# Siegel modular forms with extra twists

Debargha Banerjee and Ronit Debnath  INDIAN INSTITUTE OF SCIENCE EDUCATION AND
RESEARCH, PUNE, INDIA

###### Abstract.

In this paper, we study Siegel modular forms with extra twists. We provide
conditions on the level and genus of the forms that is necessary for the
existence of extra twists for Siegel modular forms. We also give explicit
examples of Siegel modular forms with extra twists that are different from the
complex conjugation.

###### Key words and phrases:

Modular forms, Brauer class

###### Key words and phrases:

Siegel Modular forms, Yoshida lifts

###### 2000 Mathematics Subject Classification:

Primary: 11F11, Secondary: 11F80, 11F30

###### 2000 Mathematics Subject Classification:

Primary: 11F46, Secondary: 11F80, 11F30

The first named author was partially supported by the SERB grant
MTR/2017/000357 and CRG/2020/000223. We thank Professor Abhishek Saha for
fruitful email correspondence in the initial stage of the project.

##  1\. Introduction

Let $f=\sum\limits_{n=1}a_{n}q^{n}\in S^{1}_{k}(N,\epsilon)$ be a classical
elliptic normalized cuspidal newform of weight $k\geq 2$, genus $1$ and
nebentypus character $\epsilon$ and $K={\bf Q}(a_{n})$ be the number field
generated by the Fourier coefficients of $f$ with
$[K\mathrel{\mathop{\ordinarycolon}}{\bf Q}]=d$. For all primes $l$ with
$\lambda\mid l$ , we can associate a compatible system of $\lambda$ -adic
Galois representation attached to this elliptic modular form $f$ by the
classical result due to Shimura, Deligne,

$\rho_{f,\lambda}\mathrel{\mathop{\ordinarycolon}}G_{\bf
Q}\rightarrow{\mathrm{GL}}(V_{\lambda})$

with $\mathrm{trace}(\rho_{f,\lambda}({\mathrm{Frob}}_{p}))=a_{p}$ and
$\det(\rho_{f,\lambda})=\epsilon(p)p^{k-1}$, for all primes $p\nmid lN$.
Donote by $\rho_{f,l}\mathrel{\mathop{\ordinarycolon}}=\prod_{\lambda\mid
l}\rho_{f,\lambda}$ and
$V_{l}\mathrel{\mathop{\ordinarycolon}}=\prod_{\lambda\mid l}V_{\lambda}$.

For the last fifty years, there is a considerable interest to study the image
of these Galois representations, namely $\rho_{f,l}(G_{{\bf Q}})$ with
$G_{{\bf Q}}$ given Krull topology and ${\mathrm{GL}}(V_{\lambda})$ usual
$l$-adic topology. In turn, this has several important arithmetical
applications including the proof of the Fermat’s last theorem. In this study,
certain symmetries of modular forms called inner twists or extra twists are
important ingredients. Ken Ribet pioneered the study of modular forms of extra
twists starting with the existence of the same [1, p. 48]. In this case, the
existence is guranteed by the work of Doi-Yamauchi, Birch and Koike.

For $k=2$, Ribet and Momose [13] studied the algebra
$\mathrm{Lie}(\rho_{f,l}(G_{{\bf Q}}))$ and showed that this is an explicit
central simple algebra. Eknath Ghate and his collaborators (cf. [6], [9], [5])
studied this algebra for higher weights $k\geq 2$ and they give a formulae for
local algebras in terms of slopes of modular forms (cf. [5] for details). In
the the recent past, there is a progress regarding the Galois representations
associated with more general automorphic forms using the discovery of
perfectoid spaces by Scholze [18].

The aim of this paper is to extend results to the automorphic forms for the
symplectic group ${\mathrm{Gsp}}_{2g}/{\bf Q}$ with genus $g\geq 2$. We extend
a result of Ribet [1, pg. 49 Theorem 5.7] to automorphic forms on the group
${\mathrm{Gsp}}_{2g}/{\bf Q}$ with $g\geq 2$ and give explicit examples of
Siegel modular forms with extra twists.

Using perfectoid spaces and building on work of Harris- Taylor-Thorne-Clozel-
Shin and many others, we associate to every Siegel modular forms $F$ of
arbitrary genus $g$, the compatible system of $\lambda$ -adic Galois
representation of Peter Scholze [18, p. 1034, Theorem 5.1.4]

$\rho_{F,\lambda}\mathrel{\mathop{\ordinarycolon}}G_{{\bf
Q}}\rightarrow{\mathrm{GSp}}_{2g}(K_{\lambda})$

where $K={\bf Q}(t_{p})$ is the number field obtained by adjoining the Hecke
eigenvalues $t_{p}$ of $F$ for all $p\in\mathbb{N}$. These Galois
representations are continuous, semi-simple and they encode the Satake
parameters of the automorphic representation associated to $F$.

In our first main theorem, we encapsulate a condition on the genus $g$, weight
$k$ and level $N$ of the Siegel modular form $F$ which is sufficient but not
necessary for this Siegel modular form to have an extra twist. The level
condition requires a high prime power divisibility. This is not so surprising
even for $g=1$, modular forms with extra twists are more the norm than the
exception in the case of a high prime power.

Denote by $\rho_{l}\mathrel{\mathop{\ordinarycolon}}=\prod_{\lambda\mid
l}\rho_{F,\lambda}$, ${\mathfrak{g}}_{l}=\mathrm{Lie}(\rho_{F,l}(G_{\bf Q}))$,
$\mathfrak{gsp}_{2g}=\mathrm{Lie}({\mathrm{Gsp}}_{2g})$ and

$\mathfrak{a}_{l}=\\{u\in\mathfrak{gsp}_{2g}(K\otimes{\bf
Q}_{l})|\mathrm{Tr}(u)\in{\bf Q}_{l}\\};$

where $\mathrm{Tr}(u)$ denotes the trace of the matrix $u$. Note traces as
sufficient to determine the representations as being isomorphic [20, p. 11].
We now state our first theorem:

######  Theorem 1.

Let $F$ be a non-CM (cf. Definition 15) Siegel Modular form of genus $g\geq
2$, weight $k$, level $N$ such that $|g-k|$ is odd. Let $N$ be chosen such
that $({\bf Z}/N{\bf Z})^{\times}$ has an element of order $2g$. The Siegel
modular forms $F$ admits an extra twist if and only if we then have a strict
inclusion $\mathfrak{g}_{l}\subsetneq\mathfrak{a}_{l}$ .

