01].

The periods $c^{\pm}(M)$ are interesting because of their conjectural relation
with the critical values of the $L$-function attached to $M.$ The $\ell$-adic
realizations are needed to define this $L$-function, but the definition of
$c^{\pm}(M)$ depends only on the Betti and de Rham realizations, and our
criterion for their equality can be stated and proved for systems of Betti and
de Rham realizations which are defined in Sect. 2.1 below.

Motives admit a tensor product and duals, so that one can can consider a
motive $M$ together with a multilinear algebraic structure $\mathsf{s}$. From
$(M,\mathsf{s})$, one can obtain new motives $M^{\prime}$ by multilinear
algebraic constructions. Here is a way to formalize this. Fix a realization
functor $R$ with values in ${\mathbb{Q}}$-vector spaces, for instance Betti or
de Rham. Let $G_{R}\subset{\rm GL}(M_{R})$ ($R=\mathrm{B}$ or $\mathrm{dR}$)
be the algebraic group of automorphisms of $(M_{R},\mathsf{s}_{R})$. The
comparison isomorphism $M_{\rm B}\otimes{\mathbb{C}}\to M_{\rm
dR}\otimes{\mathbb{C}}$ will induce an isomorphism
$(G_{B})_{\mathbb{C}}\to(G_{\rm dR})_{\mathbb{C}}.$ Then to any representation
$V$ of $G_{R}$ corresponds a motive $M^{V}$ deduced from $M$; this
construction being functorial and compatible with tensor product, and $M_{R}$
corresponding to $M.$ The transfer $M\mapsto M^{V}$ is the motivic analogue of
Langlands transfer on the automorphic side.

Our aim is to find structures $\mathsf{s}$ and representations $V$ for which
$c^{+}(M^{V})=c^{-}(M^{V}).$

###  2.1. Betti–de Rham systems of realizations

A weight $w$ Betti–de Rham system of realizations $M$ over ${\mathbb{Q}}$
consists of the following data (i)-(iii), obeying (iv):

  1. (i)

Betti realization: for $C$ an algebraic closure of ${\mathbb{R}}$, a
${\mathbb{Q}}$-vector space $M_{B}\\{C\\}$, depending functorially on $C$. If
${\mathbb{C}}:={\mathbb{R}}[i]/(i^{2}+1)$ is the standard algebraic closure of
${\mathbb{R}}$, this amounts to the data of $M_{B}:=M_{B}\\{{\mathbb{C}}\\},$
and of the involution $\mathsf{F}_{\infty}$ induced by the complex conjugation
automorphism of ${\mathbb{C}}$. We call the involution $\mathsf{F}_{\infty}$
the Frobenius at infinity. It gives a decomposition $M_{B}=M^{+}\oplus M^{-}$
of $M_{B}$ into its $\pm 1$ eigenspaces.

  2. (ii)

de Rham realization: a ${\mathbb{Q}}$-vector space $M_{\rm dR}$ together with
a finite decreasing filtration $\mathcal{F}$, the Hodge filtration.

  3. (iii)

comparison isomorphism: an isomorphism $M_{\rm B}\\{C\\}\otimes C\to M_{\rm
dR}\otimes C$ of $C$-vector spaces that is functorial in $C.$

Let $\sigma_{\rm B}$ (resp., $\sigma_{\rm dR}$) be the antilinear involution
of $M_{\rm B}\otimes{\mathbb{C}}$ (resp., $M_{\rm dR}\otimes{\mathbb{C}}$)
which is the tensor product of the identity and of the complex conjugation on
${\mathbb{C}}$. The action by functoriality of the complex conjugation on
$M_{B}\otimes{\mathbb{C}}$ is $\mathsf{F}_{\infty}\cdot\sigma_{\rm B}$. The
data in (iii) amounts to the data of an isomorphism $M_{\rm
B}\otimes{\mathbb{C}}\to M_{\rm dR}\otimes{\mathbb{C}}$ which carries
$\mathsf{F}_{\infty}\cdot\sigma_{\rm B}$ into $\sigma_{\rm dR}.$ In other
words, $M^{+}\otimes{\mathbb{R}}\,\oplus\,M^{-}\otimes i{\mathbb{R}}$ maps
isomorphically onto $M_{\rm dR}\otimes{\mathbb{R}}.$

  1. (iv)

One requires that if the complexified Hodge filtration is transported by the
comparison isomorphism to a filtration of $M_{B}\otimes{\mathbb{C}},$ still
denoted $\mathcal{F}$, it and its complex conjugate
$\bar{\mathcal{F}}=\sigma_{\rm B}(\mathcal{F})$ are $w$-opposite, meaning that
$M_{\rm B}\otimes{\mathbb{C}}$ is the direct sum of the
$M^{p,q}:=\mathcal{F}^{p}\cap\bar{\mathcal{F}}^{q}$ for $p+q=w.$ One calls the
decomposition $M_{\rm B}\otimes{\mathbb{C}}={\oplus}_{p+q=w}M^{p,q}$ the Hodge
decomposition. One has $\mathcal{F}^{a}=\oplus_{p\geq a}M^{p,q},$ and
$M^{p,q}$ and $M^{q,p}$ are complex conjugates. As $\mathcal{F}^{p}$ is dR-
rational, it is stable under $\sigma_{\rm B}\cdot\mathsf{F}_{\infty}$,
identified with $\sigma_{\rm dR}.$ It follows that $\mathsf{F}_{\infty}$ maps
$\mathcal{F}$ into $\bar{\mathcal{F}}$, hence $M^{p,q}$ to $M^{q,p}.$

One defines morphisms, tensor products, and duals in an obvious way. The Hodge
decomposition is functorial. It follows that any morphism is strictly
compatible with the Hodge filtrations and that the Betti–de Rham systems of
realizations of weight $w$ form an abelian category. The tensor product of
systems of realizations of weight $w$ and $w^{\prime}$ is of weight
$w+w^{\prime}$, and the dual of $M$ of weight $w$ is of weight $-w.$

#### Examples

The motivating example is $M=H^{n}(X)$, the cohomology in degree $n$ of a
smooth projective variety $X$ over ${\mathbb{Q}}$ with $M_{\rm
B}\\{C\\}:=H^{n}(X(C),{\mathbb{Q}})$, and $M_{\rm dR}$ the algebraic de Rham
cohomology of $X$; the weight is $n.$

The Tate object ${\mathbb{Q}}(1)$, of weight $-2$ and Hodge type $(-1,-1),$ is
defined by ${\mathbb{Q}}(1)_{\rm B}\\{C\\}=2\pi i{\mathbb{Q}}\subset C$, and
${\mathbb{Q}}(1)_{\rm dR}={\mathbb{Q}}$ purely of filtration $-1$. As
${\mathbb{Q}}(1)$ is of rank one, its $n^{\rm th}$ tensor power, denoted
${\mathbb{Q}}(n)$, is defined for $n\in{\mathbb{Z}}$. For $n=-1$,
${\mathbb{Q}}(-1)$ is the dual of ${\mathbb{Q}}(1)$. It is also
$H^{1}({\mathbb{G}}_{m}),$ and the $2\pi i$ comes from $\oint\frac{dz}{z}=2\pi
i.$ The $n^{\rm th}$ Tate twist of $M$ is defined as
$M(n):=M\otimes{\mathbb{Q}}(n).$

###  2.2. Critical Betti–de Rham realizations

######  Definition 2.2.1.

We say that $M$ is critical if the comparison isomorphism induces an
isomorphism

(1) $M^{+}\otimes{\mathbb{C}}\ \longrightarrow\ (M_{\rm
dR}/\mathcal{F}^{0})\otimes{\mathbb{C}}.$

The $L$-factor at infinity $L_{\infty}(M,s)$ attached to $M$ is defined as
follows ([D79, 5.2], [Se69]). One defines
$\Gamma_{\mathbb{C}}(s):=2\cdot(2\pi)^{-s}\Gamma(s),\
\Gamma_{\mathbb{R}}(s):=\pi^{-s/2}\ \Gamma(s/2),\ h^{p,q}:=\dim M^{p,q};$ for
$M$ of even weight $w,$ $w=2p,$ for $\epsilon=\pm 1$ define
$h^{p,p}(\epsilon)$ to be dimension of the $\epsilon\cdot(-1)^{p}$ eigenspace
of $\mathsf{F}_{\infty}$ acting on $M^{p,p};$ then

$L_{\infty}(M,s):=\left(\prod_{p<q}\
\Gamma_{\mathbb{C}}(s-p)^{h^{p,q}}\right)\cdot\Gamma_{\mathbb{R}}(s-p)^{h^{p,p}(+1)}\cdot\Gamma_{\mathbb{R}}(s+1-p)^{h^{p,p}(-1)},$

where the last two factors are possibly there only when $M$ is of even weight
$w=2p.$ One has $L_{\infty}(M(n),s)=L_{\infty}(M,s+n)$. Criticality of $M$ is
equivalent to finiteness of both the values $L_{\infty}(M,0)$ and
$L_{\infty}(M^{\sf v}(1),0)$.

####  2.2.2. $c^{+}(M)$ for a critical $M$

If $M$ is critical, one defines $c^{+}(M)$ to be the determinant of the
isomorphism (1) computed in terms of ${\mathbb{Q}}$-bases of $M^{+}$ and
$M_{\rm dR}/\mathcal{F}^{0}.$ It is well-defined modulo multiplication by
$\lambda\in{\mathbb{Q}}^{*}$. As $M^{+}$ maps to $M_{\rm
dR}\otimes{\mathbb{R}}$, the isomorphism in (1) is the complexification of an
isomorphism

(2) $M^{+}\otimes{\mathbb{R}}\ \longrightarrow\ (M_{\rm
dR}/\mathcal{F}^{0})\otimes{\mathbb{R}}.$

Hence, if $M$ is critical, then $c^{+}(M)$ is a nonzero real number. If
$\Lambda\subset M^{+}$ is a lattice, then $\Lambda\backslash((M_{\rm
dR}/\mathcal{F}^{0})\otimes{\mathbb{R}})$ is a torus whose Lie algebra has the
rational structure $(M_{\rm dR}/\mathcal{F}^{0})$, and $c^{+}(M)$ is the
volume of that torus for a rational volume element. If $M$ has weight $w<0$,
this torus modulo torsion, that is $M^{+}\backslash((M_{\rm
dR}/\mathcal{F}^{0})\otimes{\mathbb{R}})$ is the group of extensions of
${\mathbb{Q}}(0)$ by $M$ in the category of mixed Hodge structures with a real
Frobenius.

####  2.2.3. $c^{+}(M)$ for a critical $M$ with coefficients

More generally, if $M$ is with coefficients in $E$, that is given with
$E\to\mathrm{End}(M)$, then $M^{+}$ and $M_{\rm dR}/\mathcal{F}^{0}$ are
$E$-vector spaces. If $M$ is critical then (1) is an isomorphism of
$E\otimes{\mathbb{C}}$-modules. One then defines $c^{+}_{E}(M)$ to be the
determinant of this isomorphism, computed using $E$-bases of $M^{+}$ and
$M_{\rm dR}/\mathcal{F}^{0}$. This determinant is real, that is belongs to
$(E\otimes{\mathbb{R}})^{*}\subset(E\otimes{\mathbb{C}})^{*}$ and is well-
defined modulo $E^{*}$. The $c^{+}(M)$ previously defined is related to
$c^{+}_{E}(M)$ by

$c^{+}(M)=N_{E/{\mathbb{Q}}}\ c^{+}_{E}(M).$

When this will not lead to ambiguity, we will write simply $c^{+}(M)$ for
$c^{+}_{E}(M)$.

The $L$-factor at infinity is defined as before, using dimensions over $E$.

####  2.2.4. Conjecture on critical $L$-values

If $M$ comes from a motive ${\mathbb{M}}$, and if enough conjectures are
verified so that the $L$-function $L({\mathbb{M}},s)$ is defined, [D79]
conjectures that for a critical ${\mathbb{M}},$ $\ L({\mathbb{M}},0)$ can
vanish only for ${\mathbb{M}}$ of weight $-1$, and that if
$L({\mathbb{M}},0)\neq 0$, one has

(3) $L({\mathbb{M}},0)\ \approx\ c^{+}(M)\quad\pmod{{\mathbb{Q}}^{*}}.$

The $L$-function considered is the finite part of the global $L$-function,
i.e., without its natural factors at infinity $L_{\infty}({\mathbb{M}},s)$.

For example, ${\mathbb{Q}}(n)$ is critical for $n>0$ and even, in which case
$\mathcal{F}^{0}=0$, ${\mathbb{Q}}(n)^{+}={\mathbb{Q}}(n)_{\rm B}$, or for
$n<0$ and odd, in which case ${\mathbb{Q}}(n)_{\rm
dR}/\mathcal{F}^{0}=0={\mathbb{Q}}(n)^{+}.$ One has
$L({\mathbb{Q}}(n),s)=\zeta(s+n)$, and for ${\mathbb{Q}}(n)$ the above
conjecture is a consequence of a famous theorem of Euler.

More generally, if ${\mathbb{M}}$ is with coefficients in $E$, one gets an
$L$-function $L_{\sigma}({\mathbb{M}},s)$ for each embedding
$\sigma:E\to{\mathbb{C}}$. Using the isomorphism
$E\otimes{\mathbb{C}}\xrightarrow{\sim}{\mathbb{C}}^{{\rm
Hom}(E,{\mathbb{C}})}$, one defines $L({\mathbb{M}},s)$, with values in
$E\otimes{\mathbb{C}}$ by
$L({\mathbb{M}},s)=\\{L_{\sigma}({\mathbb{M}},s)\\}_{\sigma:E\hookrightarrow{\mathbb{C}}}$.
The conjecture in [D79] is that if $M$ is critical and that
$L({\mathbb{M}},s)\neq 0,$ one has

(4) $L({\mathbb{M}},0)\ \approx\ c^{+}_{E}({\mathbb{M}})\quad\pmod{E^{*}}.$

###  2.3. Other determinant of periods

####  2.3.1. The invariants $c^{\pm}(M)$

We will say that $M$ has vanishing middle Hodge number if either its weight
$w$ is odd, or its weight $w$ is even and $M^{w/2,w/2}=0.$ If $M$ of weight
$2n$ or $2n+1$ has vanishing middle Hodge number, then, as
$\mathsf{F}_{\infty}$ exchanges $M^{p,q}$ with $M^{q,p}$, and that
$\mathcal{F}^{n}$ contains exactly one of them, the maps

$M^{\pm}\otimes{\mathbb{C}}\ \to\ (M_{\rm
dR}/\mathcal{F}^{n})\otimes{\mathbb{C}}$

are isomorphisms. One defines $c^{\pm}(M)$ to be their determinants computed
using rational bases. As $M^{+}$ (resp., $M^{-}$) maps to $M_{\rm
dR}\otimes{\mathbb{R}}$ (resp., $M_{\rm dR}\otimes i{\mathbb{R}}$) we get

$c^{+}(M)\in{\mathbb{R}}\quad\mbox{(resp., $c^{-}(M)\in i^{\dim(M_{\rm
B})/2}{\mathbb{R}}$)}.$

If $w$ is an even integer, say $w=2n$, and $\mathsf{F}_{\infty}$ acts as the
identity (resp., by $-1$) on $M^{n,n}$, the maps

$M^{+}\otimes{\mathbb{C}}\ \to\ (M_{\rm
dR}/\mathcal{F}^{n+1})\otimes{\mathbb{C}},\quad\mbox{(resp.,
$M^{+}\otimes{\mathbb{C}}\ \to(M_{\rm
dR}/\mathcal{F}^{n})\otimes{\mathbb{C}}$),}$

and

$M^{-}\otimes{\mathbb{C}}\ \to\ (M_{\rm
dR}/\mathcal{F}^{n})\otimes{\mathbb{C}},\quad\mbox{(resp.,
$M^{-}\otimes{\mathbb{C}}\ \to\ (M_{\rm
dR}/\mathcal{F}^{n+1})\otimes{\mathbb{C}}$)}$

are isomorphisms. Again, one defines $c^{\pm}(M)$ to be their determinants
computed using rational bases, and we get

$c^{+}(M)\in{\mathbb{R}}\quad\mbox{(resp., $c^{-}(M)\in
i^{\dim(M^{-})}{\mathbb{R}}$)}.$

In the case where $M$ has vanishing middle Hodge number, the comparison
between $c^{+}(M)$ and $c^{-}(M)$ is made easier by the fact that the maps
used in their definition have the same target space.

If $M$ is with coefficients in $E$, one similarly defines invariants
$c^{\pm}_{E}(M)\in(E\otimes{\mathbb{C}})^{*}$ which are well-defined modulo
$E^{*}.$

For $M$ to be critical of weight $w$, $\mathsf{F}_{\infty}$ must, when $w$ is
even, act by $+1$ or by $-1$ on $M^{w/2,w/2},$ and $\mathcal{F}^{0}$ must
coincide with $\mathcal{F}^{n}$ appearing in the above definition of
$c^{+}(M)$, which generalizes the definition given in the critical case. If
$w\geq 0$ (resp., $w<0$) this is equivalent to the vanishing of $M^{p,q}$ for
which $p,q\geq 0$ (resp., $p,q<0$), except for $p=q$, in which case
$\mathsf{F}_{\infty}$ must act by $-1$ (resp., $+1$) on $M^{p,p}.$

####  2.3.2. Behaviour of $c^{\pm}(M)$ under Tate-twists

If the integer $n$ is even (resp., odd), so that the involution
$\mathsf{F}_{\infty}$ of ${\mathbb{Q}}(n)_{\rm B}$ is $+1$ (resp., $-1$), one
has $M(n)^{+}=M^{+}\otimes(2\pi i)^{n}{\mathbb{Q}}$ (resp.,
$M(n)^{+}=M^{-}\otimes(2\pi i)^{n}{\mathbb{Q}}$), so that

$c^{+}(M(n))=(2\pi i)^{n\dim(M^{+})}c^{+}(M),\quad\mbox{(resp.,
$c^{+}(M(n))=(2\pi i)^{n\dim(M^{-})}c^{-}(M))$}.$

The case $n=1$ can be rewritten

$c^{-}(M)=(2\pi i)^{-\dim(M^{-})}c^{+}(M(1)).$

####  2.3.3. Yoshida invariants

For $M$ a weight $w$ Betti–de Rham system of realizations over ${\mathbb{Q}}$,
with coefficients in $E$, Yoshida [Y01] defines additional invariants, again
in $(E\otimes{\mathbb{C}})^{*}/E^{*}.$ The basic ones are indexed by the $p$
such that $p\leq q:=w-p$ and $h^{p,q}\neq 0$, as well as by $p=-\infty.$ The
invariant $c_{p}(M)$ is defined to be the determinant, computed in Betti and
de Rham rational bases of a map

(5) $M_{\rm B}\otimes{\mathbb{C}}\ \longrightarrow\ M_{\rm
dR}/\mathcal{F}^{p+1}\otimes{\mathbb{C}}\ \oplus\ M_{\rm
dR}/\mathcal{F}^{q}\otimes{\mathbb{C}}.$

The map to the first (resp., second) factor is the projection of $M_{\rm dR}$
to $M_{\rm dR}/\mathcal{F}^{p+1}$ (resp., to $M_{\rm dR}/\mathcal{F}^{q}$),
complexified, composed with the comparison isomorphism ${\sf comp}$ (resp.,
${\sf comp}\circ\mathsf{F}_{\infty}$). The map (5) is an isomorphism because
the kernel of the surjective map to the first (resp., second) factor is the
sum of the $M^{a,b}$ for $a\leq p$ (resp., $a>p$, as $\mathsf{F}_{\infty}$
exchanges $M^{a,b}$ and $M^{b,a}$). For $p=-\infty$, $c_{p}(M)$ is the
determinant $\delta(M)$ of the comparison isomorphism. When $c^{\pm}(M)$ are
defined, the last $c_{p}(M)$ is the product $c^{+}(M)c^{-}(M).$

Suppose we have a family of systems $M_{\alpha}$, and a system $M$ derived
from the $M_{\alpha}$ by a multilinear algebraic construction. If
$\dim(M_{\alpha})=d_{\alpha}$; such constructions correspond to
representations of $\prod_{\alpha}{\rm GL}(d_{\alpha})$ (such that the center
acts by scalars not to mix weights). Yoshida proves that the invariant
$c_{p}(M)$ of $M$, and the invariants $c^{\pm}(M)$ when defined, are monomials
in the invariants for the $M_{\alpha}$. As an application of this theorem, he
shows that when $\dim(M^{+})=\dim(M^{-})$ and $\dim(N^{+})=\dim(N^{-})$, and
when $M\otimes N$ has vanishing middle Hodge number, then by Cor. 1 to Prop.
12 on p. 1188 of [Y01] one has:

$c^{+}(M\otimes N)\ =\ c^{-}(M\otimes N).$

To understand, it is convenient to work in a graded tensor category larger
than Betti–de Rham systems over ${\mathbb{Q}}$. The objects of weight $w$
consist of

  1. (a)

a ${\mathbb{Q}}$-vector space $H_{\rm B}$, with an involution
$\mathsf{F}_{\infty}$, giving a decomposition $H_{\rm B}=H_{\rm B}^{+}\oplus
H_{\rm B}^{-}$;

  2. (b)

a ${\mathbb{Q}}$-vector space $H_{\rm dR}$, with a decreasing filtration
$\mathcal{F}$;

  3. (c)

a comparison isomorphism ${\sf comp}:H_{\rm B}\otimes{\mathbb{C}}\to H_{\rm
dR}\otimes{\mathbb{C}}$,

such that the involution $\mathsf{F}_{\infty}$ of $H_{\rm
dR}\otimes{\mathbb{C}}$, transported by ${\rm comp}$ of the involution
$\mathsf{F}_{\infty}$ of $H_{\rm B}$, transforms the filtration $\mathcal{F}$
into a filtration that is $w$-opposite to $\mathcal{F}$ (giving us a
decomposition of $H_{\rm dR}\otimes{\mathbb{C}}$ into
$\oplus_{p+q=w}H^{p,q}$).