The sufficiency of the conditions in the theorem above stems from Proposition
16. From the Petersson inner product, $(c,\epsilon^{-1})$ is always an extra
twist for the Siegel modular forms. It is natural to ask if there are extra
twists of Siegel modular forms different from complex conjugation.

In our second theorem, we explicitly produce Siegel modular forms with extra
twists that are not coming from complex conjugation. Start with two classical
elliptic modular form $(f,g)$ with $f\in S^{1}_{k_{1}}(N,\epsilon)$, $g\in
S^{1}_{k_{2}}(N,\epsilon)$. Using a suitable embedding
${\mathrm{GL}}_{2}\times{\mathrm{GL}}_{2}\xhookrightarrow{}{\mathrm{GSp}}_{4}$,
it is possible to produce a Siegel modular form $Y(f\otimes g)$ that is a
Yoshida lift of $(f,g)$.

For two carefully chosen classical modular forms $f$ and $g$, their Yoshida
lift denoted by $Y(f\otimes g)$ is a Siegel modular form with an extra twist.

######  Theorem 2.

Let $f$ and $g$ be two classical non-CM elliptic modular forms with extra
twist associated to the same Dirichlet character $\chi$ different from complex
conjugation. Assume that the Yoshida lift $Y(f\otimes g)$ of $(f,g)$ exists,
then $Y(f\otimes g)$ is a Siegel modular form that contains extra twists
different from complex conjugation.

The above theorem says that the group of extra twists can be really large. The
method of this theorem also provides a systematic way of producing large class
of examples of Siegel modular forms with extra twists. In fact, we give few
examples of explicit Siegel modular forms with extra twists. We construct
these examples by taking the Yoshida lift of two classical modular forms of
the same level and same central nebentypus character. For families of modular
forms, group of extra twists are studied by Conti [7] in a recent paper.

Kumar et al use the extra twists for Siegel modular forms with artimetic
applications in the context of Lang-Trotter conjecture [12]. It is worth
mentioning that as of now it is very hard to compute Fourier coefficients and
hence Hecke eigenvalue for Siegel modular forms for congruence subgroups even
for $g=2$. There are several ways to produce Siegel modular forms using
various lifts like symmetric cube, Saito-Kurakawa or Yoshida lifts. It will be
really intriguing to find extra twists of non-lifted Siegel modular forms that
are not coming from complex conjugation.

##  2\. Siegel Modular Forms

###  2.1. Motivation

Consider the case of classical elliptic modular forms. These are differential
forms on the space obtained by the action of ${\mathrm{SL}}_{2}({\bf Z})$ on
the upper half plane $\mathbb{H}$. Recall, the upper half plane can be
expressed in terms of the group ${\mathrm{SL}}_{2}(R)/{\mathrm{SO}}(2)$, where
${\mathrm{SO}}(2)=U(1)$, is the stabilizer of the point $i=\sqrt{-1}$. This is
a maximal compact subgroup. The group ${\mathrm{SL}}_{2}({\bf Z})$ is the
automorphism group of the lattice ${\bf Z}^{2}$ with the standard alternating
inner product $<,>$ defined as:

$<(a,b),(c,d)>=ad-bc.$

We wish to generalise by taking for $g=2$ (the same generalisation works for
arbitrary $g$). The lattice $({\bf Z}^{2})^{2}$ of rank $2\times 2$, with
basis $e_{1},e_{2},f_{1},f_{2}$ provided with the symplectic form $<,>$
defined by :

$<e_{i},e_{j}>=0,<f_{i},f_{j}>=0,<e_{i},f_{j}>=\delta_{ij}$

with $\delta_{ij}$ being the Kronecker’s delta. The symplectic group
${\mathrm{Sp}}_{4}({\bf Z})$ is by definition the automorphism group of the
symplectic lattice

${\mathrm{Sp}}_{4}({\bf Z})\mathrel{\mathop{\ordinarycolon}}=Aut({\bf
Z}^{4},<,>)$. By using the basis of the $e^{\prime}s$ and $f^{\prime}s$ we can
write the elements as group of matrices: $\begin{bmatrix}A&B\\\
C&D\end{bmatrix}$ where $A,B,C$ and $D$ are integral $2\times 2$ matrices
satisfying the following conditions.

###  2.2. Siegel modular group

Let $R$ be a commutative ring with $1$ and fix an integer $g\geq 2$. Denote by
$I_{g}$ and $0_{g}$ be the identity and zero matrix of the ring $M_{g}(R)$ and
now consider the matrix $J_{g}=\begin{bmatrix}0_{g}&I_{g}\\\
-I_{g}&0_{g}\end{bmatrix}$. Denote by ${\mathrm{GSp}}_{2g}(R)$ the algebraic
group of symplectic similitudes with respect to $J_{g}$. Hence,

${\mathrm{GSp}}_{2g}(R)=\\{M\in{\mathrm{GL}}_{2g}(R)|M^{t}JM=\mu(M)J\\}.$

The map $M\rightarrow\mu(M)$ defines a character
$\mu\mathrel{\mathop{\ordinarycolon}}{\mathrm{GSp}}_{2g}(R)\rightarrow
R^{\times}$. We refer to $\mu$ as the similitude factor. The group

${\mathrm{Sp}}_{2g}(R)=\\{M\in{\mathrm{GL}}_{2g}(R)|M^{t}JM=J\\}$

is called the symplectic group of degree $2g$ with coefficients in $R$. Since
the above set is the automorphism group of the alternating skew-symmetric form
defined by $J_{g}$, hence ${\mathrm{Sp}}_{2g}(R)$ is a subgroup of
${\mathrm{GL}}_{2g}(R)$. For this article, we are mostly interested in $R={\bf
Z}$ or ${\bf R}$.

######  Definition 3.

[11, Page 2] For an open set $D\subset{\bf C}^{g}$, a function
$F\mathrel{\mathop{\ordinarycolon}}D\rightarrow{{\bf C}}$ is called
holomorphic if it is continuous and if for each fixed
$(z_{1}^{0},...,z_{g}^{0})\in D$, and each $j=1,...,g$, the function of a
single variable which is determined by the assignment $z_{j}\rightarrow
F(z_{1}^{0},...,z_{j-1}^{0},z_{j},z_{j+1}^{0},...,z_{g})$ is holomorphic.

The upper half plane for arbitrary genus $g\geq 1$ is the set
$\mathbb{H}_{g}=\\{X+iY|X,Y\in M_{g}({\bf C}),X=X^{t},Y=Y^{t},Y>0\\}$. The
condition $Y>0$ means it is positive definite as a matrix. We now define the
vector valued Siegel modular form in arbitrary genus $g>1$ here and
subsequently state the more explicit definition in genus $g=2$.

Suppose
$\rho\mathrel{\mathop{\ordinarycolon}}{\mathrm{GL}}_{g}\rightarrow{\mathrm{GL}}(V)$
be a finite dimensional complex representation. Then by a vector valued Siegel
modular form of genus $g$, we mean a holomorphic function
$F\mathrel{\mathop{\ordinarycolon}}\mathbb{H}_{g}\rightarrow{\bf C}$ such that
$F((A\tau+B)(C\tau+D)^{-1})=\rho(C\tau+D)F(\tau)$ for all
$\gamma=\begin{bmatrix}A&B\\\
C&D\end{bmatrix}\in{\mathrm{Sp}}_{2g}(\mathbb{{\bf Z}})$ and
$\tau\in\mathbb{H}_{g}$.

The vector valued Siegel modular form of genus $2$ can be described more
explicitly [3]. The scalar valued Siegel modular forms are just a special case
of that. For non negative integers $k$ and $j$, let
$\rho_{k,j}\mathrel{\mathop{\ordinarycolon}}{\mathrm{GL}}_{2}({\bf
C})\rightarrow{\mathrm{GL}}_{j+1}({\bf C})$ be the irreducible representation
of signature $(j+k,k)$;

$\rho_{k,j}=det^{k}\otimes Sym^{j}.$

We now define the slash operator on a function
$F\mathrel{\mathop{\ordinarycolon}}\mathbb{H}_{g}\rightarrow{\bf C}$. For a
matrix $\gamma=\begin{bmatrix}A&B\\\
C&D\end{bmatrix}\in{\mathrm{Sp}}_{2g}({\bf Z})$, and $\tau\in\mathbb{H}_{g}$,
we have the action

$(F|_{k,j}(\gamma))(\tau)=(\rho_{k,j}(C\tau+D))^{-1}F((A\tau+B)(C\tau+D)^{-1}).$

######  Definition 4.

A holomorphic function
$F\mathrel{\mathop{\ordinarycolon}}\mathbb{H}_{2}\rightarrow{\bf C}$ is called
a vector valued Siegel modular form with respect to the full subgroup
${\mathrm{Sp}}_{4}({\bf Z})$ and weight $\rho_{k,j}$ if $F|_{k,j}[\gamma]=F$
for all $\gamma\in{\mathrm{Sp}}_{4}({\bf Z})$.

The case that $j=0$, we are mainly concern with this case in this paper and
such Siegel modular forms are called scalar valued. So henceforth when we say
Siegel modular forms we mean scalar valued ones. Recall the following
definition of Siegel modular forms (respectively cusp forms) with respect to a
subgroup [2].

######  Definition 5.

Let $K$ be a subgroup of the full modular group ${\mathrm{Sp}}_{2g}({\bf Z})$.
Then $F$ is said to be a Siegel Modular Form of weight $k$ and character
$\chi$ for the subgroup $K$ if the following conditions are satisfied :

  1. (1)

$F$ is a holomorphic function on $\mathbb{H}_{g}$.

  2. (2)

For every matrix $M\in K$, the function $F$ satisfies $F|_{k}(M)=\chi(M)F$.

The set $M_{g}^{k}(K,\chi)$ denotes Siegel Modular Forms of genus $g$, weight
$k$ for the character $\chi$ and subgroup $K$.

######  Definition 6.

We define the operator $\Phi$ on $F\in M_{g}^{k}(K,\chi)$ by $(\Phi
F)(Z^{\prime})=\lim_{t\to\infty}F(\begin{bmatrix}Z^{\prime}&0\\\
0&it\end{bmatrix})$ with $Z^{\prime}\in\mathbb{H}_{g-1}$ and $t\in\mathbb{R}$.

######  Definition 7.

A Siegel modular form $F\in M_{g}^{k}(K,\chi)$ is said to be a cusp form if
$F$ Lies in the kernel of the $\Phi$ operator.

The principal congruence subgroup for the of level $N$ for the full modular
group denoted by $\Gamma^{(g)}(N)$ is defined to be

$\Gamma^{(g)}(N)=\\{M\in{\mathrm{Sp}}_{2g}({\bf Z})~{}|~{}M\equiv
I_{2g}~{}(modN)\\}.$

A congruence subgroup $K$ of ${\mathrm{Sp}}_{2g}({\bf Z})$ is any subgroup of
${\mathrm{Sp}}_{2g}({\bf Z})$ such that $K\supset\Gamma^{(g)}(N)$ for some
$N$. We are interested in the following two types of the congruence subgroups:

$\Gamma_{0}^{(g)}(N)=\\{\begin{bmatrix}A&B\\\
C&D\end{bmatrix}\in{\mathrm{Sp}}_{2g}({\bf Z})~{}|~{}C\equiv 0~{}(mod~{}N)\\}$

and

$\Gamma_{1}^{(g)}(N)=\\{\begin{bmatrix}A&B\\\
C&D\end{bmatrix}\in{\mathrm{Sp}}_{2g}({\bf Z})~{}|~{}C\equiv
0~{}(mod~{}N),A,D\equiv I_{g}~{}(mod~{}N)\\}.$

Obvious from the definitions we have the strict inclusions
$\Gamma^{(g)}(N)\subsetneq\Gamma^{(g)}_{1}(N)\subsetneq\Gamma^{(g)}_{0}(N)$.