Given $(H_{\rm B},\mathsf{F}_{\infty})$ and $(H_{\rm dR},\mathcal{F})$ of the
same dimension, the space of possible comparison isomorphisms is not empty
when $\dim{\rm Gr}_{\mathcal{F}}^{p}(H_{\rm dR})=\ \dim{\rm
Gr}_{\mathcal{F}}^{q}(H_{\rm dR})$ for $p+q=w$ and either the weight is odd,
or in even weight $2p$, $|\dim(H_{\rm B}^{+})-\dim(H_{\rm B}^{-})|\leq\dim{\rm
Gr}_{\mathcal{F}}^{p}(H_{\rm dR}).$ When this space is not empty, it is the
space of complex points of the open orbit of $Z(\mathsf{F}_{\infty})\times
P_{\rm dR}$ acting on ${\rm Hom}(H_{\rm B},H_{\rm dR})$. Here,
$Z(\mathsf{F}_{\infty})$ is the centralizer of ${\rm GL}(H^{+})\times{\rm
GL}(H^{-})$ of $\mathsf{F}_{\infty}$, and $P_{\rm dR}$ is the parabolic
subgroup of ${\rm GL}(H_{\rm dR})$ respecting $\mathcal{F}$. The existence of
an open orbit is a special case of the theorem that for any reductive group,
the subgroup fixed by an involution acts with finitely many orbits on flag
varieties.

The invariants we considered are the values at ${\sf comp}$ of functions,
defined over ${\mathbb{Q}}$, and characterized up to a constant factor by
their transformation law

$f(pxz)\ =\ \lambda_{1}(p)f(x)\lambda_{2}(z),$

with $\lambda_{1}$ (resp., $\lambda_{2}$) a character of $P_{\rm dR}$ (resp.,
$Z(\mathsf{F}_{\infty})$). The invariants deduced from such functions $f$ are
monomials, with exponents in ${\mathbb{Z}}$, in the basic invariants and
$c^{\pm}$ when defined. In [Y01], the attention is restricted to the $f$ being
polynomial in the whole of ${\rm Hom}(H_{\rm B},H_{\rm dR})$, which correspond
to monomials with nonnegative integral exponents.

In the larger category, the invariant $c_{p}(M)$ can be described as being
$c^{+}(M\otimes N)$ for $N$ of rank $2$, type $\\{(-p-1,-q),(-q,-p-1)\\},$ and
rational comparison isomorphism.

###  2.4. Two successive critical points

Let us suppose that $M$ comes from a motive ${\mathbb{M}}$ and that enough
conjectures are satisfied so that the $L$-function $L({\mathbb{M}},s)$ is
defined. One has $L({\mathbb{M}}(n),s)=L({\mathbb{M}},n+s)$. Suppose that $M$
and $M(1)$ are critical. When the weight $w$ of $M$ is even, $w=2p$, and that
$M^{p,p}\neq 0$, this happens only for $w=0$, and $\mathsf{F}_{\infty}$ has to
act by -1 on $M^{0,0}$.

The completed $L$-function is

$\Lambda({\mathbb{M}},s):=L_{\infty}(M,s)\ L({\mathbb{M}},s).$

######  Lemma 2.4.1.

If $M$ and $M$(1) are critical, then

$L_{\infty}(M,1)/L_{\infty}(M,0)\ \approx\ \pi^{-\dim(M^{-})}\
\pmod{{\mathbb{Q}}^{*}}.$

This follows from

$\Gamma_{\mathbb{C}}(s+1)/\Gamma_{\mathbb{C}}(s)\ \approx\
1/\pi\pmod{{\mathbb{Q}}^{*}},$

for $s$ an integer, and

$\Gamma_{\mathbb{R}}(2)/\Gamma_{\mathbb{R}}(1)\ =\
\pi^{-1}/(\pi^{-\frac{1}{2}}\cdot\ \pi^{\frac{1}{2}})\ =\ 1/\pi.$

######  Corollary 2.4.2.

Suppose that $M$ and $M(1)$ are critical, that $c^{+}(M)\approx
c^{-}(M)\pmod{{\mathbb{Q}}^{*}}$, and that $L({\mathbb{M}},s)$ does not vanish
for $s=0$ or for $s=1$. If the conjecture relating $c^{+}$ and $L$-values hold
for $M$ and $M(1)$, then one has

$\Lambda({\mathbb{M}},1)\ \approx\ \Lambda({\mathbb{M}},0)\
\pmod{{\mathbb{Q}}^{*}}.$

If $c^{+}(M)\ \approx\ c^{-}(M)$ then $\dim M^{-}$ is necessarily even, as
otherwise $c^{+}(M)$ would be real and $c^{-}(M)$ imaginary. From (3) and Lem.
2.4.1 one has:

$\frac{\Lambda({\mathbb{M}},0)}{\Lambda({\mathbb{M}},1)}\ =\
\frac{L_{\infty}(M,0)}{L_{\infty}(M,1)}\cdot\frac{L({\mathbb{M}},0)}{L({\mathbb{M}},1)}\
\approx\ \pi^{\dim(M^{-})}\cdot\frac{c^{+}(M)}{c^{+}(M(1))}\ \approx\
i^{\dim(M^{-})}\cdot\frac{c^{+}(M)}{c^{-}(M)}.$

Replacing ${\mathbb{Q}}^{*}$ by $E^{*}$, the same applies to motives with
coefficients in $E$.

###  2.5. Multilinear algebra constructions

Suppose $(M_{\alpha})_{\alpha\in A}$ is a finite family of Betti–de Rham
systems of realizations, with $M_{\alpha}$ of weight $w_{\alpha}.$ If
${\mathfrak{r}}=(r_{\alpha})$ and ${\mathfrak{s}}=(s_{\alpha})$ are systems of
non-negative integers, we define $T_{{\mathfrak{r}},{\mathfrak{s}}}$ to be
$(\otimes M_{\alpha}^{\otimes r_{\alpha}})\otimes(\otimes M_{\alpha}^{{\sf
v}\otimes s_{\alpha}}).$ This tensor system is of weight
$\sum_{\alpha}(r_{\alpha}-s_{\alpha})w_{\alpha}$. A multilinear algebra
structure $\mathsf{s}$ on $(M_{\alpha})$ is the data of a family of morphisms

$t_{i}:{\mathbb{Q}}(0)\ \to\
T_{{\mathfrak{r}}(i),{\mathfrak{s}}(i)}((M_{\alpha})),$

(with the targets necessarily of weight zero) and of sub-objects of sums of
tensor systems all of the same weight.

Let $\mathsf{s}$ be such a structure and let $\upsilon\in\\{{\rm B},{\rm
dR}\\}.$ We define $G(\mathsf{s})_{\upsilon}$ to be the algebraic subgroup of
$\prod_{\alpha}{\rm GL}(M_{\alpha\,\upsilon})$ that respects the
$\upsilon$-realizations $\mathsf{s}_{\upsilon}$ of $\mathsf{s}.$ One has
${\mathbb{Q}}(0)_{\upsilon}={\mathbb{Q}}$ and $G(\mathsf{s})_{\upsilon}$ is
the subgroup of $\prod_{\alpha}{\rm GL}(M_{\alpha\,\upsilon})$ which fixes the
tensors $t_{i\upsilon}(1)$ and stabilizes the $\upsilon$-realizations of the
given subobjects. Let the multiplicative group ${\mathbb{G}}_{m}$ act on
$M_{\alpha\,\upsilon}$ by $z\mapsto\mbox{multiplication by $z^{w_{\alpha}}$}.$
The morphism ${\mathbb{G}}_{m}\to\prod_{\alpha}{\rm GL}(M_{\alpha\,\upsilon})$
defining these actions factors through $\boldsymbol{w}:{\mathbb{G}}_{m}\to
G(\mathsf{s})_{\upsilon}$, the central weight morphism. A representation $V$
of $G(\mathsf{s})_{\upsilon}$ is of weight $w$ if $\boldsymbol{w}(z)$ acts on
$V$ by multiplication by $z^{w}.$

The construction which follows formalizes the notion of a system $M$
constructed by multilinear algebra from $(M_{\alpha})$ and $\mathsf{s}.$ The
construction is that of a functor $V\mapsto M^{V}$, compatible with direct
sums, tensor products, and duals, from representations of some weight $w$ of
$\mathrm{G}(\mathsf{s})_{\upsilon}$ to systems of realizations. Applied to the
representation $M_{\alpha\,\upsilon}$, it gives back $M_{\alpha}$. This
construction is a corollary of the fundamental theorem on tannakian
categories. We will first explain this, and then give a more down to earth
description of $V\mapsto M^{V}.$

####  2.5.1. Tannakian formalism giving $V\mapsto M^{V}$

Let $\mathcal{T}$ be the category of finite formal sums of Betti–de Rham
systems of realizations of different weights. In other words, let
$\mathcal{T}$ be the category of systems $N$ consisting of graded vector
spaces $N_{\rm B}=\oplus N_{\rm B}^{w}$, and $N_{\rm dR}=\oplus N_{\rm
dR}^{w}$, and for each $w$ of the data turning $(N_{\rm B}^{w},N_{\rm
dR}^{w})$ into a Betti–de Rham system of realization of weight $w$. With its
natural associative and commutative tensor product, the category $\mathcal{T}$
is tannakian, and the $\otimes$-functors $N\mapsto N_{\rm B}$ and $N\mapsto
N_{\rm dR}$ are fiber functors.

Let $\langle(M_{\alpha})\rangle$ be the subcategory of $\mathcal{T}$ generated
from the $(M_{\alpha})$ by iteratively taking direct sums, tensor products,
duals, and subquotients. It is again tannakian. The fundamental group
$\pi(\langle(M_{\alpha})\rangle,\upsilon)$ is the linear algebraic group of
automorphisms of the $\otimes$-functor $\upsilon$ from
$\langle(M_{\alpha})\rangle$ to ${\mathbb{Q}}$-vector spaces. This fundamental
group maps to $\prod{\rm GL}(M_{\alpha\,\upsilon})$ and this map identifies it
with the closed subgroup of $\prod{\rm GL}(M_{\alpha\,\upsilon})$ which
respects the $\upsilon$-realization of all multilinear algebra structures on
the $M_{\alpha}.$ Now, $\upsilon$ being a fiber functor induces an equivalence
from $\langle(M_{\alpha})\rangle$ to the category of representations of
$\pi(\langle(M_{\alpha})\rangle,\upsilon).$ The group
$\pi(\langle(M_{\alpha})\rangle,\upsilon)$ is a subgroup of
$G(\mathsf{s})_{\upsilon}$. If $V$ is a representation of
$G(\mathsf{s})_{\upsilon}$, then $M^{V}$ is obtained by applying the inverse
equivalence to the restriction of $V$ to
$\pi(\langle(M_{\alpha})\rangle,\upsilon).$

The same argument applies to systems of realizations including in addition
$\ell$-adic realizations and comparison isomorphisms with Betti realization
tensored with ${\mathbb{Q}}_{\ell}$.

####  2.5.2. Explicating the construction of $M^{V}$

Here is a more down to earth description of $V\mapsto M^{V}$, for Betti–de
Rham systems of realizations, when $\upsilon={\rm B}.$ For each $\alpha$, the
Hodge decomposition of $M_{\alpha\,{\rm B}}\otimes{\mathbb{C}}$ defines the
action of ${\mathbb{G}}_{m}\times{\mathbb{G}}_{m}$ on $M_{\alpha\,{\rm
B}}\otimes{\mathbb{C}}$ for which $(z_{1},z_{2})$ acts on $M_{\alpha}^{p,q}$
by multiplication by $z_{1}^{-p}z_{2}^{-q}.$ Each of the tensors $t_{i\,{\rm
B}}(1)$ of the structure $\mathsf{s}_{\rm B}$ is of type $(0,0)$ and fixed by
$\mathsf{F}_{\infty}$, being in the image of the morphism from
${\mathbb{Q}}(0)$ to a system of realizations. Similarly, for each subobject
$N$, part of the multilinear algebra structure, $N_{\rm B}\otimes{\mathbb{C}}$
is sum of its $N^{a,b}$ and is stable by $\mathsf{F}_{\infty}$. The action
${\mathbb{G}}_{m}\times{\mathbb{G}}_{m}\to\prod{\rm GL}(M_{\alpha\,{\rm
B}}\otimes{\mathbb{C}})$ hence factors through $G(\mathsf{s})_{\rm
B}({\mathbb{C}})$, and $\mathsf{F}_{\infty}\in\prod{\rm GL}(M_{\alpha\,{\rm
B}})$ is in $G(\mathsf{s})_{\rm B}({\mathbb{Q}})$.

Let $T$ be the scheme of isomorphisms between the collection of the
$M_{\alpha\,{\rm B}}$, provided with the structure $\mathsf{s}_{\rm B}$, and
the collection of the $M_{\alpha\,{\rm dR}}$, provided with the structure
$\mathsf{s}_{\rm dR}.$ While there are interesting cases where the scheme $T$
has no rational points, it is not empty: the collection of comparison
isomorphisms

$\mathsf{c}\ :=\ (M_{\alpha\,{\rm B}}\otimes{\mathbb{C}}\to M_{\alpha\,{\rm
dR}}\otimes{\mathbb{C}})_{\alpha\in A}$

is a complex point of $T$. As $T$ is not empty, it is a principal homogeneous
space (torsor) for the right composition action of $G(\mathsf{s})_{\rm B}$, as
well as for the left composition action of $G(\mathsf{s})_{\rm dR}.$

For a representation $(\sigma,V)$ of $G(\mathsf{s})_{\rm B}$, one defines

$M^{V}_{\rm B}\ :=\ V,\quad\mbox{$\mathsf{F}_{\infty}$ for $M^{V}$ is
$\sigma(\mathsf{F}_{\infty}).$}$

For the de Rham realization we define

$M^{V}_{\rm dR}\ :=\ \mbox{$V$ twisted by the $G(\mathsf{s})_{\rm B}$-torsor
$T$.}$

It is characterized, up to unique isomorphism, as being given with
$\rho:T\to{\rm Isom}(M^{V}_{\rm B},M^{V}_{\rm dR})$, equivariant for the right
action of $G(\mathsf{s})_{\rm B}$. Over any extension $L$ of ${\mathbb{Q}}$, a
point $p\in T(L)$ defines an isomorphism $\rho(p)$ of $L$-vector spaces from
$M^{V}_{\rm B}\otimes L=V\otimes L$ to $M^{V}_{\rm dR}\otimes L,$ and
$\rho(pg)=\rho(p)\sigma(g),$ for $g\in G(\mathsf{s})_{\rm B}$. This defines
$M^{V}_{\rm dR}\otimes L$ up to unique isomorphism when $T(L)$ is not empty,
and $M^{V}_{\rm dR}$ is defined from these $M^{V}_{\rm dR}\otimes L$ by
descent. One defines the comparison isomorphism $M^{V}_{\rm
B}\otimes{\mathbb{C}}\to M^{V}_{\rm dR}\otimes{\mathbb{C}}$ to be
$\rho(\mathsf{c})$.

The Hodge decompositions of the $M_{\alpha\,{\rm B}}\otimes{\mathbb{C}}$
define a morphism over ${\mathbb{C}}$ from
${\mathbb{G}}_{m}({\mathbb{C}})\times{\mathbb{G}}_{m}({\mathbb{C}})$ to
$G(\mathsf{s})_{\rm B}({\mathbb{C}})$. In turn, this morphism defines a Hodge
decomposition of $V\otimes{\mathbb{C}}=M^{V}_{\rm
B}\otimes{\mathbb{C}}=\oplus(M^{V})^{p,q},$ and the Hodge filtration
$\mathcal{F}^{a}=\oplus_{p\geq a}(M^{V})^{p,q}$ over ${\mathbb{C}}$ of
$M^{V}_{\rm dR}\otimes{\mathbb{C}}$.

######  Lemma 2.5.2.1.

The above filtration $\mathcal{F}^{\bullet}$ of $M^{V}_{\rm
dR}\otimes{\mathbb{C}}$ is rational, i.e., it is the complexification of a
filtration $\mathcal{F}_{\rm dR}$ of $M^{V}_{\rm dR}.$

Granting the lemma one completes the construction of $M^{V}$ by defining the
Hodge filtration of $M^{V}_{\rm dR}$ to be $\mathcal{F}_{\rm dR}.$

###### Proof of Lemma.

Let $S$ be a sum of $T_{{\mathfrak{r}}(i),\mathsf{s}(i)}(M_{\alpha})$, and let
us first suppose that $V$ is a subrepresentation of $S_{\rm B}$. The image of
$V\otimes{\mathbb{C}}$ by $t\in T({\mathbb{C}})$ is independent of $t$. As $T$
is defined over ${\mathbb{Q}}$, this image is the complexification of a
subspace $V_{\rm dR}$ of $S_{\rm dR}$. The morphism $T\to{\rm Isom}(V,V_{\rm
dR})$ identifies $M^{V}_{\rm dR}$ with $V_{\rm dR}.$ The filtration
$\mathcal{F}^{\bullet}$ is rational because it is the trace of the de Rham
filtration of $S_{\rm dR}\otimes{\mathbb{C}}$. For $V$ a subrepresentation of
$S_{\rm B}$, this exhibits $M^{V}$ as a subobject of $S$. Any $V$ can be
realized as a subquotient $V_{1}/V_{2}$ of some $S_{\rm B},$ and the
filtration $\mathcal{F}^{\bullet}$ of $M^{V_{1}/V_{2}}_{\rm
dR}\otimes{\mathbb{C}}$ is rational because it is the image of the rational
filtration of $M^{V_{1}}\otimes{\mathbb{C}}.$ ∎

####  2.5.3. The image of $G(\mathsf{s})_{\rm B}$ under a representation.

######  Proposition 2.5.3.1.

If $(\sigma,V)$ is a representation of $G(\mathsf{s})_{\rm B}$, and if $M^{V}$
is the corresponding system of realizations deduced from the data
$((M_{\alpha}),\mathsf{s},(\sigma,V))$, then the image of $G(\mathsf{s})_{\rm
B}$ in ${\rm GL}(M^{V}_{\rm B})={\rm GL}(V)$ is the group $G(\mathsf{t})_{\rm
B}$ for some multilinear algebra structure $\mathsf{t}$ on $M^{V}.$

This results from the theorem that any algebraic subgroup of ${\rm GL}(V)$ is
the subgroup which respects some multilinear algebra structure $\mathsf{t}$ on
$V$. For an algebraic subgroup of ${\rm GL}(V)$ containing the image of
$G(\mathsf{s})_{\rm B}$, such a structure $\mathsf{t}$ is necessarily the
Betti realization of a multilinear algebra stucture on $M^{V}$.

###  2.6. A criterion for $c^{+}(M)\approx_{E}c^{-}(M)$

Let $M$ be a Betti–de Rham system of realizations, and $\mathsf{s}$ be a
multilinear algebra structure on $M.$ The subgroup $G(\mathsf{s})_{\rm B}$ of
${\rm GL}(M_{\rm B})$ respecting the Betti realization $\mathsf{s}_{\rm B}$ of
$\mathsf{s}$ contains the Frobenius at infinity $\mathsf{F}_{\infty}.$ Let
$Z(\mathsf{F}_{\infty})$ be its centralizer. It acts on the $+1$ and $-1$
eigenspaces $M^{+}$ and $M^{-}$ of $\mathsf{F}_{\infty}.$ Let $\chi^{+}$ and
$\chi^{-}$ be the characters of $Z(\mathsf{F}_{\infty})$ which are the
determinants of the action of $Z(\mathsf{F}_{\infty})$ on $M^{+}$ and $M^{-}.$
If $M$ is with coefficients in $E$, and if $E\to{\rm End}(M)$ is part of the
structure $\mathsf{s}$, then $G(\mathsf{s})_{\rm B}$ is contained in ${\rm
GL}_{E}(M_{\rm B}),$ and one similarly defines algebraic homomorphisms
$\chi_{E}^{+}$ and $\chi^{-}_{E}$ from $Z(\mathsf{F}_{\infty})$ to $E^{*}:$

$\textstyle{Z(\mathsf{F}_{\infty})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi_{E}^{\pm}}$$\textstyle{{\rm
GL}_{E}(M^{\pm})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\det}$$\textstyle{E^{*}}$

To view $E\to{\rm End}(M)$ as a multilinear algebra structure, one encodes it
by endomorphisms of $M$ indexed by a basis of $E$ and viewed as morphisms
${\mathbb{Q}}(0)\to M^{\sf v}\otimes M;$ this uses that ${\rm End}(M)={\rm
Hom}({\mathbb{Q}}(0),M^{\sf v}\otimes M).$ Let $P$ be the subgroup of
$G(\mathsf{s})_{\rm B}({\mathbb{C}})$ which respects the Hodge filtration of
$M_{\rm B}\otimes{\mathbb{C}}.$

As in 2.3.2, let $T$ be the scheme of isomorphisms from $(M_{\rm B},s_{\rm
B})$ to $(M_{\rm dR},s_{\rm dR})$, and $P_{\rm dR}$ be the algebraic subgroup
of $G(s)_{\rm dR}$ respecting $\mathcal{F}_{\rm dR}$. For
$\overline{{\mathbb{Q}}}$ the algebraic closure of ${{\mathbb{Q}}}$ in
${\mathbb{C}}$, ${\rm Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$ acts on set
$\pi_{0}(T({\mathbb{C}}))$ on connected components of $T({\mathbb{C}})$. Let
$K_{0}$ be the field of definition of the connected component
$T({\mathbb{C}})_{0}$ containing the comparison isomorphism $\mathsf{c}\in
T({\mathbb{C}})$. It is the finite extension of ${\mathbb{Q}}$ in
${\mathbb{C}}$ corresponding to the stabilizer of
$T({\mathbb{C}})_{0}\in\pi_{0}(T({\mathbb{C}}))$. Let $K_{1}\subset K_{0}$ the
field of definition of the union $P_{\rm dR}({\mathbb{C}})\
T({\mathbb{C}})_{0}\ Z(\mathsf{F}_{\infty})({\mathbb{C}})$ of connected
components. The most interesting case is when the union is all of
$T({\mathbb{C}})$ (it is so when $G(s)_{\rm B}$ is connected). In that case,
$K_{1}={\mathbb{Q}}$.

######  Theorem 2.6.1.

Suppose that $M$ is with coefficients in $E$, and $E\to{\rm End}(M)$ is part
of the multilinear algebra structure $\mathsf{t}$. If the weight $w$ of $M$ is
even, then suppose also that then $M^{w/2,w/2}=0.$

  1. (i)

If the restrictions of $\chi^{+}_{E}$ and $\chi^{-}_{E}$ to the connected
component $Z(\mathsf{F}_{\infty})^{0}$ of $Z(\mathsf{F}_{\infty})$ are equal,
then $c^{+}(M)/c^{-}(M)$ is algebraic. More precisely,
$c^{+}(M)/c^{-}(M)\in(E\otimes K_{0})^{*}.$

  2. (ii)

If $\chi^{+}_{E}=\chi^{-}_{E}$, then $c^{+}(M)/c^{-}(M)\in(E^{*}\otimes
K_{1})^{*}.$ Moreover, if $G(\mathsf{s})_{\rm B}$ is connected, then
$c^{+}(M)/c^{-}(M)\in E^{*}$.

###### Proof.

Fix bases over $E$ of $M^{\pm}$, and of $M_{\rm dR}/\mathcal{F}^{n}$ where
$2n$ or $2n-1$ is the weight of $M.$ Given $t\in T$, define $t^{\pm}$ by the
diagram:

$\textstyle{M^{\pm}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t^{\pm}}$$\textstyle{M_{\rm
dR}/\mathcal{F}^{n}}$$\textstyle{M_{\rm
B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t}$$\textstyle{M_{\rm
dR}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Define algebraic functions $f^{\pm}:T\to E$ as $f^{\pm}(t):=\det(t^{\pm}).$
Here, $E$ is viewed as the scheme over ${\mathbb{Q}}$ whose points with
coefficients in a ${\mathbb{Q}}$-algebra $R$ is $E\otimes R,$ and $f^{\pm}$
maps $t\in T(R)$ to the determinant of $t^{\pm}:M^{\pm}\otimes R\to M_{\rm
dR}/\mathcal{F}^{n}\otimes R,$ computed with respect to the $E$-bases fixed
above; $f^{\pm}(t)$ is in $E\otimes R$ and as an element of $(E\otimes
R)/E^{*}$ it is independent of choice of bases.

Let $\chi_{\rm dR}:P_{\rm dR}\to E^{*}$ be the determinant of the action of
$P_{\rm dR}$ on $M_{\rm dR}/\mathcal{F}^{n}.$ One has

(6) $c^{\pm}(M)\ =\ f^{\pm}(\mathsf{c}),$

where $\mathsf{c}\in T({\mathbb{C}})$ is the comparison isomorphism.
Furthermore, for all $t\in T$, $p\in P_{\rm dR}$ and $z\in
Z(\mathsf{F}_{\infty})$ we have the equivariance:

(7) $f^{\pm}(ptz)\ =\ \chi_{\rm dR}(p)f^{\pm}(t)\chi^{\pm}(z).$

######  Lemma 2.6.2.

Under the assumption (i), on the connected component of $T({\mathbb{C}})$
containing $\mathsf{c}$, $f^{+}$ and $f^{-}$ are generically invertible and
$f^{+}/f^{-}$ is a constant.

###### Proof.

By (6), $f^{+}$ and $f^{-}$, with values in $E\otimes{\mathbb{C}},$ are
invertible at the point $\mathsf{c}\in T({\mathbb{C}})$, and hence in a
Zariski neighbourhood of $\mathsf{c}$. In this neighbourhood, it follows from
(7) that $f^{+}/f^{-}$ is constant on the double coset $P_{\rm
dR}({\mathbb{C}})\mathsf{c}Z(\mathsf{F}_{\infty})^{0}({\mathbb{C}}).$ It
suffices to check that this double coset contains an open neighbourhood of
$\mathsf{c}$. If we identify $G(\mathsf{t})_{\rm B}({\mathbb{C}})$ with
$T({\mathbb{C}})$ by $g\mapsto\mathsf{c}g,$ then this means showing that ${\rm
Lie}(G(\mathsf{t})_{\rm B}({\mathbb{C}}))$ is the sum of ${\rm
Lie}(Z(\mathsf{F}_{\infty})({\mathbb{C}}))$ and ${\rm Lie}(P_{\rm
dR}({\mathbb{C}})).$ Indeed, the adjoint representation of $G(\mathsf{t})_{\rm
B}$ is the Betti realization of the system of realizations of weight $0$ it
defines; call it $N$. In it, ${\rm Lie}(Z(\mathsf{F}_{\infty}))$ is $N^{+}$
while ${\rm Lie}(P_{\rm dR})$ is $\mathcal{F}^{0}.$ One concludes by observing
that in any system of realizations $N$ of weight $0$, $N_{\rm
B}\otimes{\mathbb{C}}$ is the sum of $N_{\rm B}^{+}\otimes{\mathbb{C}}$ and
$\mathcal{F}^{0}.$ ∎

Proof of Thm. 2.6.1 (i). The scheme $T$ is over ${\mathbb{Q}}$. The connected
component $T({\mathbb{C}})_{0}$ of $T({\mathbb{C}})$ containing $\mathsf{c}$
is defined over $K_{0}\subset{\mathbb{C}},$ meaning that $T_{K_{0}}$ has a
connected component $T_{K_{0}}^{0}$ with
$T_{K_{0}}^{0}({\mathbb{C}})=T({\mathbb{C}})_{0}.$ On the scheme
$T_{K_{0}}^{0}$ over $K$, $f^{+}/f^{-}$ is constant. The constant value must
be in $(E\otimes K_{0})^{*}.$

Proof of Thm. 2.6.1 (ii). The union $U:=P_{\rm
dR}({\mathbb{C}})T({\mathbb{C}})_{0}Z(\mathsf{F}_{\infty})({\mathbb{C}})$ of
connected components of $T({\mathbb{C}})$ is defined over $K_{1}$, and it
follows from (2.6.2) that $f^{+}$ and $f^{-}$ are generically invertible on
this union. As $f^{+}$ and $f^{-}$ are defined over ${\mathbb{Q}}$, the union
$U_{1}$ of the connected components of $U$ on which $f^{+}/f^{-}$ is constant
is again defined over $K_{1}$. On it, $f^{+}/f^{-}$ is constant. The constant
value must be in $E\otimes K_{1}.$ ∎

######  Remark 2.6.3.

Suppose that $M$ has even weight 0, that $M^{0,0}\neq 0$, and that
$\mathsf{F}_{\infty}$ acts on $M^{0,0}$ either by $+1$, or by $-1$, so that
$c^{\pm}(M)$ is defined. Let $\varphi:P_{\rm dR}\rightarrow E^{*}$ be the
determinant character of the action of the action $P_{\rm dR}$ on
$Gr^{0}_{\mathcal{F}}(M_{\rm dR})$. Then, the conclusion of theorem continues
to hold, under the additional assumption, for (i), that $\varphi$ is trivial
on the connected component $P^{0}_{\rm dR}$ of $P_{\rm dR}$, and for (ii),
that $\varphi$ is trivial. This is a special case of the results of the next
section, and the proof is essentially the same.

######  Remark 2.6.4.

In all the examples that we consider in the next section, $G(\mathsf{s})^{0}$
is reductive. When this is the case, one could have argued that $P_{\rm
dR}\cap G(\mathsf{s})^{0}({\mathbb{C}})$ is a parabolic subgroup, and that
$Z(\mathsf{F}_{\infty})\cap G(\mathsf{s})^{0}$ is the fixed points of an
involution, to obtain that on any connected component of $T({\mathbb{C}})$,
some double coset for the left action of $P_{\rm dR}({\mathbb{C}})^{0}$ and
the right action of $Z(\mathsf{F}_{\infty})^{0}({\mathbb{C}})$ is dense.

###  2.7. Multiplicative relations between determinants of periods

#### 2.7.1.

As in 2.5, let $(M_{\alpha})_{\alpha\in A}$ be a finite family of Betti–de
Rham realizations, let $\mathsf{s}$ be a multilinear structure, and define
$G(s)_{\rm B}$ to be its automorphism group in the Betti realization.

Let $(V_{i})$ be a finite family of representations of $G(s)_{\rm B}$ of
weights $w_{i}$ on $E$-vector spaces. The $M_{i}:=M^{V_{i}}$ are with
coefficients in $E$, and one can attach to each of them the following
determinants of periods, in $(E^{*}\otimes{\mathbb{C}})/E^{*}$ :

  1. (a)

$c^{\pm}_{E}(M_{i}).$ When $w_{i}$ is even, $w_{i}=2p$, they are defined only
under the condition that $\mathsf{F}_{\infty}$ acts on $(M^{V_{i}})^{p,p}$
either by $+1$, or by $-1$.

  2. (b)

The determinant $\delta_{E}(M_{i})$ of the comparison isomorphism $M_{i,{\rm
B}}\otimes{\mathbb{C}}\xrightarrow{\sim}M_{i,{\rm dR}}\otimes{\mathbb{C}}$,
computed using $E$-bases of $M_{i,{\rm B}}$ and $M_{i,{\rm dR}}.$

To each of those is associated characters
$\chi^{\pm}:Z(\mathsf{F}_{\infty})\to E^{*}$ and a character $\varphi:P_{\rm
dR}\to E^{*}.$ The character $\chi^{\pm}$ of $Z(\mathsf{F}_{\infty})$ is
respectively the determinant of the action of $Z(\mathsf{F}_{\infty})$ on
$M^{+}_{i}$ and $M^{-}_{i}.$ The character of $P_{\rm dR}$ is the determinant
of the action of $P_{\rm dR}$ on $M_{i_{\rm dR}}/\mathcal{F}^{p}_{\rm dR}$,
where this quotient of $M_{i_{\rm dR}}$, complexified, is the target used in
the definition of the respective $c^{\pm}$ or $\delta$.

Let now $m$ be a monomial (with exponents in ${\mathbb{Z}}$) in the
$c^{\pm}_{E}(M_{i})$ and $\delta(M_{i})$, and let $\chi_{m}$ and $\varphi_{m}$
be the corresponding product of powers of the associated characters of
$Z(\mathsf{F}_{\infty})$ and $P_{\rm dR}$. With the same definition of $K_{0}$
and $K_{1}$ as before, one has

######  Theorem 2.7.2.

The image of $m$ in $(E^{*}\otimes{\mathbb{C}})/(E^{*}\otimes K^{0})^{*}$
(resp., in $(E^{*}\otimes{\mathbb{C}})/(E\otimes K^{1})^{*}$) is uniquely
determined the restriction of $\chi_{m}$ and $\varphi_{m}$ to
$Z(\mathsf{F}_{\infty})^{0}$ and $P^{0}$ (resp., by $\chi_{m}$ and
$\varphi_{m}$). Special cases: If $G(s)_{\rm B}$ is connected, the image of
$m$ in $(E^{*}\otimes{\mathbb{C}})/E^{*}$ is uniquely determined by $\chi_{m}$
and $\varphi_{m}$. If further $\chi_{m}$ and $\varphi_{m}$ are trivial, $m$ is
in $E^{*}$.

The proof is the same as before: $m$ is the value at $\mathsf{c}$ of a
function $f$ on $U\subset T$, with values in $E$, with the covariance property

$f(ptz)=\varphi_{m}(p)f(t)\chi_{m}(z).$

####  2.7.3. Grothendieck period conjecture.

If the $M_{\alpha}$’s come from motives ${\mathbb{M}}_{\alpha}$’s, and if
$\mathsf{s}$ consists of all multilinear structures on the motives
${\mathbb{M}}_{\alpha}$, so that $G(\mathsf{s})$ is the “motivic Galois group”
of the family of ${\mathbb{M}}_{\alpha}$, Grothendieck conjectures that
$\mathsf{c}$ is Zarisky dense in $T$. This conjecture implies that if
monomials $m(j)$ as in Sect. 2.7 give rise to multiplicatively independent
characters $(\chi(j),\varphi(j)),$ the $m(j)$ are algebraically independent.

######  Example 2.7.4.

Let $M$ be a Betti–de Rham system of realizations of dimension 2, weight $w$
and two nonzero Hodge numbers $h^{pq}=h^{qp}=1.$ We give it no multilinear
structure, so that $G_{\rm B}:=G_{\rm B}(\phi)$ is $\rm GL(M_{\rm B}).$ In a
suitable basis $e^{+},\ e^{-}$ of $M_{\rm B},$ $\mathsf{F}_{\infty}$ is the
diagonal matrix ${\rm diag}(+1,-1)$ and its centralizer
$Z(\mathsf{F}_{\infty})$ is, in that basis, the group of diagonal matrices.
The characters of $Z(\mathsf{F}_{\infty})$ are the ${\rm diag}(a,b)\mapsto
a^{m}b^{n},$ for integers $m,n.$ Suppose $p>q.$ In a basis $(e,f)$ of $M_{\rm
dR},$ with $e$ spanning $\mathcal{F}^{p}_{\rm dR},P_{\rm dR}$ is the group of
upper triangular matrices. Its characters are

$\begin{pmatrix}a&b\\\ 0&d\end{pmatrix}\longmapsto a^{r}\,d^{s}$

for integers $r,s$. If $V$ is an irreducible representation of $G_{\rm B}$,
the corresponding characters of $Z(\mathsf{F}_{\infty})$ and $P_{\rm dR}$ are
hence given by four integers $(m,n,r,s).$ Expressing that the scalar matrices
are in both $Z(\mathsf{F}_{\infty})$ and $P$, one finds that $m+n=r+s.$ For
the representation $M_{\rm B}$ itself, one finds that the characters
corresponding to $c^{+},c^{-}$ and $\delta$ are respectively given by the
$4$-tuples

$c^{+}\ :\ (1,0,0,1),\quad\ c^{-}\ :\ (0,1,0,1),\quad\ \delta\ :\ (1,1,1,1).$

The ${\mathbb{Z}}$-span of these three $4$-tuples is the additive group of all
integral $4$-tuples $(m,n,r,s)$ for which $m+n=r+s.$ It follows that for any
irreducible representation $V$ of $\rm GL(M_{\rm B}),$ $c^{+}(M^{V}),$
$c^{-}(M^{V}),$ and $\delta(M^{V})$ are, mod ${\mathbb{Q}}^{*},$ monomials in
$c^{+}(M),$ $c^{-}(M),$ and $\delta(M).$ Computing the relevant $m,n,r,s,$ one
finds the formula of [D79, Prop. 7.7] computing the $c^{\pm}({\rm Sym}^{n}M).$

If $M$ comes from a motive ${\mathbb{M}}$ and if ${\sf s}$ is all multilinear
structures of $M.$ The group $G_{\rm B}({\mathbb{C}})$ is at least as big as a
Cartan normalizer. Indeed, it contains the
${\mathbb{G}}_{m}\times{\mathbb{G}}_{m}$ giving the Hodge decomposition for
$M_{\rm B}\otimes{\mathbb{C}},$ as well as $F_{\infty}$ which permutes the two
factors ${\mathbb{G}}_{m}.$ If it is strictly bigger, it must be the full $\rm
GL.$ In this non-CM case, Grothendieck period conjecture implies that
$c^{+}(M)$ and $c^{-}(M)$ are algebraically independent.