##  3\. Extra twists for Siegel modular forms

In this section, we define extra twists for Siegel modular forms and state
some properties of the same.

######  Proposition 8.

[14] A Siegel modular form $F$ has the expansion

(1) $F(Z)=\sum_{A\in E^{2g},A\geq 0}t(A)e^{\pi itr(AZ)}$

where $t(A)$ denotes the coefficients of the expansion, $E^{2g}$ denotes the
set of $2g\times 2g$ half integral matrices, and $\mathrm{tr}$ denotes the
trace.

As recalled in the introduction, we associate to every Siegel modular forms
$F$ of arbitrary genus $g\geq 1$, the compatible system of $\lambda$ -adic
Galois representations [18, p. 1034, Theorem 5.1.4]

$\rho_{F,\lambda}\mathrel{\mathop{\ordinarycolon}}G_{{\bf
Q}}\rightarrow{\mathrm{GSp}}_{2g}(K_{\lambda})$

where $K={\bf Q}(t_{p})$ is the number field obtained by adjoining the Hecke
eigenvalues $t_{p}$ of $F$ for all $p\in\mathbb{N}$. These Galois
representations are continuous, semi-simple and they encode the Satake
parameters of the automorphic representation associated to $F$.

Let $\Gamma=Aut(K)$ and consider the set
$D\mathrel{\mathop{\ordinarycolon}}=\\{\varepsilon\mathrel{\mathop{\ordinarycolon}}G_{\bf
Q}\rightarrow K^{\times}\\}$ be the set of characters from $G_{{\bf Q}}$ to
$K^{\times}$. Let $V$ be the corresponding module over $K\otimes{\bf Q}_{l}$
for which $Aut(V)={\mathrm{GSp}}_{2g}(K\otimes{\bf Q}_{l})$.

Following [4], we define the Siegel modular forms with extra twists using
Galois representation. Following lemma proves that the image of the Galois
representation associated to a Siegel modular form is non-abelian.

######  Lemma 9.

The image $\rho_{F,\lambda}(G_{{\bf Q}})$ is non abelian.

###### Proof.

Let $c$ be the complex conjugation in $G_{{\bf Q}}$. Since $c^{2}=I$ and
$\det\rho_{\lambda}(c)=-1$, the eigenvalues of $\rho_{\lambda}(c)$ are
$+1,-1$. Hence the matrix $\rho_{\lambda}(c)$ is similar to its Jordan
canonical form with $+1$’s and $-1$’s in the diagonal. So the elements of
${\mathrm{GSp}}_{2g}(K_{\lambda})$ which commute with it form a block diagonal
subgroup T. Since the representation $\rho_{\lambda}$ is irreducible the image
cannot be a subset of $T$. Hence there are elements in the image that do not
commute with $\rho_{\lambda}(c)$ and so the image is non abelian. ∎

######  Definition 10.

A Siegel modular form $F$ is said to have an extra twist if there exists a
tuple $(\gamma,\chi_{\gamma})$ with $\gamma\in\Gamma$, $\chi_{\gamma}\in D$
such that $\rho_{F}\cong\gamma(\rho_{F})\otimes\chi_{\gamma}$.

We now state some properties about extra twists.

######  Lemma 11.

The extra twists for $\rho_{F}$ over ${\bf Q}$ form a group.

###### Proof.

Let $\gamma_{1},\gamma_{2}\in\Gamma$ be two extra twists of Siegel modular
forms. This means that $\rho_{F}=\gamma_{1}(\rho_{F})\otimes\chi_{\gamma_{1}}$
and $\rho_{F}=\gamma_{2}(\rho_{F})\otimes\chi_{\gamma_{2}}$. Hence
$\rho_{F}=\gamma_{1}\gamma_{2}(\rho_{F})\otimes\chi_{\gamma_{1}}\chi_{\gamma_{2}}=\gamma_{1}\gamma_{2}(\rho_{F})\otimes\chi_{\gamma_{1}.\gamma_{2}}$
where the character $\chi_{\gamma_{1}.\gamma_{2}}$ denotes the product of the
characters $\chi_{\gamma_{1}}$ and $\chi_{\gamma_{2}}$. Further the identity
element, inverse are trivial and we have already cHecked closure. So the set
of self twists for $\rho_{F}$ over ${\bf Q}$ form a group. ∎

We prove two lemmas similar to [5].

######  Lemma 12.

For every extra twist $\gamma$, the character $\chi_{\gamma}$ satisfying the
equivalence is uniquely determined.

###### Proof.

Let $\chi_{\gamma_{1}}$ and $\chi_{\gamma_{2}}$ be two characters associated
to the automorphism $\gamma$. Hence
$\rho_{F}=\gamma(\rho_{F})\otimes\chi_{\gamma_{1}}=\gamma(\rho_{F})\otimes\chi_{\gamma_{2}}$.
This means that
$\gamma(\rho_{F})\otimes(\chi_{\gamma_{1}}-\chi_{\gamma_{2}})=0$. If $c$
denotes complex conjugation, $\rho_{F}(c)\neq 0,$ and hence
$\chi_{\gamma_{1}}=\chi_{\gamma_{2}}$. ∎

######  Lemma 13.

The association $\delta\rightarrow\chi^{\delta}$ defines a cocycle on the
group of self-twist with values in $K^{\times}$.

###### Proof.

For $\gamma,\delta\in\Gamma$, the identity
$\chi_{\gamma\delta}\rightarrow\chi_{\gamma}\chi_{\delta}^{\gamma}$ shows that
$\gamma\rightarrow\chi_{\gamma}$ is a $1-$ cocycle. Specializing to $g\in
G_{{\bf Q}}$, we see that $\gamma\rightarrow\chi_{\gamma}(g)$ is a $1-$
cocycle. By Hilbert’s Theorem 90, $H^{1}(\Gamma,E^{\times})$ is trivial and
there is an element $\alpha(g)\in E^{\times}$ such that

(2) $\frac{\gamma(\alpha(g))}{\alpha(g)}=\chi_{\gamma}(g)$

for all $\gamma\in\Gamma$. Clearly $\alpha(g)$ is completely determined (upto
multiplication) by elements of $F^{\times}$. Varying $g\in G_{{\bf Q}}$ we
obtain the well defined map

$\overline{\alpha}\mathrel{\mathop{\ordinarycolon}}G_{{\bf Q}}\rightarrow
E^{\times}/F^{\times}.$

Since each $\chi_{\gamma}$ is a character, $\overline{\alpha}$ is a
homomorphism. ∎

The following proposition closely follows from [4, Proposition (2.15)].