From the automorphic point of view, we are here considering classical
holomorphic new cuspforms of weight $k\geqslant 2.$ Such forms give rise to
$M$ as above, with $h^{k-1,0}=h^{0,k-1}=1.$

##  3\. Examples

###  3.1. Tensor product motives over ${\mathbb{Q}}$

Let $M^{\prime}$ and $M^{\prime\prime}$ be Betti–de Rham systems of
realizations over ${\mathbb{Q}}$ with coefficients in $E,$ and $M$ their
tensor product over $E.$

######  Proposition 3.1.1.

Suppose that $M^{\prime}$ and $M$ have vanishing middle Hodge numbers. For
$M^{\prime\prime}$, let
$\delta^{\prime\prime}=\dim(M^{\prime\prime+})-\dim(M^{\prime\prime-}).$ Then

$\frac{c^{+}(M)}{c^{-}(M)}\ \approx_{E}\
\left(\frac{c^{+}(M^{\prime})}{c^{-}(M^{\prime})}\right)^{\delta^{\prime\prime}}.$

In particular, if we assume
$\dim(M^{{}^{\prime\prime}+})=\dim(M^{{}^{\prime\prime}-})$, which is the case
if $M^{\prime\prime}$ also has vanishing middle Hodge number, then
$c^{+}(M)\approx_{E}c^{-}(M).$

This result is due to Yoshida [Y01], and our Sect. 2.6 formalizes his
argument. In Sect. 3.5, we will show that the same result holds for systems of
realizations over a totally real field.

###### Proof.

As multilinear structures over $M^{\prime}$ and $M^{\prime\prime}$, we take
only the structures of $E$-modules. The group $G(\mathsf{s})_{\rm B}$ to
consider is ${\rm GL}_{E}(M^{\prime}_{\rm B})\times{\rm
GL}_{E}(M^{\prime\prime}_{\rm B})$, and $M$ corresponds to the representation
$M^{\prime}_{\rm B}\otimes M^{\prime\prime}_{\rm B}$. The centralizer of
$\mathsf{F}_{\infty}$ is the product of its centralizers in ${\rm
GL}_{E}(M^{\prime}_{\rm B})$ and ${\rm GL}_{E}(M^{\prime\prime}_{\rm B}).$ The
assumption of vanishing middle Hodge number of $M^{\prime}$ implies that its
$E$-dimension is even, say, $2n^{\prime}$; one has
$\dim(M^{\prime+})=\dim(M^{\prime-})=n^{\prime}$. For $M^{\prime\prime}$,
define

$n^{\prime\prime}(+):=\dim(M^{\prime\prime+}),\quad n^{\prime\prime}(-)\ :=\
\dim(M^{\prime\prime-}),$

hence $\delta^{\prime\prime}=n^{\prime\prime}(+)-n^{\prime\prime}(-).$ If $V$
and $W$ are vector spaces (over $E$), then the determinant $\det(A\otimes B)$
of the action of $(A,B)\in{\rm GL}(V)\otimes{\rm GL}(W)$ on $V\otimes W$ is
given by $\det(A)^{\dim(W)}\det(B)^{\dim(V)}.$ From the isomorphisms

$M^{+}\ =\ M^{\prime+}\otimes M^{\prime\prime+}\ \oplus\ M^{\prime-}\otimes
M^{\prime\prime-},\quad M^{-}\ =\ M^{\prime+}\otimes M^{\prime\prime-}\
\oplus\ M^{\prime-}\otimes M^{\prime\prime+},$

we get for the determinant of the action of $Z(\mathsf{F}_{\infty})$

$\displaystyle\chi_{M}^{+}$ $\displaystyle=$
$\displaystyle(\chi_{M^{\prime}}^{+})^{n^{\prime\prime}(+)}(\chi_{M^{\prime}}^{-})^{n^{\prime\prime}(-)}\cdot(\chi_{M^{\prime\prime}}^{+}\chi_{M^{\prime\prime}}^{-})^{n^{\prime}}$
$\displaystyle\chi_{M}^{-}$ $\displaystyle=$
$\displaystyle(\chi_{M^{\prime}}^{+})^{n^{\prime\prime}(-)}(\chi_{M^{\prime}}^{-})^{n^{\prime\prime}(+)}\cdot(\chi_{M^{\prime\prime}}^{+}\chi_{M^{\prime\prime}}^{-})^{n^{\prime}},$

so that
$\chi_{M}^{+}/\chi_{M}^{-}=(\chi_{M^{\prime}}^{+}/\chi_{M^{\prime}}^{-})^{\delta^{\prime\prime}}.$
We deduce the following relation from Thm. 2.7.2:

(8) $\frac{c^{+}(M)}{c^{-}(M)}\ \approx_{E}\
\left(\frac{c^{+}(M^{\prime})}{c^{-}(M^{\prime})}\right)^{\delta^{\prime\prime}}.$

It remains to check that both sides correspond to the same character of
$P_{\rm dR}$, which is the product of the $P_{\rm dR}$ for ${\rm
GL}(M^{\prime}_{\rm dR})$ and ${\rm GL}(M^{\prime\prime}_{\rm dR}).$ Indeed,
as $M$ as well as $M^{\prime}$ have vanishing middle Hodge numbers, the
relevant characters are trivial.

Furthermore, if we assume $\delta^{\prime\prime}=0$, then the second assertion
in the proposition trivially follows from the first. ∎

#### 3.1.2.

Let us now assume only that $c^{+}$ and $c^{-}$ are defined for $M^{\prime}$,
$M^{\prime\prime}$, and $M$, and that both $M^{\prime}$ and $M^{\prime\prime}$
have nonvanishing middle Hodge numbers. Let $2p^{\prime}$ and
$2p^{\prime\prime}$ be the weights of $M^{\prime}$ and $M^{\prime\prime}$, and
define $p=p^{\prime}+p^{\prime\prime}.$ By assumption, $\mathsf{F}_{\infty}$
acts by a sign, i.e., by either $+1$ or $-1$, on $M^{\prime
p^{\prime}p^{\prime}};$ call $s(M^{\prime})$ this sign. Similarly for
$M^{\prime\prime}.$ The assumption that $c^{\pm}(M)$ are defined implies that
$M^{\prime p^{\prime}p^{\prime}}\otimes M^{\prime\prime
p^{\prime\prime}p^{\prime\prime}}\simeq M^{pp},$ and that
$s(M)=s(M^{\prime})s(M^{\prime\prime}).$ With the notations of Prop. 3.1.1,
one has $\delta^{\prime\prime}=s(M^{\prime\prime})\dim(M^{\prime\prime
p^{\prime\prime}p^{\prime\prime}}).$ Defining similarly $\delta^{\prime}$ for
$M^{\prime}$, we get the following proposition:

######  Proposition 3.1.3.

Under the assumptions of 3.1.2, one has:

$\frac{c^{+}(M)}{c^{-}(M)}\ \approx_{E}\
\left(\frac{c^{+}(M^{\prime})}{c^{-}(M^{\prime})}\right)^{\delta^{\prime\prime}}\left(\frac{c^{+}(M^{\prime\prime})}{c^{-}(M^{\prime\prime})}\right)^{\delta^{\prime}}.$

Concerning the relevant characters of $Z(\mathsf{F}_{\infty})$, the proof is
the same, but the relevant characters of $P_{\rm dR}=P^{\prime}_{\rm dR}\times
P^{\prime\prime}_{\rm dR}$ are now nontrivial. For $c^{+}(M)/c^{-}(M)$, it is
given by $\det{\rm Gr}^{p}_{\mathcal{F}}(M_{\rm dR})^{s(M)}.$ Similarly for
$c^{+}(M^{\prime})/c^{-}(M^{\prime})$ and
$c^{+}(M^{\prime\prime})/c^{-}(M^{\prime\prime}).$ One concludes by using that

$\displaystyle\det{\rm Gr}^{p}_{F}(M_{\rm dR})^{s(M)}$ $\displaystyle=$
$\displaystyle\left(\det{\rm Gr}^{p^{\prime}}_{\mathcal{F}}(M^{\prime}_{\rm
dR})^{\dim(M^{\prime\prime p^{\prime\prime}p^{\prime\prime}})}\cdot\det{\rm
Gr}^{p^{\prime\prime}}_{\mathcal{F}}(M^{\prime\prime}_{\rm
dR})^{\dim(M^{\prime p^{\prime}p^{\prime}})}\right)^{s(M)}$ $\displaystyle=$
$\displaystyle\left[\det{\rm Gr}^{p^{\prime}}_{\mathcal{F}}(M^{\prime}_{\rm
dR})^{s(M^{\prime})}\right]^{\delta^{\prime\prime}}\cdot\left[\det{\rm
Gr}^{p^{\prime\prime}}_{\mathcal{F}}(M^{\prime\prime}_{\rm
dR})^{s(M^{\prime\prime})}\right]^{\delta^{\prime}}.$

#### 3.1.4.

As explained in [BR13], Prop. 3.1.1 gives a direct motivic explanation, via
[D79], of the main results of [HR20] when the base field is ${\mathbb{Q}}$.

It is worth noting that if $M^{\prime}$ (resp., $M^{\prime\prime}$) is the
motive of even (resp., odd) dimension, say, $2n^{\prime}$ (resp.,
$2n^{\prime\prime}+1$), conjecturally attached to a cohomological cuspidal
automorphic representation $\sigma^{\prime}$ (resp., $\sigma^{\prime\prime}$)
of ${\rm GL}(2n^{\prime})$ (resp., ${\rm GL}(2n^{\prime\prime}+1)$), then
because the Langlands parameter of the representation at infinity is regular,
one has $\delta^{\prime}=0$ and $\delta^{\prime\prime}=\pm 1$. This explains
why in [HR20], for the theorem on ratios of $L$-values for a Rankin–Selberg
$L$-function $L(s,\sigma^{\prime}\times\sigma^{\prime\prime})$ for ${\rm
GL}(2n^{\prime})\times{\rm GL}(2n^{\prime\prime}+1)$, one sees the relative
period $\Omega(\sigma^{\prime})$ of $\sigma^{\prime}$ or its reciprocal; the
relative period $\Omega(\sigma^{\prime})$ corresponds to the ratio
$c^{+}(M^{\prime})/c^{-}(M^{\prime})$ in the right hand side of Prop. 3.1.1,
and whether it or its reciprocal appears is dictated by
$\delta^{\prime\prime}=\pm 1$.

####  3.1.5. Example: The tensor product of two rank-two motives-I

For $i=1,2$, let $\varphi_{i}\in S_{k_{i}}(N,\omega_{i})$ be a primitive
holomorphic cuspidal modular form of weight $k_{i}\geq 2$, for
$\Gamma_{0}(N)$, and nebetypus character $\omega_{i}$; primitive means that it
is an eigenform, a newform, and normalized $a_{1}(\varphi_{i})=1.$ Assume
(only for convenience) that $k_{1}$ and $k_{2}$ are even. Let $E$ be a number
field which is large enough to contain all the Fourier coefficients of
$\varphi_{1}$ and $\varphi_{2}$. Let $M_{i}=M(\varphi_{i})$ be the rank-two
motive over ${\mathbb{Q}}$ with coefficients in $E$ associated to
$\varphi_{i}$ (see [Sc90]). Put $M=M_{1}\otimes M_{2}.$ Assume that $k_{1}\neq
k_{2}$. Then $M_{1}$, $M_{2},$ and $M$ have vanishing middle Hodge numbers;
furthermore, the $L$-function of $M$, which is the degree-4 Rankin–Selberg
$L$-function, has at least two critical points. We know from Prop. 3.1.1 that
$c^{+}(M)\approx_{E}c^{-}(M).$ This is compatible with the statement that for
the completed Rankin–Selberg $L$-function
$L(m,\varphi_{1}\times\varphi_{2})\approx_{E}L(m+1,\varphi_{1}\times\varphi_{2}),$
for $\min\\{k_{1},k_{2}\\}\leq m<m+1<\max\\{k_{1},k_{2}\\}$, which follows
from Shimura [Shi77, Thm. 4]. See also Blasius [Bl87]. This example also works
Hilbert modular forms.

#### 3.1.6.

When the base field is a general number field the situation is more delicate.
See below for results that hold for systems of realizations over number fields
$L$ which require additional arguments: as is the case for $L$-functions,
$c^{+}$ and $c^{-}$ are defined by first restricting from $L$ to
${\mathbb{Q}}$, but this operation does not commute with taking a tensor
product or with the construction of group of automorphisms $G(\mathsf{s}).$

###  3.2. Artin Motives

Let us review Artin motives, which will be used in the next section. If, in
Grothendieck’s definition of motives, one considers only varieties of
dimension zero, one obtains the Artin motives. They give rise to the Artin
$L$-functions. An equivalent, more down to earth definition is that an Artin
motive is a representation of ${\rm
Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$ on a ${\mathbb{Q}}$-vector space,
for $\overline{{\mathbb{Q}}}$ an algebraic closure of ${\mathbb{Q}}$. This
category does not depend on a choice of $\overline{{\mathbb{Q}}}$. Let us take
for $\overline{{\mathbb{Q}}}$ the algebraic closure of ${\mathbb{Q}}$ in
${\mathbb{C}}.$ The Betti–de Rham system of realizations attached to the
representation $V$ is then given as follows:

  * •

$V_{\rm B}$: the vector space $V$; $\mathsf{F}_{\infty}$ is the action of
complex conjugation.

  * •

$V_{\rm dR}$: $(V\otimes\overline{{\mathbb{Q}}})^{{\rm
Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})},$ purely of Hodge filtration $0.$

  * •

Comparison isomorphism: the inclusion of $V_{\rm dR}$ in
$V\otimes\overline{{\mathbb{Q}}}$ extends to an isomorphism of $V_{\rm
dR}\otimes\overline{{\mathbb{Q}}}$ to $V\otimes\overline{{\mathbb{Q}}}$, which
one complexifies.

Example: Let $L$ be a finite extension of ${\mathbb{Q}}.$ Then $M:=H^{0}({\rm
Spec}\,L)$ of Sect. 2.1 is an Artin motive. The set of complex points of ${\rm
Spec}\,L$ is the set ${\rm Hom}(L,{\mathbb{C}})$ of complex embeddings of $L$;
$M_{\rm B}$ is hence ${\mathbb{Q}}^{{\rm Hom}(L,{\mathbb{C}})}$, the sum of
copies of ${\mathbb{Q}}$ indexed by ${\rm Hom}(L,{\mathbb{C}}).$ The de Rham
cohomology $M_{\rm dR}$ is $L.$ That $L\otimes{\mathbb{C}}$ is a direct sum of
copies of ${\mathbb{C}}$ indexed by ${\rm Hom}(L,{\mathbb{C}})$ gives the
comparison isomorphism. This turns $H^{0}({\rm Spec}\,L)$ into a ring in the
category of Artin motives. (The reader is also referred to [D79, Sect. 6].)

######  Proposition 3.2.1.

The functor from Artin motives (i.e., representations $V$ of ${\rm
Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$) to Betti–de Rham system of
realizations is fully faithful.

###### Proof.

One recovers the action of ${\rm Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$
on $V=V_{\rm B}$ by embedding $V_{\rm B}$ in $V_{\rm
dR}\otimes\overline{{\mathbb{Q}}}$ by the comparison isomorphism, and by
restricting to $V_{B}$ the action of ${\rm
Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$ on $V_{\rm
dR}\otimes\overline{{\mathbb{Q}}}$. ∎

######  Remark 3.2.2.

Let $M$ be the Betti–de Rham system defined by the representation $V$ of ${\rm
Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$, and let ${\rm Gal}$ be the image
of ${\rm Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$ in ${\rm GL}(V)$. From
the Tannakian point of view of Sect. 2.5.1, what Prop. 3.2.1 tells us is that
if on $M$ one puts all the multilinear structures it admits, the corresponding
$G(\mathsf{s})$ is ${\rm Gal}.$ Special case: if $M=H^{0}({\rm Spec}\,L),$
where $L$ is a finite extension of ${\mathbb{Q}}$, and if $N$ is the normal
closure of $L$ in $\overline{{\mathbb{Q}}}\subset{\mathbb{C}}$, generated by
the images of the embeddings of $L$ in $\overline{{\mathbb{Q}}}$, one has
$G(\mathsf{s})={\rm Gal}(N/{\mathbb{Q}}).$

###  3.3. Systems of Betti–de Rham realizations over a number field $L$

The motivating example is the cohomology $H^{n}(X)$ of a projective
nonsingular variety $X$ over $L$: for such a variety, if $\sigma$ is an
embedding of $L$ into an algebraic closure $C$ of ${\mathbb{R}}$, $X(C)$ is
defined and so is $H^{n}(X(C),{\mathbb{Q}}).$ One also has the de Rham
cohomology $H_{\rm dR}^{n}(X)$, a vector space over $L.$

#### 3.3.1.

A weight $w$ Betti–de Rham system of realization over $L$ consists of the
following data $({\rm i}_{L})-({\rm iii}_{L})$ obeying $({\rm iv}_{L})$:

  1. $({\rm i}_{L})$

For $\sigma$ an embedding of $L$ into an algebraic closure $C$ of
${\mathbb{R}}$, a ${\mathbb{Q}}$-vector space $M_{\sigma}$, depending
functorially on $C.$ If for $C$ we take the standard algebraic closure
${\mathbb{C}}$ of ${\mathbb{R}}$, the functoriality in $C$ amounts to an
involutive system $\mathsf{F}_{\infty}:M_{\sigma}\to M_{c\sigma}$ of
isomorphisms. If $\sigma$ is real, i.e., when $\sigma=c\sigma$, the involution
$\mathsf{F}_{\infty}$ of $M_{\sigma}$ defines a decomposition
$M_{\sigma}=M_{\sigma}^{+}\oplus M_{\sigma}^{-}$ into the $\pm 1$ eigenspaces.