######  Proposition 14.

Let ${\bf Q}[Tr(Ad(\rho))]$ denote the ring generated over ${\bf Q}$ by the
set $Tr(Ad(\rho)(g))$ for $g\in G_{{\bf Q}}$. Then every element of ${\bf
Q}[\mathrm{Tr}(Ad(\rho))]$ is fixed by all self twists for $\rho$ over ${\bf
Q}$.

###### Proof.

From [4, Proposition (2.14)] we get that there is an equivalent representation
to $\rho$ which we call $\rho^{\prime}$ satisfying :

  1. (1)

$\rho^{\prime}(g)\in{\mathrm{GSp}}_{4}(K_{\rho})$ where the field
$K_{\rho}={\bf Q}(tr(Ad(\rho)))$.

  2. (2)

$\gamma(\rho^{\prime})=\rho^{\prime}\otimes\epsilon$.

For $g\in G_{{\bf Q}}$, we consider the $1$-cocycle
$c_{g};\Gamma_{[g]}\rightarrow L^{\times}$ defined by
$c_{g}(\gamma)=s^{-1}((\rho^{\prime})^{\gamma}\rho^{\prime}(g)^{-1})$, where
$s\mathrel{\mathop{\ordinarycolon}}L^{\times}\rightarrow{\mathrm{GSp}}_{4}(L)$
denotes the scalar morphism. By Hilbert’s 90,
$H^{1}(\Gamma_{[\rho^{\prime}]},L^{\times})$ is the trivial module. Hence
$c_{g}(\gamma)=\frac{a_{g}}{\gamma(a_{g})}$ with some $a_{g}\in L^{\times}$.
Then for any $g\in G$ and $\gamma\in\gamma_{[\rho^{\prime}]}$, we obtain

$\rho^{\prime}(g)^{\gamma}\otimes\frac{\gamma(a_{g})}{a_{g}}=\rho^{\prime}(g).$

The identity $\gamma(\rho^{\prime}(g)a_{g})=\rho^{\prime}(g)a_{g}$ shows that
$\rho(g)a_{g}\in{\mathrm{GSp}}_{4}(K_{\rho})$. Hence the field ${\bf
Q}(Tr(Ad(\rho(g))))$ is fixed by all self twists for $\rho$ over ${\bf Q}$. ∎

We first define the Siegel modular form without complex multiplication below
which is important for the main theorems as they true hold for forms without
complex multiplication. The definition is taken from [4, Definition 2.3].

######  Definition 15.

A Siegel modular form $F$ is said to admit complex multiplication (or be cm)
if there exists non trivial $\epsilon$ such that
$\rho_{F}\cong\rho_{F}\otimes\epsilon^{-1}$.

Denote the matrix $dU_{g}=$ ( $\begin{matrix}d^{-1}I_{g}&0\\\
0&dI_{g}\end{matrix})$. Corresponding to a congruence subgroup $K$ and a
character $\psi\mathrel{\mathop{\ordinarycolon}}({\bf Z}/N{\bf
Z})^{\times}\rightarrow{\bf C}^{\times}$, we have the space
$S_{k}(K,\chi,\psi)=\\{u\in S_{k}(K,\chi)~{}|~{}F<d>=\psi(d)F\\}$, where
$<d>=dU_{g}$. For the next proposition, we assume that $g=2$.

It is worth mentioning that [2, Lemma 4.14], to our knowledge holds for scalar
valued Siegel modular forms and hence we haven’t generalised this result to
vector valued Siegel modular forms here. In the next Proposition we see how
the same result can be generalised to Siegel modular forms of higher genus
putting similar conditions connecting the level and the genus.

######  Proposition 16.

Let $F\in S_{k}^{g}(K,\chi,\psi)$ be a Siegel cusp form of level $N$ and
weight $k$. Let the level $N$ be chosen so that $({\bf Z}/N{\bf Z})^{\times}$
has an element of order $2g$. Assume that $|g-k|$ is odd. There exists
$\sigma\in H=\ker(\psi)$, such that the characteristic polynomial of $\sigma$
has distinct roots.

###### Proof.

We assume $g\geq 2$ as we already know the result for $g=1$ . Recall that we
start with an $N$ for which $({\bf Z}/N{\bf Z})^{\times}$ has an element of
order $2g$, which we call $a$. So $a^{g}=-1$. Rewriting the equation from [2,
Lemma 4.14], we get $\chi(aU_{g})^{g}a^{gk-g(g+1)}F=\psi(a)F$.

Hence, we deduce that $\chi(-I)(-1)^{k-g+1}F=\psi(a)F$. For $gk\in 2{\bf Z}$
(which is ensured when $F\neq 0$), this further gives
$(-1)^{gk+k-g+1}=(-1)^{k-g+1}=\psi(a)$. Given $g$ and $k$ are of different
parity, we are able to find a $\sigma\in H$ having $2g$ distinct roots such
that $\rho_{\psi}(\sigma)=1$ with $\rho_{\psi}$ as in [8, Pg 385]. From this
we conclude that $\sigma$ is an element such that it satisfies the equation
$X^{2g}-1=0$. This equation has $2g$ distinct roots. ∎

######  Corollary 17.

If $g=2$, we choose $N$ such that $({\bf Z}/N{\bf Z})^{\times}$ has an element
of order $4$. Assume that weight is odd. There exists $\sigma\in
H=\ker(\psi)$, such that $\sigma$ the characteristic polynomial of $\sigma$
has distinct roots.

Furthermore, we conclude that $\pi_{N}(conj)=-1=\sigma^{2}$. From [21, Lemma
1.1], we thus deduce again the following corollary.

######  Corollary 18.

We have an inclusion $\rho_{l}(H)\subset GSp_{2g}(L\otimes Q_{l})$ with
$L=K^{<\sigma>}$ and $L\subset\mathbb{R}$.

From [21, Lemma 1.1], we deduce that $\rho_{l}(H)\subset GSp_{4}(L\otimes
Q_{l})$, where $L=K^{<\sigma>}$. Now as $(conj)\in\pi_{N}^{-1}(<\sigma>)$,
$L\subset\mathbb{R}$.

For an embedding
$\sigma\mathrel{\mathop{\ordinarycolon}}K\hookrightarrow\overline{{\bf
Q}}_{l}$, let $\rho_{\sigma}$ be the map

$\rho_{\sigma}\mathrel{\mathop{\ordinarycolon}}G_{{\bf
Q}}\rightarrow{\mathrm{GSp}}_{2g}(K\otimes\overline{{\bf
Q}_{l}})\rightarrow{\mathrm{GSp}}_{2g}(\overline{{\bf Q}_{l}});$

where the first map is given by $\rho_{l}$ and the latter map is induced by
$\sigma$ entry wise by
$\sigma(k,t)\mathrel{\mathop{\ordinarycolon}}=\sigma(k)t$. Let
$\mathfrak{gsp}_{2g}({\overline{{\bf Q}}_{l,\sigma}})=\mathfrak{g}_{\sigma}$
denotes the Lie algebra of the image of $\rho_{\sigma}(G_{{\bf Q}})$, and so
similarly for $\tau$, we have $\mathfrak{gsp}_{4}(\overline{{\bf
Q}}_{l,\tau})=\mathfrak{g}_{\tau}$. Also let
$\mathfrak{g}_{l}^{\prime}\mathrel{\mathop{\ordinarycolon}}=\\{(u,u^{\prime})\in\mathfrak{g}_{\sigma}\times\mathfrak{g}_{\tau}\\}$.
We show that $\mathfrak{g_{l}^{\prime}}$ is surjective onto each of its
components.

We state the following Proposition that we use in the proof of the main
theorem. The idea of this proof essentially follows from [15, Step 1, Pg 790].

######  Proposition 19.

Given $\mathfrak{g}_{l}^{\prime}$ as above, the projective maps
$p_{\sigma}\mathrel{\mathop{\ordinarycolon}}\mathfrak{g}_{l}^{\prime}\rightarrow\mathfrak{g}_{\sigma}$
and
$p_{\tau}\mathrel{\mathop{\ordinarycolon}}\mathfrak{g}_{l}^{\prime}\rightarrow\mathfrak{g}_{\tau}$
are surjective.

###### Proof.

Since the $\lambda$-adic Galois representation associated to Siegel modular
form is irreducible, we have
$End_{\mathfrak{g}_{\sigma}}(V_{\sigma})=\overline{{\bf Q}_{l}}$. Also
${\mathfrak{g}}_{\sigma}$ is reductive with a centre that is diagonalisable as
$G_{\bf Q}$ acts semi simply on $V$. It follows that
$\mathfrak{g}_{\sigma}\subset\mathfrak{gsp}_{2g}(V_{\sigma})$. Now
$\mathfrak{gsp}_{2g}(V_{\sigma})=\mathfrak{sp}_{2g}(V_{\sigma})\oplus\overline{{\bf
Q}_{l}}^{\times}$. Now as $\mathfrak{sp}_{2g}(V_{\sigma})$ is simple, and
${\mathfrak{g}}_{\sigma}$ is semisimple,
${\mathfrak{g}}_{\sigma}=\mathfrak{sp}_{2g}(V_{\sigma})$ or
${\mathfrak{g}}_{\sigma}=\mathfrak{gsp}_{2g}(V_{\sigma})$. The former is
impossible as $\chi(Frob_{l})\neq 1$ and hence
${\mathfrak{g}}_{\sigma}=\mathfrak{gsp}_{2g}(V_{\sigma})$. ∎

Goursat’s lemma states that if $G_{1}$ and $G_{2}$ are groups such that $H$ is
a subgroup of $G_{1}\times G_{2}$, such that the two projections
$p_{1}\mathrel{\mathop{\ordinarycolon}}H\rightarrow G_{1}$ and
$p_{2}\mathrel{\mathop{\ordinarycolon}}H\rightarrow G_{2}$ are surjective. If
$N_{1}$ is the kernel of $p_{2}$ and $N_{2}$, the kernel of $p_{1}$, then $H$
is the graph of an isomorphism from $G_{1}/N_{1}\cong G_{2}/N_{2}$.

We now prove a lemma about the kernels of the Goursat’s lemma in our context.
For each embedding
$\sigma\mathrel{\mathop{\ordinarycolon}}E\rightarrow\overline{{\bf Q}_{l}}$,
define $G_{\sigma}={\mathrm{GSp}}_{2g}(V_{\sigma})$ and
$G_{l}\mathrel{\mathop{\ordinarycolon}}=\prod_{\sigma\mathrel{\mathop{\ordinarycolon}}E\rightarrow\overline{{\bf
Q}_{l}}}{\mathrm{GSp}}_{2g}(V_{\tau})$. Let
$J=\\{(u,u^{\prime})\in{\mathrm{GSp}}_{2g}(V_{\sigma})\times{\mathrm{GSp}}_{2g}(V_{\tau})\\}$
where $u=\rho_{\sigma}(g)$ and $u^{\prime}=\rho_{\tau}(g)$ for some $g\in
G_{{\bf Q}}$. Hence $J\subset G\times G^{\prime}$ on which we can apply
Goursat’s lemma as conditions of the same are satisfied due to Proposition19.

######  Lemma 20.

With $G_{\sigma},G_{\tau}$ and $J$ as above, and if $G_{l}\subsetneq A_{l}$,
then $J$ is the graph of an isomorphism $G_{\sigma}\rightarrow G_{\tau}$ for
some choice of $\sigma$ and $\tau$.