  2. $({\rm ii}_{L})$

An $L$-vector space $M_{\rm dR}$, endowed with a finite decreasing filtration
$\mathcal{F}_{\rm dR}$, the Hodge filtration.

  3. $({\rm iii}_{L})$

For $\sigma$ as in $({\rm i}_{L})$, a comparison isomorphism
$M_{\sigma}\otimes_{\mathbb{Q}}C\to M_{\rm dR}\otimes_{L,\sigma}C$, which is
functorial in $C$.

  4. $({\rm iv}_{L})$

One requires that for $\sigma$ as in $({\rm i}_{L})$, if the complexified
Hodge filtration is transported by the comparison isomorphism to a filtration,
still denoted $\mathcal{F}$, of $M_{\sigma}\otimes_{\mathbb{Q}}C$, it and its
complex conjugate are $w$-opposite.

By $({\rm iv}_{L})$, for each $\sigma$, $M_{\sigma}$ is endowed with a Hodge
structure of weight $w$. Its Hodge numbers $h_{\sigma}^{p,q}=\dim_{L}{\rm
Gr}_{\mathcal{F}}^{p}(M_{\rm dR})$ are independent of $\sigma.$

The restriction of base field, from $L$ to ${\mathbb{Q}}$, of such a system is
defined by:

$\displaystyle{\rm Res}_{L/{\mathbb{Q}}}(M)_{\rm B}\ $ $\displaystyle:=$
$\displaystyle\bigoplus_{\sigma:L\to{\mathbb{C}}}M_{\sigma},$
$\displaystyle{\rm Res}_{L/{\mathbb{Q}}}(M)_{\rm dR}$ $\displaystyle:=$
$\displaystyle M_{\rm dR}\ \mbox{viewed as a ${\mathbb{Q}}$-vector space.}$

One has ${\rm Res}_{L/{\mathbb{Q}}}(M)_{\rm
dR}\otimes_{\mathbb{Q}}{\mathbb{C}}\ \simeq\
\oplus_{\sigma:L\to{\mathbb{C}}}\,M_{\rm dR}\otimes_{L,\sigma}{\mathbb{C}},$
and the comparison isomorphism for ${\rm Res}_{L/{\mathbb{Q}}}(M)$ is the sum
over all $\sigma:L\to{\mathbb{C}}$ of the comparison isomorphisms of $M.$

One defines $c^{\pm}(M)$ to be $c^{\pm}({\rm Res}_{L/{\mathbb{Q}}}(M))$. This
parallels the fact that when $M$ comes from a motive ${\mathbb{M}}$ with an
$L$-function $L({\mathbb{M}},s)$ (which is defined as an Euler product over
all places of $L$), one has

$L({\mathbb{M}},s)\ =\ L({\rm Res}_{L/{\mathbb{Q}}}({\mathbb{M}}),s).$

In the case of the motivating example, if for $X\to{\rm Spec}\,L$, a variety
over $L$, one defines $\amalg_{L/{\mathbb{Q}}}(X)$ to be the variety over
${\mathbb{Q}}$ as $X\to{\rm Spec}\,L\to{\rm Spec}\,{\mathbb{Q}}$, then one has

${\rm Res}_{L/{\mathbb{Q}}}(H^{n}(X))\ =\ H^{n}(\amalg_{L/{\mathbb{Q}}}(X)).$

For the Betti realization, $\amalg_{L/{\mathbb{Q}}}(X)({\mathbb{C}})$ is the
disjoint union over embeddings $\sigma:L\to{\mathbb{C}}$ of the
${\mathbb{C}}$-points of $X_{\sigma}:=X\times_{L,\sigma}{\mathbb{C}}$; hence,
$H^{n}(\amalg_{L/{\mathbb{Q}}}(X)({\mathbb{C}}),{\mathbb{Q}})=\oplus_{\sigma:L\to{\mathbb{C}}}H^{n}(X_{\sigma}({\mathbb{C}}),{\mathbb{Q}}).$

####  3.3.2. Description as module

The functor ${\rm Res}_{L/{\mathbb{Q}}}$ identifies systems of realizations
over $L$ with systems of realizations $M$ over ${\mathbb{Q}}$ provided with a
module structure over $H^{0}({\rm Spec}\,L)$. The inverse functor is described
as follows. The module structure of $M_{\rm B}$ over ${\mathbb{Q}}^{{\rm
Hom}(L,{\mathbb{C}})}$ gives a decomposition of $M_{\rm B}$ into pieces
$M_{\sigma}$ indexed by the embedding of $L$ into ${\mathbb{C}}.$ The module
structure of $M_{\rm dR}$ over $L$ turns $M_{\rm dR}$ into the required de
Rham realization.

Let $M$ be a system of realizations over $L$. With the notations as in 3.2.2,
if on ${\rm Res}_{L/{\mathbb{Q}}}(M)$ and $H^{0}({\rm Spec}\,L)$ we take as
multilinear structures all possible structures on $H^{0}({\rm Spec}\,L)$ and
the module structure of ${\rm Res}_{L/{\mathbb{Q}}}(M)$, the resulting group
$G(\mathsf{s})_{\rm B}$ is an extension of ${\rm Gal}(N/{\mathbb{Q}})$ by
$\prod_{\sigma}{\rm GL}(M_{\sigma})$, the product is over all embeddings
$\sigma:L\to{\mathbb{C}},$ where $N$ is the normal closure of $L$ in
$\overline{{\mathbb{Q}}}\subset{\mathbb{C}}$. The group elements above
$\alpha\in{\rm Gal}(N/{\mathbb{Q}})$ are the systems of isomorphisms
$M_{\sigma}\to M_{\alpha\sigma}.$

####  3.3.3. A remark on Hodge numbers

As before, a system with coefficients in $E$ is a system $M$ provided with an
$E$-module structure $E\to{\rm End}(M).$ A crucial complication is that the
$L\otimes E$-modules ${\rm Gr}^{p}(M_{\rm dR})$ are usually not free. A
consequence is that when $M$ comes from a motive and that for each
$\sigma:L\to{\mathbb{C}}$ a corresponding $L$-function $L^{\sigma}$ is
defined, the factors $L^{\sigma}_{v}$ at infinite places $v$ of $L$ depend on
$\sigma.$ But their product over $v$ is independent of $\sigma.$ Very briefly,
a weight $w$ Betti–de Rham system of realization $M$ over a number field $L$
with coefficients in a number field $E$ consists of:

  * •

(Betti) For each embedding $\sigma$ of $L$ into ${\mathbb{C}}$, an $E$-vector
space $M_{\sigma}$; together with a Hodge decomposition
$M_{\sigma}\otimes{\mathbb{C}}=\oplus_{p+q=w}M_{\sigma}^{p,q}$ of
$E\otimes{\mathbb{C}}$-modules;

  * •

(Real Frobenius) an involutive isomorphism $\mathsf{F}_{\infty}:M_{\sigma}\to
M_{c\sigma}$ of $E$-vector spaces which upon complexification maps
$M_{\sigma}^{p,q}$ to $M_{c\sigma}^{q,p};$

  * •

(de Rham) an $E\otimes L$-module $M_{\rm dR}$, endowed with a Hodge filtration
$\mathcal{F}_{\rm dR}$; and

  * •

(Comparison) a comparison isomorphism
$M_{\sigma}\otimes_{\mathbb{Q}}{\mathbb{C}}\to M_{\rm
dR}\otimes_{L,\sigma}{\mathbb{C}}$ of $E\otimes{\mathbb{C}}$-modules, inducing
$M_{\sigma}^{p,q}={\rm Gr}^{p}\mathcal{F}_{\rm
dR}\otimes_{L,\sigma}{\mathbb{C}}.$

In the Hodge decomposition, the summand $M_{\sigma}^{p,q}$ being an
$E\otimes{\mathbb{C}}$-module amounts to specifying a ${\mathbb{C}}$-vector
space $M_{\sigma,\tau}^{p,q}$ for every embedding $\tau:E\to{\mathbb{C}}.$
Define the Hodge numbers of $M$ as:

(9) $h_{\sigma,\tau}^{p,q}\ =\ \dim_{\mathbb{C}}(M_{\sigma,\tau}^{p,q}).$

######  Proposition 3.3.4.

With notations as above, the Hodge number $h_{\sigma,\tau}^{p,q}$ depends only
on the restriction of $\sigma$ to the largest CM or totally real subfield of
the base field $L$, and only on the restriction of $\tau$ to the largest CM or
totally real subfield of the coefficient field $E$.

###### Proof.

From $\mathsf{F}_{\infty}$, one has

(10) $h_{\sigma,\tau}^{p,q}\ =\ h_{c\sigma,\tau}^{q,p}.$

Since $L$ and $E$ are number fields, $\sigma$ and $\tau$ take values in the
algebraic closure $\overline{{\mathbb{Q}}}$ of ${\mathbb{Q}}$ in
${\mathbb{C}}$. There is an action of $L\otimes E$ on ${\rm
Gr}^{p}\mathcal{F}_{\rm dR}$, and
$h_{\sigma,\tau}^{p,q}=\dim_{\overline{{\mathbb{Q}}}}({\rm
Gr}^{p}\mathcal{F}_{\rm dR}\otimes_{L,\sigma}\overline{{\mathbb{Q}}});$ hence,
for any $\gamma\in{\rm Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$ one has

(11) $h_{\sigma,\tau}^{p,q}\ =\ h_{\gamma\sigma,\gamma\tau}^{p,q}.$

From (10) and (11),

$h_{\sigma,\tau}^{p,q}\ =\ h_{\gamma\sigma,\gamma\tau}^{p,q}\ =\
h_{c\gamma\sigma,\gamma\tau}^{q,p}\ =\
h_{\gamma^{-1}c\gamma\sigma,\tau}^{q,p}\ =\
h_{c\gamma^{-1}c\gamma\sigma,\tau}^{p,q}.$

Hence, $h_{\sigma,\tau}^{p,q}=h_{\eta\sigma,\tau}^{p,q}$ for all $\eta$ fixing
the maximal CM subfield of $\overline{{\mathbb{Q}}}.$ The Galois set ${\rm
Hom}(L,\overline{{\mathbb{Q}}})$, modulo the action of all such $\eta$
corresponds to ${\rm Hom}(L_{1},\overline{{\mathbb{Q}}})$ for the maximal CM
or totally real subfield $L_{1}$ of $L$. Applying (11) to $\gamma=c$ we get

(12) $h_{\sigma,\tau}^{p,q}\ =\ h_{\sigma,c\tau}^{q,p},$

and the same argument applies to $\tau.$ ∎

######  Corollary 3.3.5.

If $L$ or $E$ admits a real embedding, one has

$h_{\sigma,\tau}^{p,q}\ =\ h_{\sigma,\tau}^{q,p}.$

###### Proof.

If $L$ (resp., $E$) has a real embedding, then its largest CM or totally real
subfield of $L$ is necessarily totally real, so that $h_{\sigma,\tau}^{p,q}\
=\ h_{c\sigma,\tau}^{q,p}$ (resp., $h_{\sigma,\tau}^{p,q}\ =\
h_{\sigma,c\tau}^{q,p}$), and one applies (10) (resp., (12)). ∎

######  Corollary 3.3.6.

If $L$ has a real embedding $\sigma_{0}$, and suppose the weight is even, say
$2p$, and the involution $\mathsf{F}_{\infty}$ of $M_{\sigma_{0}}$ acts on
$M_{\sigma_{0}}^{p,p}$ by either $+1$ or $-1$, then

$\sum_{r<s}h_{\sigma,\tau}^{r,s}\ \ \ {\rm and}\ \ \ \sum_{r\leq
s}h_{\sigma,\tau}^{r,s}$

are independent of $\sigma$ and $\tau.$

In the case of an even weight $2p$ (resp, odd weight $2p-1$), the conclusion
can be rephrased as ${\rm Gr}^{p}(M_{\rm dR})$ and $\mathcal{F}^{p+1}(M_{\rm
dR})$ (resp., $\mathcal{F}^{p}(M_{\rm dR})$) is a free $L\otimes E$-module.

###### Proof.

In the even weight $2p$ case, for the Hodge structure of $M_{\sigma_{0}}$,
$M_{\sigma_{0}}^{+}$ (resp., $M_{\sigma_{0}}^{-}$), complexified, maps
isomorphically to $(M_{\rm
dR}/\mathcal{F}^{p})\otimes_{L,\sigma_{0}}{\mathbb{C}}$ (resp., $(M_{\rm
dR}/\mathcal{F}^{p+1})\otimes_{L,\sigma_{0}}{\mathbb{C}}$). Since
$M_{\sigma_{0}}^{\pm}$ are free $E\otimes{\mathbb{C}}$-modules, so are
$(M_{\rm dR}/\mathcal{F}^{p})\otimes_{L,\sigma_{0}}{\mathbb{C}}$ and $(M_{\rm
dR}/\mathcal{F}^{p+1})\otimes_{L,\sigma_{0}}{\mathbb{C}}$, meaning that
$\sum_{r<p}h_{\sigma_{0},\tau}^{r,s}$ and $h_{\sigma_{0},\tau}^{p,p}$ are
independent of $\tau$; one concludes by (11). The argument is similar in the
odd weight case. ∎

####  3.3.7. Highest weights

Prop. 3.3.4 has an analogue on automorphic side: in [R22c], a condition called
strong-purity is identified as a necessary condition on a dominant integral
weight $\lambda$ to support cuspidal cohomology for ${\rm GL}(n)$ over a
number field $L.$ Roughly speaking it says that such a highest weight
$\lambda$ is the base-change from $L_{1}$ of a pure dominant integral weight
for ${\rm GL}(n)/L_{1}$; see [R22c, Prop. 2.6]; whereas the discussion in loc.
cit. is for a totally imaginary base field $L$, it applies mutatis mutandis
for a general number field.

###  3.4. Systems of Betti–de Rham realizations over a totally imaginary field

Let $M$ be a Betti–de Rham system of realizations over a totally imaginary
field $K$, with coefficients in a number field $E$. We assume vanishing middle
Hodge number. Our aim is to express $c^{+}_{E}(M)/c^{-}_{E}(M)$, an element of
$({\mathbb{C}}\otimes E)^{*}/E^{*}$, in terms of the invariant $\delta(K)$ of
$K$ defined in 3.4.1 below. This invariant is the image in
$({\mathbb{C}}\otimes E)^{*}/E^{*}$ of an element of order dividing $2$ of
${\mathbb{C}}^{*}/{\mathbb{Q}}^{*}$, by the injection:
${\mathbb{C}}^{*}/{\mathbb{Q}}^{*}\ =\
({\mathbb{C}}\otimes{\mathbb{Q}})^{*}/{\mathbb{Q}}^{*}\ \hookrightarrow\
({\mathbb{C}}\otimes E)^{*}/E^{*}.$

#### 3.4.1.

Suppose first that $K$ is a CM field. Let $K^{+}$ be the totally real field of
which it is a quadratic extension: $K=K^{+}(\sqrt{D})$ with $D$ totally
negative. As $D\in K^{+}$ is well-defined up to multiplication by a square,
the square root in ${\mathbb{C}}$ of $N_{K^{+}/{\mathbb{Q}}}(D)$ is well-
defined modulo ${\mathbb{Q}}^{*}.$ Define $\delta(K)$ to be its image in
${\mathbb{C}}^{*}/{\mathbb{Q}}^{*}$, as well as in $({\mathbb{C}}\otimes
E)^{*}/E^{*}.$

In general, let $K_{1}$ be the largest CM field or totally real subfield of
$K.$ If $K_{1}$ is totally real then define $\delta(K):=1.$ If $K_{1}$ is CM,
then define

$\delta(K)\ :=\ \delta(K_{1})^{[K:K_{1}]}.$

This formula holds more generally for $K_{1}$ any CM field contained in $K.$

######  Theorem 3.4.2.

Let $M$ be a Betti–de Rham system of realizations over a totally imaginary
field $K$, with coefficients in $E$. Assume that $M$ has no middle Hodge
number. If $M$ is of dimension $n$ over $E$ (i.e., $n=\dim_{E}(M_{\sigma})$
for any embedding $\sigma:K\to{\mathbb{C}}$), then one has

$c^{+}_{E}(M)/c^{-}_{E}(M)\ =\ \delta(K)^{n}.$

###### Proof.

Let ${}_{0}M$ be deduced from $M$ by restriction of ground field from $K$ to
${\mathbb{Q}}$. By definition, $c_{E}^{\pm}(M)=c_{E}^{\pm}({}_{0}M).$ One has

${}_{0}M_{\rm B}\ =\ \bigoplus_{\sigma:K\to{\mathbb{C}}}M_{\sigma},$

and for $c$, the complex conjugation, the involution $\mathsf{F}_{\infty}$ of
${}_{0}M_{\rm B}$ permutes $M_{\sigma}$ and $M_{c\sigma}.$ Let $\Phi$ be a CM
type for $K$, i.e., it is a set of complex embeddings containing exactly one
element of each pair $\\{\sigma,c\sigma\\}$. One has then:

(13) $\begin{split}{}_{0}M^{+}&=\ \bigoplus_{\sigma\in\Phi}\ (\mbox{graph of
$\mathsf{F}_{\infty}:M_{\sigma}\to M_{c\sigma}$})\\\ {}_{0}M^{-}&=\
\bigoplus_{\sigma\in\Phi}\ (\mbox{graph of $-\mathsf{F}_{\infty}:M_{\sigma}\to
M_{c\sigma}$}),\end{split}$

and the projections to the $M_{\sigma},$ for $\sigma\in\Phi$, induce an
isomorphism of $E$-vector spaces:

(14) ${}_{0}M_{\rm B}^{\pm}\ =\ \bigoplus_{\sigma\in\Phi}M_{\sigma}.$

Let $p$ be the integer such that $M$ is of weight $2p-1$ or $2p-2$. The
comparison isomorphism induces ${\mathbb{C}}\otimes E$-linear isomorphisms
between the complexifications of the $E$-vector spaces ${}_{0}M^{\pm}$ and
${}_{0}M_{\rm dR}/\mathcal{F}^{p}{}_{0}M_{\rm dR},$ and the periods
$c_{E}^{\pm}(M)$ are their determinants, computed in bases over $E$ of
${}_{0}M^{\pm}$ and ${}_{0}M_{\rm dR}/\mathcal{F}^{p}{}_{0}M_{\rm dR}.$

The ratio $c_{E}^{+}(M)/c_{E}^{-}(M)$ is the image in $({\mathbb{C}}\otimes
E)^{*}/E^{*}$ of the determinant, in $({\mathbb{C}}\otimes E)^{*},$ of the
automorphism of ${\mathbb{C}}\otimes(\oplus_{\sigma\in\Phi}M_{\sigma}),$ which
in the diagram below is the left vertical arrow, defined as the composite of
the various isomorphisms or their inverses:

(15)
$\textstyle{{\mathbb{C}}\otimes(\oplus_{\sigma\in\Phi}M_{\sigma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{C}}\otimes{}_{0}M^{+}={\mathbb{C}}\otimes(\bigoplus_{\sigma\in\Phi}\
(\mbox{graph of $\mathsf{F}_{\infty}:M_{\sigma}\to
M_{c\sigma}$}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{C}}\otimes({}_{0}M_{\rm
dR}/\mathcal{F}^{p}{}_{0}M_{\rm
dR})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{C}}\otimes(\oplus_{\sigma\in\Phi}M_{\sigma})}$$\textstyle{{\mathbb{C}}\otimes{}_{0}M^{-}={\mathbb{C}}\otimes(\bigoplus_{\sigma\in\Phi}\
(\mbox{graph of $-\mathsf{F}_{\infty}:M_{\sigma}\to
M_{c\sigma}$}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

As we now have an automorphism, the choices of bases are irrelevant; one
should just use the same basis at the source and target. When we identify
${\mathbb{C}}\otimes E$ with a product of copies of ${\mathbb{C}}$ indexed by
the complex embeddings $\tau:E\to{\mathbb{C}}$, the determinant of the
isomorphism in the left vertical arrow in (15) becomes a family of elements of
${\mathbb{C}}$ indexed by $\tau:E\to{\mathbb{C}}.$

Both ${}_{0}M_{\rm dR}$ and $\mathcal{F}^{p}{}_{0}M_{\rm dR}$ are
$K$-$E$-bimodules. The first is free as a $K\otimes E$-module. Indeed, if we
choose $\sigma:K\to{\mathbb{C}}$, it becomes free after extension of scalars
from $K\otimes E$ to $K\otimes{\mathbb{C}}$ by $\sigma$, becoming isomorphic
to ${\mathbb{C}}\otimes M_{\sigma}$. The $K\otimes E$-module
$\mathcal{F}^{p}{}_{0}M_{\rm dR}$ is usually not free. Define
$\mathcal{F}^{+}:={\mathbb{C}}\otimes\mathcal{F}^{p}{}_{0}M_{\rm dR}.$ As a
${\mathbb{C}}$-algebra ${\mathbb{C}}\otimes K\otimes E$ is the product of
copies of ${\mathbb{C}}$ indexed by pairs $(\sigma,\tau)$ of complex
embeddings $\sigma$ of $K$ and $\tau$ of $E$; the module structure of
$\mathcal{F}^{+}$ amounts to a decomposition

$\mathcal{F}^{+}\ =\ \bigoplus_{\sigma,\tau}\mathcal{F}^{+}_{\sigma,\tau},$

where $\mathcal{F}^{+}_{\sigma,\tau}$ is a finite-dimensional complex vector;
we define

$m(\sigma,\tau)\ :=\ \dim_{\mathbb{C}}\mathcal{F}^{+}_{\sigma,\tau}.$

The set of integers $\\{m(\sigma,\tau)\\}_{\sigma,\tau}$ determine the
isomorphism class of the $K$-$E$-bimodule $\mathcal{F}^{p}{}_{0}M_{\rm dR}.$

######  Lemma 3.4.3.

When we identify ${\mathbb{C}}\otimes E$ with a product of copies of
${\mathbb{C}}$ indexed by embeddings $\tau:E\to{\mathbb{C}}$, the determinant,
in ${\mathbb{C}}\otimes E,$ of the automorphism of
${\mathbb{C}}\otimes(\oplus_{\sigma\in\Phi}M_{\sigma}),$ in (15), is:

$\left(\prod_{\sigma\in\Phi}\
(-1)^{m(\sigma,\tau)}\right)_{\tau:E\to{\mathbb{C}}}.$

###### Proof.

We will identify $\mathcal{F}^{+}$ with its image by the comparison
isomorphism between ${\mathbb{C}}\otimes{}_{0}M_{\rm B}$ and
${\mathbb{C}}\otimes{}_{0}M_{\rm dR}.$ It is the sum of its intersections
$\mathcal{F}^{+}_{\sigma}$ with the summands ${\mathbb{C}}\otimes M_{\sigma}$,
and $\mathsf{F}_{\infty}(\mathcal{F}^{+}_{\sigma})$ is a supplement to
$\mathcal{F}^{+}_{c\sigma}$ in ${\mathbb{C}}\otimes M_{c\sigma}$. Indeed,
$\mathcal{F}^{+}_{\sigma}$ is the sum over $\tau$ of the
$\mathcal{F}_{\sigma,\tau}$, and $\mathsf{F}_{\infty}(\mathcal{F}^{+})$ is the
complex conjugation of $\mathcal{F}^{p}({\mathbb{C}}\otimes(\oplus
M_{\sigma})).$

The automorphism of ${\mathbb{C}}\otimes(\oplus_{\sigma\in\Phi}M_{\sigma})$ in
(15) is the product of the following automorphisms of the ${\mathbb{C}}\otimes
M_{\sigma}$ ($\sigma\in\Phi$):

(16) $\textstyle{{\mathbb{C}}\otimes
M_{\sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{C}}\otimes(\mbox{graph
of $\mathsf{F}_{\infty}:M_{\sigma}\to
M_{c\sigma}$})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({\mathbb{C}}\otimes
M_{\sigma})/\mathcal{F}^{+}_{\sigma}+({\mathbb{C}}\otimes
M_{c\sigma})/\mathcal{F}^{+}_{c\sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{C}}\otimes
M_{\sigma}}$$\textstyle{{\mathbb{C}}\otimes(\mbox{graph of
$-\mathsf{F}_{\infty}:M_{\sigma}\to
M_{c\sigma}$})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

We will conclude the proof of the lemma by showing that the determinant of the
automorphism of ${\mathbb{C}}\otimes M_{\sigma},$ the left vertical arrow in
(16), is $\\{(-1)^{m(\sigma,\tau)}\\}_{\tau:E\to{\mathbb{C}}}.$

Let us decompose ${\mathbb{C}}\otimes M_{\sigma}$ as the direct sum of
$\mathcal{F}_{\sigma}^{+}$ and
$\mathsf{F}_{\infty}(\mathcal{F}^{+}_{c\sigma}).$ The automorphism of
${\mathbb{C}}\otimes M_{\sigma}$ in (16) is $-1$ on the first summand and $+1$
on the second. Indeed, as $x\in\mathcal{F}^{+}_{\sigma}$ maps to
$(x,\mathsf{F}_{\infty}(x))$, equal modulo $\mathcal{F}^{+}_{\sigma}$ to
$(-x,\mathsf{F}_{\infty}(x))$ in the graph of $-\mathsf{F}_{\infty}$,
projecting to $x$; and for $y\in\mathcal{F}^{+}_{c\sigma}$,
$\mathsf{F}_{\infty}(y)$ maps to $\mathsf{F}_{\infty}(y)+y$, congruent modulo
$\mathcal{F}^{+}_{c\sigma}$ to $\mathsf{F}_{\infty}(y)-y$ projecting to
$\mathsf{F}_{\infty}(y)$. As $\mathcal{F}^{+}_{\sigma}$ is the product of the
$\mathcal{F}_{\sigma,\tau}$, this gives the formula for the determinant of the
automorphism in (16). ∎

#### 3.4.4.

Let $A\subset{\mathbb{C}}$ be a Galois extension of ${\mathbb{Q}},$ large
enough to contain the images of $K$ and $E$ by any complex embedding. One has

$m(\sigma,\tau)\ =\ \dim_{A}(\mathcal{F}^{p}{}_{0}M_{\rm
dR}\,\otimes_{K\otimes E,\sigma\otimes\tau}\,A).$

It follows that the function $(\sigma,\tau)\mapsto m(\sigma,\tau)$ is ${\rm
Gal}(A/{\mathbb{Q}})$-invariant. We go through the steps in the proof of Prop.
3.3.4 one more time. The ${\mathbb{C}}\otimes E$-module ${\mathbb{C}}\otimes
M_{\sigma}$ is free of rank $n$ and is the direct sum of
$\mathcal{F}^{+}_{\sigma}$ and $\mathsf{F}_{\infty}\mathcal{F}^{+}_{c\sigma}$.
It follows that

(17) $m(\sigma,\tau)+m(c\sigma,\tau)\ =\ n.$

For any $c^{\prime}=gcg^{-1}$ conjugate in ${\rm Gal}(A/{\mathbb{Q}})$ of the
complex conjugation $c$, one has

$m(\sigma,\tau)+m(c^{\prime}\sigma,\tau)\ =\ n.$

Indeed,

$m(c^{\prime}\sigma,\tau)\ =\ m(gcg^{-1}\sigma,\tau)\ =\
m(cg^{-1}\sigma,g^{-1}\tau)\ =\ n-m(g^{-1}\sigma,g^{-1}\tau)\ =\
n-m(\sigma,\tau).$

It follows that, as a function of $\sigma$, $m(\sigma,\tau)$ is invariant by
the normal subgroup $H$ generated by $c^{\prime}c^{\prime\prime-1},$ for
$c^{\prime}$ and $c^{\prime\prime}$ any conjugates of $c$. The quotient of
${\rm Gal}(A/{\mathbb{Q}})$ by $H$ is the largest quotient in which $c$
becomes central, and invariance of $m$ means that $m(\sigma,\tau)$ depends
only, as a function of $\sigma$, on the restriction of $\sigma$ to the
subfield $K_{1}$ of $K.$

If $K_{1}$ is totally real, (17) shows that $m(\sigma,\tau)=n/2$, so that
$\prod_{\sigma\in\Phi}(-1)^{m(\sigma,\tau)}$ is independent of $\tau$; it is
either $+1$ or $-1$ for all $\tau,$ and is the image of $\pm 1\in E^{*}$ in
$({\mathbb{C}}\otimes E)^{*}$. As claimed in Thm. 3.4.2, the image of
$c^{+}_{E}(M)/c^{-}_{E}(M)$ in $({\mathbb{C}}\otimes E)^{*}/E^{*}$ is trivial.

We now suppose that $K_{1}$ is a CM field. A subset $\Phi_{1}$ of the set of
complex embeddings $\sigma_{1}$ of $K_{1}$ containing exactly one element of
each pair $\\{\sigma_{1},c\sigma_{1}\\}$ defines a subset $\Phi$ of complex
embeddings of $K$: the extensions of elements of $\Phi_{1}$. Let
$m_{1}(\sigma_{1},\tau)$ be $m(\sigma,\tau)$ for any extension $\sigma$ of
$\sigma_{1}.$ One has

$\prod_{\sigma\in\Phi}(-1)^{m(\sigma,\tau)}\ =\
\left(\prod_{\sigma_{1}\in\Phi_{1}}(-1)^{m_{1}(\sigma_{1},\tau)}\right)^{[K:K_{1}]}.$

To prove Thm. 3.4.2, it remains to show that the image of

(18)
$\left(\prod_{\sigma_{1}\in\Phi_{1}}(-1)^{m_{1}(\sigma_{1},\tau)}\right)_{\tau}\
\in\ ({\mathbb{C}}\otimes E)^{*}$

in $({\mathbb{C}}\otimes E)^{*}/E^{*}$ coincides with $\delta(K_{1})^{n}.$
This relies on the identity
$m_{1}(\sigma_{1},\tau)+m_{1}(c\sigma_{1},\tau)=n,$ which implies that if we
modify $\Phi_{1}$ to $\Phi_{1}^{\prime}$, by replacing $\sigma_{1}\in\Phi_{1}$
by $c\sigma_{1}$, the product (18) gets multiplied, for each $\tau$, by
$(-1)^{n}.$

Let $K_{1}^{+}$ be the totally real field of which $K_{1}=K_{1}^{+}(\sqrt{D})$
is a quadratic extension with $D$ a totally negative element of $K_{1}^{+}$,
and let $d=N_{K^{+}_{1}/{\mathbb{Q}}}(D).$ We have a commutative square:

$\textstyle{A\otimes E\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\
{\mathbb{C}}\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\oplus_{\tau}A\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\
\oplus_{\tau}{\mathbb{C}}.}$

Let $d^{1/2}$ be a square root of $d$ in $A$; note that
$d^{1/2}=\delta(K_{1}).$ We need to show that the ratio $(d^{1/2})^{n}\otimes
1\in A\otimes E$ and (18) in $A\otimes E$ is in $E^{*},$ i.e., that the ratio
is invariant by ${\rm Gal}(A/{\mathbb{Q}}).$

Let $\beta$ be the following character of ${\rm Gal}(A/{\mathbb{Q}})$: for
$\Phi_{1}$ as before, and $g\in{\rm Gal}(A/{\mathbb{Q}})$, define $\beta(g)$
to be $(-1)^{k}$, where $k$ is the number of $\sigma_{1}\in\Phi_{1}$ which
need to be replaced by $c\sigma_{1}$ to transform $\Phi_{1}$ to $g\Phi_{1}$.
If we apply $g$ to the element in (18)

$\tau\ \mapsto\ \prod_{\sigma_{1}\in\Phi_{1}}(-1)^{m(\sigma_{1},\tau)}$

of $A\otimes E$, identified with a product of copies of $A$, one obtains

$g\tau\ \mapsto\ \prod_{\sigma_{1}\in\Phi_{1}}(-1)^{m(\sigma_{1},\tau)}\ =\
\prod_{\sigma_{1}\in\Phi_{1}}(-1)^{m(g\sigma_{1},g\tau)},$

that is

$\tau\ \mapsto\ \prod_{\sigma_{1}\in g\Phi_{1}}(-1)^{m(\sigma_{1},\tau)}.$

This is simply (18) multiplied by $\beta(g)^{n}.$ On the other hand, a square
root of $d$ in $A$ is obtained by taking

$d^{1/2}\ =\ \prod_{\sigma\in\Phi_{1}}\sigma(\sqrt{D}).$

For this square root we have $g(d^{1/2})\ =\ \beta(g)d^{1/2}.$ Hence, the
ratio of $(d^{1/2})^{n}\otimes 1$ and (18) in $A\otimes E,$ being invariant by
${\rm Gal}(A/{\mathbb{Q}}),$ is in $E^{*}.$ This completes the proof of Thm.
3.4.2. ∎

The reader should compare the above theorem with some of the motivic
calculations in Harris [Ha97] and Hida [Hi94].

####  3.4.5. Rankin–Selberg $L$-functions over a totally imaginary field

The period relation in Thm. 3.4.2 has a direct bearing on the rationality
results in [R22c] for the Rankin–Selberg $L$-functions for ${\rm
GL}(n)\times{\rm GL}(n^{\prime})$ over a totally imaginary field $F$. Suppose
$\sigma$ (resp., $\sigma^{\prime}$) is a cohomological cuspidal automorphic
representation of ${\rm GL}(n)$ (resp., ${\rm GL}(n^{\prime})$) over $F$, and
$E$ is a Galois extension of ${\mathbb{Q}}$ inside $\overline{{\mathbb{Q}}}$
which contains a copy of $F$ and the Hecke eigenvalues of $\sigma$ and
$\sigma^{\prime}$ then the main theorem of that article [R22c, Thm. 5.16]
states that when we have two successive critical points, say $m$ and $m+1$,
then

$|\delta_{F/{\mathbb{Q}}}|^{-\tfrac{nn^{\prime}}{2}}\cdot\frac{L(m,\sigma\times\sigma^{\prime})}{L(m+1,\sigma\times\sigma^{\prime})}\in
E,$

where $L(s,\sigma\times\sigma^{\prime})$ is the completed Rankin–Selberg
$L$-function, and $\delta_{F/{\mathbb{Q}}}$ is the absolute discriminant of
$F/{\mathbb{Q}}$. Furthermore, for all $\gamma\in{\rm
Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$ we have the reciprocity law

$\gamma\left(|\delta_{F/{\mathbb{Q}}}|^{-\tfrac{nn^{\prime}}{2}}\cdot\frac{L(m,\sigma\times\sigma^{\prime})}{L(m+1,\sigma\times\sigma^{\prime})}\right)\
=\
\varepsilon(\gamma)\cdot\varepsilon^{\prime}(\gamma)\cdot|\delta_{F/{\mathbb{Q}}}|^{-\tfrac{nn^{\prime}}{2}}\cdot\frac{L(m,{}^{\gamma}\sigma\times{}^{\gamma}\sigma^{\prime})}{L(m+1,{}^{\gamma}\sigma\times{}^{\gamma}\sigma^{\prime})}.$

where $\varepsilon(\gamma),\varepsilon^{\prime}(\gamma)\in\\{\pm 1\\}$ are
certain signatures that appear due to Galois action on cohomology of the
boundary of the Borel–Serre compactification of a locally symmetric space for
${\rm GL}(N)$, with $N=n+n^{\prime}.$ If we take $M$ to be the tensor product
over $E$ of the motives $M(\sigma)$ and $M(\sigma^{\prime})$ that
conjecturally correspond to $\sigma$ and $\sigma^{\prime}$, then Thm. 3.4.2
applied to this $M$ is compatible with the above mentioned [R22c, Thm. 5.16]
after the following equality of signatures

$\frac{\gamma(|\delta_{F/{\mathbb{Q}}}|^{nn^{\prime}/2})}{|\delta_{F/{\mathbb{Q}}}|^{nn^{\prime}/2}}\cdot\varepsilon(\gamma)\cdot\varepsilon^{\prime}(\gamma)\
=\
\frac{\gamma((i^{[F:{\mathbb{Q}}]/2}\,\delta(F))^{nn^{\prime}})}{(i^{[F:{\mathbb{Q}}]/2}\,\delta(F))^{nn^{\prime}}}$

that is proved in [R22c, Prop. 5.26]. It is interesting to reflect on the idea
that the two sides of the above equality of signatures have very different
origins: the signature on the left hand side comes from
$|\delta_{F/{\mathbb{Q}}}|^{nn^{\prime}/2}$ (that is used to normalize a
measure–[R22c, 5.2.3.5]) and $\varepsilon(\gamma),\
\varepsilon^{\prime}(\gamma)$ which arise from Galois action in boundary-
cohomology, or what amounts to the same, on Eisenstein-cohomology; the
signature on the right hand side is intrinsic to the base field as in the
proof of Thm. 3.4.2 above.

###  3.5. Tensor product motives over a number field.

In this section, we will extend the results of Sect. 3.1 to the case of
systems of realizations over a totally real number field, at least in the case
of vanishing middle Hodge numbers.

Let $M^{\prime}$ and $M^{\prime\prime}$ be Betti–de Rham systems of
realizations over a number field $L$ (3.3.1). A tensor product is defined in
the obvious way. It does not commute with restriction from $L$ to
${\mathbb{Q}}$. To interpret ${\rm Res}_{L/{\mathbb{Q}}}(M^{\prime}\otimes
M^{\prime\prime})$, in terms of which $c^{+}$ and $c^{-}$ of
$M^{\prime}\otimes M^{\prime\prime}$ are defined, one should view ${\rm
Res}_{L/{\mathbb{Q}}}(M^{\prime})$ and ${\rm
Res}_{L/{\mathbb{Q}}}(M^{\prime\prime})$ as modules over $H^{0}({\rm
Spec}\,L)$ (3.3.2). As $H^{0}({\rm Spec}\,L)$-modules,

(19) ${\rm Res}_{L/{\mathbb{Q}}}(M^{\prime}\otimes M^{\prime\prime})\ =\ {\rm
Res}_{L/{\mathbb{Q}}}(M^{\prime})\otimes_{H^{0}({\rm Spec}\,L)}{\rm
Res}_{L/{\mathbb{Q}}}(M^{\prime\prime}).$

The same applies for systems of realizations with coefficients in $E$, and
tensor product of $M^{\prime}$ and $M^{\prime\prime}$ over $E$; in (19),
$H^{0}({\rm Spec}\,L)$ should be replaced by $H^{0}({\rm Spec}\,L)\otimes E.$

######  Proposition 3.5.1.

Let $M^{\prime}$ and $M^{\prime\prime}$ be systems of Betti–de Rham
realizations over a totally real number field $L$, with coefficients in $E$.
If $M^{\prime}$, $M^{\prime\prime},$ and
$M^{\prime}\otimes_{E}M^{\prime\prime}$ have vanishing middle Hodge numbers,
then as elements of $({\mathbb{C}}\otimes E)^{*}/E^{*}$ we have

$c^{+}(M^{\prime}\otimes_{E}M^{\prime\prime})\ =\
c^{-}(M^{\prime}\otimes_{E}M^{\prime\prime}).$

###### Proof.

We will consider ${\rm Res}_{L/{\mathbb{Q}}}(M^{\prime}\otimes
M^{\prime\prime})$ as derived by the multilinear algebra construction (19)
from ${\rm Res}_{L/{\mathbb{Q}}}(M^{\prime})$, ${\rm
Res}_{L/{\mathbb{Q}}}(M^{\prime\prime})$, and $H^{0}({\rm Spec}\,L)$. The
structures used are the ring structure of $H^{0}({\rm Spec}\,L),$ and the
$H^{0}({\rm Spec}\,L)-$module and $E-$module structures of ${\rm
Res}_{L/{\mathbb{Q}}}(M^{\prime})$ and ${\rm
Res}_{L/{\mathbb{Q}}}(M^{\prime\prime})$.