###### Proof.

We know that

${\mathrm{GSp}}_{2g}(V_{\sigma})={\mathrm{Sp}}_{2g}(V_{\sigma})\bigoplus\overline{{\bf
Q}_{l}}^{\times},{\mathrm{GSp}}_{2g}(V_{\tau})={\mathrm{Sp}}_{2g}(V_{\tau})\bigoplus\overline{{\bf
Q}_{l}}^{\times}.$

We show that the kernel of the homomorphism is trivial. The kernel is an
normal subgroup of ${\mathrm{Sp}}_{2g}(V_{\sigma})\bigoplus\overline{{\bf
Q}_{l}}^{\times}$. Recall ${\mathrm{Sp}}$ is a simple group. Following [13, p.
106], recall that if you resrict to certain open subgroup, the Galois
representation remains irreducible. Hence, $G_{\sigma}$ contains
${\mathrm{Sp}}_{2g}(V_{\sigma})$. However, the determinant which is an open
map. Hence the only possibilities for the second component of the kernel can
be trivial or whole of $\overline{{\bf Q}_{l}}^{\times}$. Since we have the
assumption that $G_{l}\subsetneq A_{l}$, there exists at least one embedding
$\sigma$ for which $G_{\sigma}\neq A_{\sigma}$. Hence for that $\sigma$, the
normal subgroup $N_{\sigma}=\\{\pm I\\}\times 1$. So the graph is from
$G_{\sigma}/N_{\sigma}\rightarrow G_{\tau}/N_{\tau}$ where for the sake of
isomorphism, both $N_{\sigma}$ and $N_{\tau}$ are trivial because for any
other choice of $N_{\tau}$, the isomorphism would not exist. ∎

##  4\. The Main Theorem

In this section we prove our main Theorem 1 for general $g\geq 1$. By
extension of scalars, we regard $\rho_{l}$ as the $K\otimes\overline{{\bf
Q}_{l}}$ representation,

$\rho_{l}\mathrel{\mathop{\ordinarycolon}}G_{{\bf
Q}}\rightarrow{\mathrm{GSp}}_{2g}(K\otimes{\bf
Q}_{l})\rightarrow{\mathrm{GSp}}_{2g}(K\otimes\overline{{\bf Q}_{l}})$

Changing bases if necessary we proved in Corollary 18, that there exists $H$
such that $\rho_{l}(H)\subset{\mathrm{GSp}}_{2g}(L\otimes{\bf Q}_{l})$. Here
we assume that $N\in\mathbb{N}$ such that $({\bf Z}/N{\bf Z})^{\times}$ has an
element of order $4$. Our aim is to calculate Lie algebra of $\rho_{l}(G_{\bf
Q})$ which we have denoted as $\mathfrak{g_{l}}$. On $H$,
$\rho_{\sigma}=\rho_{\tau}\iff\sigma|_{L}=\tau|_{L}$. We list the following
properties of the $l$-adic Galois representation:

  1. (1)

$\rho_{l}$ is semisimple.

  2. (2)

$det(\rho_{l})=\chi(l)^{2k-3}$

  3. (3)

No representation $\rho_{\lambda}$ with $\lambda|l$ becomes abelian on an open
subgroup of $G_{\bf Q}$.

Because of the aforementioned conditions, $\rho_{l}$ satisfies the conditions
for [15, Theorem 4.4.10].

We now prove our main Theorem 1. The main theorem says if certain conditions
are satisfied then the Siegel modular forms contain extra twist if and only if
${\mathfrak{g}}_{l}\subsetneq\mathfrak{a_{l}}$. In other word, proving main
theorem is equivalent to showing the following statements are equivalent as in
the classical cases of elliptic modular forms:

  1. (1)

${\mathfrak{g}}_{l}\subsetneq\mathfrak{a_{l}}$

  2. (2)

$\exists~{}\sigma,\tau$ s.t. $\sigma|_{K}\neq\tau|_{K}$ but there $\exists$ an
open subset $H_{0}$ of $G_{\bf Q}$ such that
$\rho_{\sigma}\mathrel{\mathop{\ordinarycolon}}H_{0}\rightarrow{\mathrm{GSp}}_{2g}(\overline{{\bf
Q}_{l}})$ and
$\rho_{\tau}\mathrel{\mathop{\ordinarycolon}}H_{0}\rightarrow{\mathrm{GSp}}_{2g}(\overline{{\bf
Q}_{l}})$ are isomorphic.

  3. (3)

There exists a finite order character
$~{}\phi\mathrel{\mathop{\ordinarycolon}}G_{\bf Q}\rightarrow\overline{{\bf
Q}_{l}}^{\times}$ s.t. $\rho_{\tau}\cong\rho_{\sigma}\otimes\phi$.

###### Proof.

We establish $(1)\implies(2)$. We use the same argument as in [19, Lemma 7]
and modify it for our case. Recall that
$\mathfrak{g}_{l}^{\prime}=\\{(u,u^{\prime})\in\mathfrak{g}_{\sigma}\times\mathfrak{g}_{\tau}\\}$.
At Lie algebra level,
$\mathfrak{g}_{l}^{\prime}\subset\mathfrak{gsp}_{2g}(V_{\sigma})\times\mathfrak{gsp}_{2g}(V_{\tau})$.
Suppose $\mathfrak{g}_{l}\subsetneq\mathfrak{a}_{l}$, as the projections are
surjective by Proposition 19. By applying Goursat’s Lemma 3, we deduce that
$\mathfrak{g}_{l}^{\prime}$ is the graph of an isomorphism 20
$\alpha\mathrel{\mathop{\ordinarycolon}}\mathfrak{gsp}_{2g}(V_{\sigma})\rightarrow\mathfrak{gsp}_{2g}(V_{\tau})$
which takes $1$ to $1$. Note that as the projections are surjective, the graph
always exists. However only when is the inclusion proper are the kernels
trivial.

Recall that $\mathfrak{gsp}_{2g}(V)=\mathfrak{sp}_{2g}(V)\oplus\overline{{\bf
Q}_{l}}^{\times}I_{n}$. Hence, any automorphism of $\mathfrak{gsp}_{2g}(V)$ is
determined by what it does to $\mathfrak{sp}_{2g}(V)$. Hence our automorphism
is determined by its restriction to $\mathfrak{sp}_{2g}(V)$. So by [10],
$\alpha(u)=f\circ u\circ f^{-1}$ where $u\in\mathfrak{gsp}_{2g}(V_{\sigma})$
and $f\in V_{\sigma}\rightarrow V_{\tau}$. Here $f$ is an isomorphism of
$\mathfrak{g}_{l}^{\prime}$ modules.