The Betti realization of $H^{0}({\rm Spec}\,L)$, with its ring structure
$\mathsf{s}_{L}$, is the sum of copies of ${\mathbb{Q}}$ indexed by the set
$\Sigma_{L}:={\rm Hom}(L,{\mathbb{C}})$ of the complex embeddings of $L$. The
corresponding group $G(\mathsf{s}_{L})_{B}$ is the symmetric group
$S_{\Sigma_{L}}$ of permutations of $\Sigma_{L}.$ Using more multilinear
algebra structures on $H^{0}({\rm Spec}\,L)$, one could reduce it to the
Galois group of the normal closure of $L$ in ${\mathbb{C}}$, but this will not
be needed.

For ${\rm Res}_{L/{\mathbb{Q}}}(M^{\prime})$ with its $H^{0}({\rm
Spec}\,L)-$module and $E-$module structures $\mathsf{s}^{\prime}$, the Betti
realization is $\bigoplus_{\sigma\in\Sigma_{L}}M^{\prime}_{\sigma}$, and the
group $G(\mathsf{s}^{\prime})_{B}$ is an extension of $S_{\Sigma_{L}}$ by the
product of the ${\rm GL}_{E}(M^{\prime}_{\sigma}).$ The real Frobenius
$\mathsf{F}_{\infty}$ is in this product group. One has

${\rm Res}_{L/{\mathbb{Q}}}(M^{\prime})^{\pm}\ =\
\bigoplus_{\sigma\in\Sigma_{L}}M^{\prime}_{\sigma}{}^{\pm},$

and for each $\sigma$,
$\dim(M^{\prime}_{\sigma}{}^{\pm})=\dim(M^{\prime}_{\sigma})/2.$ It follows
that the centralizer $Z(\mathsf{F}_{\infty})$ maps onto $S_{\Sigma_{L}}$.

For ${\rm Res}_{L/{\mathbb{Q}}}(M^{\prime}),$ ${\rm
Res}_{L/{\mathbb{Q}}}(M^{\prime\prime})$ and $H^{0}({\rm Spec}\,L)$, and the
ring and module structures $\mathsf{s}$, one has similarly,

$G(\mathsf{s})_{B}\ =\ S_{\Sigma_{L}}\ltimes\left(\prod_{\sigma}{\rm
GL}_{E}(M^{\prime}_{\sigma})\times\prod_{\sigma}{\rm
GL}_{E}(M^{\prime\prime}_{\sigma})\right).$

We need to prove that the determinants of the actions of
$Z(\mathsf{F}_{\infty})$ on the $+$ and $-$ parts of ${\rm
Res}_{L/{\mathbb{Q}}}(M^{\prime}\otimes M^{\prime\prime})$ are the same. For
the connected component

$\prod_{\sigma\in\Sigma_{L}}\left({\rm
GL}_{E}(M^{\prime}_{\sigma}{}^{+})\times{\rm
GL}_{E}(M^{\prime}_{\sigma}{}^{-})\times{\rm
GL}_{E}(M^{\prime\prime}_{\sigma}{}^{+})\times{\rm
GL}_{E}(M^{\prime\prime}_{\sigma}{}^{-})\right)$

the proof is the same as in Sect. 3.1. Let us pick bases
$e^{\prime+}_{i,\sigma}$, $e^{\prime-}_{i,\sigma}$,
$e^{\prime\prime+}_{j,\sigma}$, and $e^{\prime\prime-}_{j,\sigma}$, where
$1\leq i\leq\dim(M^{\prime})/2$ and $1\leq j\leq\dim(M^{\prime})/2$, of the
spaces $M^{\prime}_{\sigma}{}^{\pm}$ and $M^{\prime\prime}_{\sigma}{}^{\pm}$.
We can lift $S_{\Sigma_{L}}$ in $G(\mathsf{s})_{B}$ so that for each
decoration ′, ′′, +, -, $i$ or $j$, it permutes the $e_{\sigma}.$ For this
choice, the action of $S_{\Sigma_{L}}$ on ${\rm
Res}_{L/{\mathbb{Q}}}(M^{\prime}\otimes M^{\prime\prime})^{\pm}$ is the sum of
$\dim(M^{\prime}\otimes M^{\prime\prime})/2$ permutation representation: the
same for $+$ and $-$, hence the same determinant. ∎

###  3.6. Motives via multiplication induction: Asai motives.

####  3.6.1. Definition

Consider an extension $K/F$ of number fields, and a Betti–de Rham realization
$M$ over $K$ of rank $n$ and weight $w$. There is a notion of multiplicative
induction from $K$ to $F$ of $M$:

$M\ \mapsto\ \bigotimes_{K/F}M,$

which is a Betti–de Rham realization over $F$ of rank $n^{[K:F]}$ and weight
$[K:F]w$. To define $\otimes_{K/F}M$ over $F$, it suffices to define it
locally for the etale topology over ${\rm Spec}(F).$ This means defining what
should be its base-change to ${\rm Spec}(F^{\prime})$, for $F^{\prime}$ a
large enough finite extension of $F$, defining a compatibility isomorphism for
an iterated extension $F^{\prime\prime}/F^{\prime}/F$, and a compatibility
between them for $F^{\prime\prime\prime}/F^{\prime\prime}/F^{\prime}/F$. Here,
large enough means that it contains a copy of the normal closure of $K/F$. For
$F^{\prime}$ large enough, there are $[K:F]$ embeddings over $F$ of $K$ in
$F^{\prime}$, and the base-change to $F^{\prime}$ of $\bigotimes_{K/F}M$ is
the tensor product of the corresponding base-changes of $M$. More generally,
one can take for $K$ an etale algebra over $F$, that is a finite product of
finite extensions $K_{i}$ of $F$. A Betti–de Rham realization $M$ over $K$ is
a family $M_{i}$ of realizations over $K_{i}$. The definition above continues
to make sense, and

$\bigotimes_{K/F}M\ =\ \bigotimes_{i}\bigotimes_{K_{i}/F}M_{i}$

Statements true for tensor products can often be generalized to multiplicative
induction.

######  Theorem 3.6.2.

Let $K/F$ be an extension of totally real number fields of degree $[K:F]>1.$
Let $M$ be a Betti–de Rham realization over $K$ of rank $n$ with coefficients
in a number field $E$. Assume that $M$ and $\otimes_{K/F}M$ have vanishing
middle Hodge numbers. Necessarily, $n$ is even. Then as elements of
$(E\otimes{\mathbb{C}})^{*}/E^{*}$ one has

$c^{+}({\rm Res}_{F/{\mathbb{Q}}}(\otimes_{K/F}M))=c^{-}({\rm
Res}_{F/{\mathbb{Q}}}(\otimes_{K/F}M))$

unless $[K:F]=2$ and $n/2$ is odd, in which case if $K=F(\sqrt{D})$ then

$c^{+}({\rm Res}_{F/{\mathbb{Q}}}(\otimes_{K/F}M))/c^{-}({\rm
Res}_{F/{\mathbb{Q}}}(\otimes_{K/F}M))\ =\ \sqrt{N_{F/{\mathbb{Q}}}(D)},$

where $\sqrt{N_{F/{\mathbb{Q}}}(D)}$ is viewed as an element of
${\mathbb{C}}^{*}/{\mathbb{Q}}^{*}\hookrightarrow({\mathbb{C}}\otimes
E)^{*}/E^{*}.$

###### Proof.

The method of proof is the same as in Sect. 3.5, starting this time with
$H^{0}({\rm Spec}(F))$ and $H^{0}({\rm Spec}(K)),$ their ring structures, and
the morphism

$H^{0}({\rm Spec}(F))\ \to\ H^{0}({\rm Spec}(K)),$

$M$ and its $H^{0}({\rm Spec}(K))-$module and $E$-module structures.

Let $\Sigma_{K}={\rm Hom}(K,{\mathbb{C}})={\rm Hom}(K,{\mathbb{R}})$ be the
set of all complex (real) embeddings of $K$, and similarly $\Sigma_{F}$.
Restriction to $F$ gives the surjective map

$\textstyle{J:=\Sigma_{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{I:=\Sigma_{F}.}$

If we consider just $H^{0}({\rm Spec}(F))$ and $H^{0}({\rm Spec}(K)),$ their
ring structures, and the morphism $H^{0}({\rm Spec}(F))\to H^{0}({\rm
Spec}(K)),$ one has $G(\mathsf{s})_{B}={\rm Aut}(J\to I)$, the group of all
permutations of $J$ compatible with $\pi:J\to I.$ All the fibres of $\pi$ have
cardinality $[K:F]$, and ${\rm Aut}(J\to I)$ is a wreath product: an extension

$1\ \longrightarrow\ \prod_{i\in I}S_{\pi^{-1}(i)}\ \longrightarrow\ {\rm
Aut}(J\to I)\ \longrightarrow\ S_{I}\ \longrightarrow\ 1.$

If we consider, in addition, $M$ and its module structures, one has for the
Betti realizations:

${\rm Res}_{K/{\mathbb{Q}}}(M)_{\rm B}\ =\ \bigoplus_{j\in J}M_{j},\quad{\rm
and}\quad{\rm Res}_{F/{\mathbb{Q}}}(\otimes_{K/F}M)_{\rm B}\ =\
\bigoplus_{i\in I}\bigotimes_{\pi(j)=i}M_{j}.$

and $G(\mathsf{s})_{B}$ is an extension

$1\ \longrightarrow\ \prod_{j\in J}{\rm GL}_{E}(M_{j})\ \longrightarrow\
G(\mathsf{s})_{B}\ \longrightarrow\ {\rm Aut}(J\to I)\ \longrightarrow\ 1.$

This is as in Sect. 3.5, except that we have used an additional structure on
$H^{0}({\rm Spec}(K))$ to reduce $S_{J}$ to ${\rm Aut}(J\to I).$ Here
$\mathsf{F}_{\infty}$ is the family of involutions in $\prod_{j\in J}{\rm
GL}(M_{j})$, defining decompositions $M_{j}=M_{j}^{+}\oplus M_{j}^{-}$. Since
$M$ has vanishing middle Hodge number, one has
$\dim(M_{j}^{+})=\dim(M_{j}^{-})=\dim(M)/2=m$ (say). The equality of the
dimensions of all the $M_{j}^{+}$, $j\in J$, ensures that
$Z(\mathsf{F}_{\infty})$ maps onto ${\rm Aut}(J\to I)$.

We need to compare the characters of $Z(\mathsf{F}_{\infty})$ acting on

${\rm Res}_{F/{\mathbb{Q}}}(\otimes_{K/F}M)^{+}\ =\ \ \bigoplus_{i\in
I}\bigotimes_{\begin{subarray}{c}\varepsilon:\pi^{-1}(i)\to\\{0,1\\}\\\
\sum_{j\in\pi^{-1}(i)}\varepsilon(j)=0\end{subarray}}M_{j}^{\varepsilon(j)},$
${\rm Res}_{F/{\mathbb{Q}}}(\otimes_{K/F}M)^{-}\ =\ \ \bigoplus_{i\in
I}\bigotimes_{\begin{subarray}{c}\varepsilon:\pi^{-1}(i)\to\\{0,1\\}\\\
\sum_{j\in\pi^{-1}(i)}\varepsilon(j)=1\end{subarray}}M_{j}^{\varepsilon(j)},$

where, $M_{j}^{0}=M_{j}^{+}$ and $M_{j}^{1}=M_{j}^{-}.$

For the action of the connected component $Z(\mathsf{F}_{\infty})=\prod{\rm
GL}(M_{j}^{+})\times{\rm GL}(M_{j}^{-})$, only the $\pi(j)$-summand matters
for the determinant of the action of ${\rm GL}(M_{j}^{+})$ (resp., ${\rm
GL}(M_{j}^{-})$). The action is on

$M_{j}^{+}\otimes({\rm something})$

with $\dim({\rm something})$ being the same for $+$ or $-$. The same for ${\rm
GL}(M_{j}^{-})$. It remains to see what happens on the representatives of the
connected components.

Fix bases $\\{e_{j,1}^{\pm},\dots,e_{j,m}^{\pm}\\}$ for $M_{j}^{\pm}.$ As a
representative for $\sigma$, we take
$\\{e_{j,1}^{\pm},\dots,e_{j,m}^{\pm}\\}\to\\{e_{\sigma(j),1}^{\pm},\dots,e_{\sigma(j),m}^{\pm}\\}$.
We need to compare the actions of $\det(\sigma)$ on ${\rm
Res}_{F/{\mathbb{Q}}}(\otimes_{K/F}M)^{\pm}.$

Action of $S_{\pi^{-1}(i)}$, the permutation group of the fibre above $i$: It
suffices to compare the actions on $\otimes_{\pi(j)=i}M_{j}^{\varepsilon(j)}.$
For a transposition of two $j$’s; call them $1$ and $2$; if $[K:F]>2$, we
have—for some $W^{\pm}$ of the same dimension—to consider

$W^{+}\otimes\left((M_{1}^{+}\otimes M_{2}^{+})\oplus(M_{1}^{-}\otimes
M_{2}^{-})\right)\ \oplus\ W^{-}\otimes\left((M_{1}^{+}\otimes
M_{2}^{-})\oplus(M_{1}^{-}\otimes M_{2}^{+})\right),$

and

$W^{-}\otimes\left((M_{1}^{+}\otimes M_{2}^{+})\oplus(M_{1}^{-}\otimes
M_{2}^{-})\right)\ \oplus\ W^{+}\otimes\left((M_{1}^{+}\otimes
M_{2}^{-})\oplus(M_{1}^{-}\otimes M_{2}^{+})\right).$

The determinant is the same since $\dim(W^{+})=\dim(W^{-}).$ Only the case
$[K:F]=2$ remains to be considered. Here, we have to compare

$(M_{1}^{+}\otimes M_{2}^{+})\oplus(M_{1}^{-}\otimes M_{2}^{-})\quad{\rm
and}\quad(M_{1}^{+}\otimes M_{2}^{-})\oplus(M_{1}^{-}\otimes M_{2}^{+}).$

For the latter, we have an involution with no fixed points on the basis
vectors, the determinant of the involution is $(-1)^{m^{2}};$ on the former,
we have $\dim(M)$ many fixed-points: $e_{1,a}^{+}\otimes e_{2,a}^{+}$ and
$e_{1,a}^{-}\otimes e_{2,a}^{-}$, hence the determinant of the involution is
$(-1)^{m^{2}-m}(-1)^{m^{2}-m}=1;$ whence, the ratio of the two determinants is
$(-1)^{m^{2}}=(-1)^{m}=(-1)^{\dim(M)/2}.$ For $\dim(M)/2$ even, we get the
same trivial determinant on $\prod_{i\in I}S_{\pi^{-1}(i)}$, and if
$\dim(M)/2$ is odd then we get $\prod_{i\in I}{\rm sgn}.$

Let us choose an isomorphism between $J\to I$ and $(J_{0}\times I)\to I$ so
that

${\rm Aut}(J\to I)\ =\ S_{I}\times\prod_{i\in I}S_{J_{0}}.$

Except in the case of $[K:F]=2$, and $\dim(M)/2$ is odd, we have an equality
of characters on $\prod_{i\in I}S_{J_{0}}.$ We claim that in this case we have
an equality of characters on all of ${\rm Aut}(J\to I)$ (and hence
$c^{+}\approx c^{-}$). Indeed, the $(\otimes_{\pi(j)=i}M_{i})^{+}$ for all $i$
have been identified, and this gives the same action. Same for $-$, and we
have the same dimension, hence the same character.

The same applies to the case $[K:F]=2$ and $\dim(M)/2$ odd, in which case the
ratio of the two characters is

$\begin{cases}\mbox{$1$ on $S_{I}$,}\\\ \mbox{$\prod_{i\in I}{\rm sgn}$ on
$\prod_{i\in I}S_{\pi^{-1}(i)}$}\end{cases}$

This character does not depend on how the $\pi^{-1}(i)$ have all been
identified with a fixed $J_{0}$; two such identifications are conjugate by an
element of $\prod_{i\in I}S_{\pi^{-1}(i)}$. We have inside ${\rm Aut}(J\to
I)$, the Galois group of the normal closure of $F/{\mathbb{Q}}.$ Restricting
the character to this Galois group gives us a quadratic extension
${\mathbb{Q}}$; if $K=F(\sqrt{D})$, then this quadratic extension is
${\mathbb{Q}}(\sqrt{N_{F/{\mathbb{Q}}}(D)}).$ It gives an Artin motive $A$ of
rank $1$ over ${\mathbb{Q}}$, and in the case $[K:F]=2$ and $\dim(M)/2$ odd,
we get

$c^{+}({\rm Res}_{F/{\mathbb{Q}}}(\otimes_{K/F}M))\ \approx\ c^{-}({\rm
Res}_{F/{\mathbb{Q}}}(\otimes_{K/F}M))c^{+}(A).$

∎

####  3.6.3. Motivating example-I

Consider a projective nonsingular variety $X$ over $K$, and the variety
$\prod_{K/F}X$ over $F$ given by Weil restriction of scalars. For any
$F$-algebra $A$, one defines $(\prod_{K/F}X)(A)=X(K\otimes_{F}A).$ The
motivating example for multiplicative induction comes from relating the
cohomology of $\prod_{K/F}X$ and the cohomology of $X$. For example, for the
Betti realizations, take $A={\mathbb{C}}$, fix an embedding
$\sigma:F\to{\mathbb{C}}$, and fix an ordering $\\{\eta_{1},\dots,\eta_{d}\\}$
on the set of embeddings $K\to{\mathbb{C}}$ that restrict to $\sigma$, where
$d=[K:F]$. For the ${\mathbb{C}}$-points, one has:

$(\prod_{K/F}X)_{\sigma}({\mathbb{C}})\ =\
\prod_{j=1}^{d}X_{\eta_{j}}({\mathbb{C}}),$

where, $(\prod_{K/F}X)_{\sigma}=(\prod_{K/F}X)\times_{F,\sigma}{\mathbb{C}}$
and $X_{\eta_{j}}:=X\times_{K,\eta_{j}}{\mathbb{C}}.$ By the Künneth theorem,
one has

(20) $H^{p}((\prod_{K/F}X)_{\sigma}({\mathbb{C}}),{\mathbb{Q}})\ =\
\bigoplus_{\sum
p_{j}=p}\bigotimes_{j=1}^{d}H^{p_{j}}(X_{\eta_{j}}({\mathbb{C}}),{\mathbb{Q}}).$

Now consider the Betti–de Rham realization for the motive $H^{q}(X)$ over $K$
with coefficients in ${\mathbb{Q}}$. If $q$ is even, the Betti realization of
the multiplicative induction $\bigotimes_{K/F}H^{q}(X)$ is the piece in
$H^{q[K:F]}((\prod_{K/F}X)({\mathbb{C}}),{\mathbb{Q}})$ corresponding to the
summand in (20) with $p_{1}=\cdots=p_{d}=q.$

####  3.6.4. Motivating example-II: Asai motives

To a Hilbert modular form $f$ for a real quadratic extension $K/{\mathbb{Q}}$,
Asai [As77] attached a degree-$4$ $L$-function which has all the usual
analytic properties of an automorphic $L$-function under suitable assumptions
on $f$. This $L$-function is now called the Asai $L$-function, or the twisted
tensor $L$-function. As is now well-known, the Asai $L$-function is the
standard $L$-function of a suitable automorphic form ${\rm As}(f)$ on ${\rm
GL}(4)/{\mathbb{Q}}$ obtained by Langlands’s principle of functoriality which
on the Galois side is given by multiplicative induction: a two-dimensional
irreducible representation $V$ of ${\rm Gal}(\overline{{\mathbb{Q}}}/K)$
corresponding to $f$ gives after multiplicative induction a four-dimensional
representation ${\rm As}(V)=\otimes_{K/{\mathbb{Q}}}V$ of ${\rm
Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$ that corresponds to ${\rm As}(f)$;
see [Kr03]. The Asai $L$-function is one of the $L$-functions in the
Langlands-Shahidi family of $L$-functions. More generally, given a quadratic
extension of totally real number fields $K/F$, and given a cuspidal
automorphic representation $\pi$ of ${\rm GL}_{n}/K$, the principle of
functoriality predicts an automorphic representation ${\rm As}(\pi)$ of ${\rm
GL}_{n^{2}}/F$, which is cuspidal under some regularity assumption on $\pi$.
Furthermore, suppose that $\pi$ is motivic and $M=M(\pi)$ is the conjectural
motive associated to $\pi$, then conjecturally, ${\rm As}(\pi)$ is also
motivic, and one expects $\otimes_{K/{\mathbb{Q}}}(M(\pi))\ =\ M({\rm
As}(\pi)).$ The period relation in Thm. 3.6.2 then gives a motivic explanation
to the rationality results for the ratios of critical values of Asai
$L$-functions in the forthcoming [KR].

###  3.7. Symplectic and Orthogonal Motives

####  3.7.1. The case when the base field is ${\mathbb{Q}}$

We consider a Betti–de Rham system of realizations $M$ over ${\mathbb{Q}}$
with coefficients in $E$. In the even weight case we assume the vanishing of
the middle Hodge number. We assume $M$ is given a linear algebra structure
$\mathsf{s}$ such that $G(\mathsf{s})$ is contained in a group ${\rm GSp}$ of
symplectic similitudes, or in the connected component ${\rm GO}^{\circ}$ of a
group ${\rm GO}$ of orthogonal similitudes. The group ${\rm GO}^{\circ}$ is
also sometimes denoted ${\rm GSO}.$

######  Theorem 3.7.2.

If the multiplier of $\mathsf{F}_{\infty}$ is $1$, then
$c^{+}_{E}(M)=c^{-}_{E}(M)$.

Let us elaborate our setting a bit more. One starts with $M$ of rank $2n$ over
$E,$ and with an $L$ of rank $1$ over $E$. In practice (motivic case), $L$
will be $L(\chi)(m),$ with $L(\chi)$ the system of realizations defined by a
Dirichlet character $\chi$ with values in $E$ and with $m\in{\mathbb{Z}}.$ The
assumption of ‘the multiplier of $\mathsf{F}_{\infty}$ is $1$’ translates to
$\chi(-1)=(-1)^{m}.$ In the symplectic case, the linear algebra structure
$\mathsf{s}$ is a non-degenerate alternating $E$-bilinear pairing

$B:M\otimes_{E}M\to L.$

In the orthogonal case, one has to give more than a non-degenerate symmetric
$E$-bilinear pairing

$B:M\otimes_{E}M\to L.$

Such a pairing induces a similar pairing on $\det(M):=\wedge^{2n}M$:

$\det(B):\det(M)^{\otimes^{2}}\to L^{\otimes^{2n}}.$

This new pairing amounts to a triviliazation of

$(\det(M)\otimes L^{\otimes^{-n}})^{\otimes^{2}}.$

The required multilinear algebra structure $\mathsf{s}$ consists of the $B$,
together with a trivialization of $\det(M)\otimes L^{\otimes^{-n}}$
(isomorphism with the unit object) inducing the above trivialization of its
tensor square.

###### Proof.

For $x\in M^{+}$ and $y\in M^{-}$, on the one hand, one has
$B(\mathsf{F}_{\infty}x,\mathsf{F}_{\infty}y)=B(x,-y)=-B(x,y),$ but on the
other hand one also has $B(\mathsf{F}_{\infty}x,\mathsf{F}_{\infty}y)=B(x,y),$
hence, $B(x,y)=0;$ whence, $M_{B}$ is the symplectic or orthogonal direct sum
of $M^{+}$ and $M^{-}.$

We have to show that $Z(\mathsf{F}_{\infty})$ acts with the same determinant
on $M^{+}$ and $M^{-}$. The dimension $n$ of $M^{+}$ and $M^{-}$ is even. In
the symplectic case, because $M^{+}$ and $M^{-}$ are symplectic. In the
orthogonal case, the connected component of the group of similitudes is given
by

$\det(g)={\rm multiplier}(g)^{n},$

forcing $n$ to be even since the multiplier of $\mathsf{F}_{\infty}$ is $1$
and determinant $(-1)^{n}.$

An element $g$ in $G(\mathsf{s})$ centralizing $\mathsf{F}_{\infty}$ is the
sum of an $g^{+}$ acting on $M^{+}$ and $g^{-}$ acting on $M^{-}$. In the
symplectic case, $g^{+}$ and $g^{-}$ are symplectic similitudes with the same
multipliers, hence the same determinant

$\det(g^{\pm})={\rm multiplier}(g^{\pm})^{n/2}.$

In the orthogonal case, they are orthogonal similitudes with the same
multiplier. The ratios

$\det(g^{\pm})/{\rm multiplier}(g^{\pm})^{n/2}\in\\{\pm 1\\},$

but $\det(g)/{\rm multiplier}(g)^{n}=1$ means that they are equal: for some
sign $\epsilon$, one has

$\det(g^{\pm})={\rm multiplier}(g^{\pm})^{n/2}\cdot\epsilon,$

proving that $\det(g^{+})=\det(g^{-}).$ ∎

####  3.7.3. The case when the base field is totally real

Thm. 3.7.2 will continue to hold for $M$ and $L$ Betti–de Rham systems of
realizations over a totally real field $F$. The parity assumption is now that
for each complex embedding $\sigma$ of $F$, the involution
$\mathsf{F}_{\infty}$ of $L_{\sigma}$ is trivial. In the motivic case, one
will want $L$ to be $L(\chi)(m)$ for $\chi$ an abelian character of ${\rm
Gal}(\overline{F}/F)$ with values in $E^{*}$, and the parity assumption is
that for each infinite Frobenius $\mathsf{F}_{\infty}$ one has
$\chi(\mathsf{F}_{\infty})=(-1)^{m}.$ The relevant group $G(\mathsf{s})$ for
${\rm Res}_{F/{\mathbb{Q}}}(M)$ is now an extension of the symmetric group
$S_{T},$ of the set $T$ of complex embeddings of $F$, by the product of the
(connected components) of the groups of similitudes of the $M_{\sigma}$’s. The
group $Z(\mathsf{F}_{\infty})$ is the extension of $S_{T}$ by the product of
the centralizers. The proof that the action of this product of centralizers on

${\rm Res}_{F/{\mathbb{Q}}}(M)^{+}=\prod M_{\sigma}^{+},\quad{\rm
and}\quad{\rm Res}_{F/{\mathbb{Q}}}(M)^{-}=\prod M_{\sigma}^{-},$

have the same determinant is as before. It remains to compute a ratio of
determinants on the quotient $S_{T}$. For this, we may first extend the field
of coefficients to split the forms $B$ on $M_{\sigma}$. One can then find
basis $e^{\pm}_{i,\sigma}$ of the $M_{\sigma}^{\pm}$ and representatives for
elements of $S_{T}$ which for each $(\pm,i)$, permute the
$e^{\pm}_{i,\sigma}$. As $n$ is even, for such a choice both determinants are
trivial.

####  3.7.4. Example: The tensor product of two rank-two motives-II

We revisit the example discussed in 3.1.5. Using the notations therein, if
$\varphi\in S_{k}(\Gamma_{0}(N),\omega)$ is a primitive cuspform and
$M(\varphi)$ the associated rank-two motive, then $M(\varphi)$ is symplectic
but does not satisfy the parity condition in the hypothesis of Thm. 3.7.2. The
symplectic form $B$ is an alternating form $B_{\varphi}:M(\varphi)\otimes
M(\varphi)\to{\mathbb{Q}}(\omega)(1-k).$ Since the Dirichlet character
$\omega$ is the nebentypus character of a weight $k$ modular form, one has
$\omega(-1)=(-1)^{k}$. Hence $\mathsf{F}_{\infty}$ acts on the rank-one motive
${\mathbb{Q}}(\omega)(1-k)$ via $\omega(-1)\cdot(-1)^{1-k}=-1.$ In particular,
Thm. 3.7.2 does not apply to $M(\varphi)$–but this is expected since the two
periods for a general (non-CM) cuspform $\varphi\in
S_{k}(\Gamma_{0}(N),\omega)$ are expected to be algebraically independent.

Now suppose, for $j=1,2$ we have $\varphi_{j}\in
S_{k_{j}}(\Gamma_{0}(N_{j}),\omega_{j}).$ Assume $k_{1}\neq k_{2}.$ The
orthogonal form $B_{\varphi_{1}}\otimes B_{\varphi_{2}}$ on the tensor product
motive $M(\varphi_{1})\otimes M(\varphi_{2})$ takes values in the rank-one
motive ${\mathbb{Q}}(\omega_{1}\omega_{2})(2-k_{1}-k_{2})$, on which the
action of $\mathsf{F}_{\infty}$ is via
$(\omega_{1}\omega_{2})(-1)\cdot(-1)^{2-k_{1}-k_{2}}=1,$ satisfying the
hypothesis of the theorem. It remains to check that $G(\mathsf{s})$ is
contained in the connected component ${\rm GO}^{\circ}$. This is a special
case of the following general lemma.

######  Lemma 3.7.5.

Let $(V_{1},B_{1})$ and $(V_{2},B_{2})$ be nondegenerate symplectic vector
spaces over a field $F$. Then $V=V_{1}\otimes_{F}V_{2}$ equipped with
$B=B_{1}\otimes B_{2}$ is a nondegenerate orthogonal space over $F$. Under the
canonical map ${\rm GL}(V_{1})\times{\rm GL}(V_{2})\to{\rm GL}(V_{1}\otimes
V_{2})$, the image of ${\rm GSp}(V_{1},B_{1})\times{\rm GSp}(V_{2},B_{2})$ is
contained inside ${\rm GSO}(V,B).$ (In particular, ${\rm GL}(2)\times{\rm
GL}(2)$ maps inside ${\rm GSO}(4).$)

###### Proof.

Suppose the dimension of $V_{1}$ is $2n_{1}$ and ${\rm GSp}(V_{1},B_{1})$ has
a similitude character $\mu_{1}$ then for $g_{1}\in{\rm GSp}(V_{1},B_{1})$ one
has $\det(g_{1})^{2}=\mu_{1}(g_{1})^{2n_{1}}.$ Similarly, for $g_{2}\in{\rm
GSp}(V_{2},B_{2}),$ one has $\det(g_{2})^{2}=\mu_{2}(g_{2})^{2n_{2}}.$ For the
element $g_{1}\otimes g_{2}$ acting on $(V,B)$, the similitude character $\mu$
and the determinant are given by $\mu(g_{1}\otimes
g_{2})=\mu_{1}(g_{1})\mu_{2}(g_{2})$ and $\det(g_{1}\otimes
g_{2})=\det(g_{1})^{2n_{2}}\det(g_{2})^{2n_{1}},$ respectively. One checks
that $\det(g_{1}\otimes g_{2})\cdot\mu(g_{1}\otimes g_{2})^{\dim(V_{1}\otimes
V_{2})/2}=1$ which implies that $g_{1}\otimes g_{2}\in{\rm GSO}(V,B).$ ∎

####  3.7.6. Example: The conjectural motive attached to representations of
${\rm O}(2n)$

In this paragraph we discuss the relation of Thm. 3.7.2 with the main result
of [BR20]. Suppose ${\rm O}(2n)$ is a split orthogonal group over
${\mathbb{Q}}$, and $\sigma$ is a cuspidal automorphic representation of ${\rm
O}(2n)({\mathbb{A}}_{\mathbb{Q}})$ satisfying the conditions:

  1. (i)

the Arthur parameter $\Psi$ of $\sigma$ is a cuspidal automorphic
representation of ${\rm GL}(2n)/{\mathbb{Q}}$;

  2. (ii)

$\sigma$ is globally generic with respect to some Whittaker datum $\psi$;

  3. (iii)

$n$ is even, and the $\sigma_{\infty}$ is a discrete series representation of
${\rm O}(n,n)$ which is locally generic with respect to $\psi_{\infty}.$

We will see below that Thm. 3.7.2 applies to the conjectural motive $M$
attached to $\sigma$ giving us then a motivic explanation of the main theorem
of [BR20] recalled in (LABEL:eqn:BR).

The motive is attached not so much as to $\sigma$, but rather to its Arthur
parameter $\Psi$, which by (i) is a cuspidal automorphic representation of
${\rm GL}(2n)/{\mathbb{Q}}$. However, $\Psi$ is not a representation that is
of motivic type, in as much as it is not algebraic, and as in [C90], one
remedies the situation by considering:

$\Psi^{T}:=\Psi\otimes|\\!|\ |\\!|^{1/2}$

a half-integral Tate-twist of $\Psi$ (one may also consider $\Psi\otimes|\\!|\
|\\!|^{m/2}$ for any odd integer $m$) which is of motivic type. Let
$M=M(\Psi^{T})$ be the conjectural rank $2n$ motive associated to $\Psi^{T}.$
For the expected details of this dictionary between motives and automorphic
forms, the reader is referred to [HR20, Chap. 7].

To get the Hodge types of $M$, recall from [BR20, 2.2.8] that the Langlands
parameter of the discrete series representation $\sigma_{\infty}$, which by
definition of the transfer $\sigma\mapsto\Psi$ is the same as the Langlands
parameter $\varphi(\Psi_{\infty})$ of $\Psi_{\infty},$ is of the form:
$\oplus_{j=1}^{n}I(\chi_{\ell_{j}})$, for a decreasing sequence of even
positive integers $\ell_{1}>\ell_{2}>\cdots>\ell_{n}$, where $\chi_{\ell}$ is
the character of ${\mathbb{C}}^{*}$ that maps $z$ to $(z/\bar{z})^{\ell/2}$,
and $I(\chi_{\ell})$ is its induction to a $2$-dimensional irreducible
representation of the Weil group $W_{\mathbb{R}}$ of ${\mathbb{R}}$. We can
read off the exponents of the characters of ${\mathbb{C}}^{*}$ appearing in
$\varphi(\Psi^{T}_{\infty})$. One aspect of the dictionary
$\Psi^{T}\leftrightarrow M$ is that if $(p,q)$ is a Hodge type for $M$ if and
only if $z^{-p}\bar{z}^{-q}$ appears in
$\varphi(\Psi^{T}_{\infty}\otimes|\\!|\ |\\!|^{(1-2n)/2}$). One concludes that
the Hodge types of $M(\Psi^{T})$ are of the form:

$\left\\{\left(\frac{\pm\ell_{j}+2n-2}{2},\frac{\mp\ell_{j}+2n-2}{2}\right)\right\\}_{1\leq
j\leq n}.$

The purity weight of the motive is $w=w(M)=2n-2.$

Next, we look at $L$-functions. One has the following relations: $L(s,\sigma)\
=\ L(s,\Psi),$ i.e., the degree-$2n$ $L$-function of $\sigma$ on ${\rm O}(2n)$
corresponding to the inclusion ${\rm O}(2n,{\mathbb{C}})\hookrightarrow{\rm
GL}(2n,{\mathbb{C}})$ of dual groups, is the standard $L$-function $L(s,\Psi)$
attached to $\Psi.$ Of course, by definition of Tate-twist, one also has
$L(s,\Psi^{T})\ =\ L(s+1/2,\Psi).$ Another aspect of the dictionary
$\Psi^{T}\leftrightarrow M$ is that $L(s,\Psi^{T}\otimes|\\!|\
|\\!|^{(1-2n)/2})\ =\ L(s,M).$ Hence, $L(s,\Psi)=L(s+n-1,M).$ The same
relations hold for local and global $L$-functions. One deduces for the
symmetric square $L$-functions the relation $L(s,{\rm
Sym}^{2},\Psi)=L(s+2n-2,{\rm Sym}^{2}(M)).$ Since $\Psi$ is a transfer from
${\rm O}(2n)$, it is part of the Arthur classification, that $L(s,{\rm
Sym}^{2},\Psi)$ has a pole at $s=1$, or that $L(s,{\rm Sym}^{2}(M))$ has a
pole at $s=2n-1.$ This happens when the Tate motive ${\mathbb{Q}}(2-2n)$
appears in ${\rm Sym}^{2}(M)$; in other words, we have an orthogonal structure
$\mathsf{s}$:

$M\otimes M\to{\mathbb{Q}}(2-2n).$

If $g\in G(\mathsf{s})$ the group preserving $\mathsf{s}$, then its multiplier
is

$\mu(g)=|\\!|g|\\!|^{2-2n}.$

Also, the multiplier of $\mathsf{F}_{\infty}$ on ${\mathbb{Q}}(2-2n)$ is
$(-1)^{2-2n}=1$ satisfying the parity condition in Thm. 3.7.2. The determinant
of $M$ is the central character of $\Psi^{T}\otimes|\\!|\ |\\!|^{(1-2n)/2}$.
The central character of $\Psi$ is trivial since it is a transfer from the
split orthogonal group, hence the determinant character of $M$ is

$\det(M)(g)=|\\!|g|\\!|^{(2-2n)n}.$

It follows that $\det(M)(g)\mu(g)^{-n}=1$, i.e., $G(\mathsf{s})$ is contained
in ${\rm GSO}(M)$. Hence, Thm. 3.7.2 applies to the motive $M$ as claimed.

Acknowledgements: The second author is grateful to the Charles Simonyi
Endowment that funded his membership at the Institute for Advanced Study,
Princeton, during the Spring and Summer terms of 2018, when the authors got
started on this project; he also acknowledges the warm hospitality of the IAS
during several subsequent short visits.

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