Shifting to Lie group level, we deduce that $\overline{f}$ is an isomorphism
of $U$ modules. So if $H_{0}=\rho_{l}^{-1}(U)$ where $\overline{f}$ is an
isomorphism of $U$ modules, it satisfies the required conditions of (2).

We now prove $(3)\implies(1)$. We start by assuming that there exists a finite
order character $\phi$ for which $\rho_{\sigma}\cong\rho_{\tau}\otimes\phi$.
Hence $det(\rho_{\sigma})=det(\rho_{\tau}\otimes\phi)$. Since $\phi$ is only a
finite order character,
$det(\rho_{\sigma}\rho_{\tau}^{-1})=\phi^{2g}\subsetneq\overline{{\bf
Q}_{l}}^{\times}$. Hence
${\mathfrak{g}}_{\sigma}{\mathfrak{g}}_{\tau}^{-1}\subsetneq\mathfrak{a}_{\sigma}\mathfrak{a}_{\tau}$,
which gives us ${\mathfrak{g}}_{l}\subsetneq\mathfrak{a}_{l}$.

To prove $(3)\implies(2)$ is easy. We start with the assumption that there is
a finite order character such that $\rho_{\tau}\cong\rho_{\sigma}\otimes\phi$.
Hence the kernel of $\phi$ will be a finite index subgroup of $G_{{\bf Q}}$
which will be our $H_{0}$. So, as $\phi\equiv 1$ on $H_{0}$,
$\rho_{\tau}|_{H_{0}}\cong\rho_{\sigma}|_{H_{0}}$ and $(2)$ holds true.

We now prove $(2)\implies(3)$. For any finite index subgroup $H_{0}$ of
$G_{\bf Q}$, the image $\rho_{\sigma}(H_{0})$ has commutant consisting of
scalar matrices in ${\mathrm{GSp}}_{2g}(\overline{{\bf Q}_{l}})$ by Schur’s
lemma. The same is also true for the representation $\rho_{\tau}$. Hence there
exists $\phi\mathrel{\mathop{\ordinarycolon}}G_{\bf
Q}\rightarrow\overline{{\bf Q}_{l}}^{\times}$ of finite order as $H_{0}$ being
open is a finite index subgroup of $G_{\bf Q}$ such that
$\rho_{\sigma}\cong\rho_{\tau}\otimes\phi$.

∎

##  5\. Examples of Siegel modular forms with extra twists

###  5.1. Yoshida lifts of classical modular Forms with characters

We briefly recall the theory of Yoshida lifts [17]. This is basically coming
from the embedding
${\mathrm{GL}}_{2}\times{\mathrm{GL}}_{2}\xhookrightarrow{}{\mathrm{GSp}}_{4}$.
Given two classical elliptic, weight $2$ modular forms $f$ and $g$ of level
$N_{1},N_{2}$ respectively (two automorphic representations for
${\mathrm{GL}}_{2}$) associated to the same primitive character $\chi$, the
Yoshida lift $F$ is the automorphic representation for ${\mathrm{GSp}}_{4}$.
Recall that for the pair $(f,g)$, automorphic representation
$F\mathrel{\mathop{\ordinarycolon}}=Y(f\otimes g)$ is the Yoshida lift if the
following conditions are satisfied:

  1. (1)

The adelization of $F$ generates an irreducible cuspidal automorphic
represenation $\pi_{F}$ of ${\mathrm{GSp}}_{4}(\mathbb{A})$.

  2. (2)

The local $L-$ parameter for $\pi_{F,v}$ at each place $v$ is the direct sum
of the $L$-parameters for $\pi_{f,v}$ and $\pi_{g,v}$.

The Yoshida lift is a special case of Langlands functoriality coming from the
embedding of dual groups

$\\{(g_{1},g_{2})\in{\mathrm{GL}}_{2}({\bf C})\times{\mathrm{GL}}_{2}({\bf
C})|\det(g_{1})=\det(g_{2}))\\}\rightarrow{\mathrm{GSp}}_{4}({\bf C})$

given by

$(\begin{bmatrix}a&b\\\ c&d\end{bmatrix}$,
$\begin{bmatrix}a^{\prime}&b^{\prime}\\\ c^{\prime}&d^{\prime}\end{bmatrix})$
$\rightarrow$ $\begin{bmatrix}a&0&b&0\\\ 0&a^{\prime}&0&b^{\prime}\\\
c&0&d&0\\\ 0&c^{\prime}&0&d^{\prime}\end{bmatrix}$.

Note that the determinant condition of the embedding is satisfied since in
both cases the determinant is equal to $\chi(p)p^{k-1}$ as the elliptic
modular forms are associated to the same character $\chi$. We now define
Yoshida lifts of two given classical modular forms associated to the same
character.

Following Roberts [16], we list below the necessary conditions for the
existence of the Yoshida lifts. Given two classical newforms $f$ and $g$, we
say the pair satisfies the conditions of a Yoshida lift if

  1. (1)

The modular form $f$ is not a scalar multiple of $g$

  2. (2)

The characters of $f$ and g arise from the same primitive Dirichlet character

  3. (3)

One of the weights is $2$ and the other weight has to be an even integer
greater than $2$.

  4. (4)

There exists a finite prime $p$ at which $\pi_{f,p}$ and $\pi_{g,p}$ are both
discrete series.

######  Definition 21.

Suppose these conditions are satisfied and $f,g\in S^{1}_{2}(N,\chi)$, there
is a unique representation $\Pi_{F,p}$ of ${\mathrm{Gsp}}_{4}({\bf Q}_{p})$
satisfying

$L(\Pi_{F,p})=L(\pi_{f,p})\oplus L(\pi_{g,p}).$

In this case $F$ is said to be a Yoshida lift of $f$ and $g$.

If $f,g$ are two classical modular forms with coefficient fields $K_{1},K_{2}$
and $\Gamma_{1}$ and $\Gamma_{2}$ are the respective group of extra twists,
then we denote their Yoshida lift by $Y(f\otimes g)$. We also denote the group
of extra twists for the Yoshida lift as $\Gamma_{Y}$

######  Lemma 22.

If two classical modular forms $f$ and $g$ satisfy the conditions listed in
5.1, then $\Gamma_{Y}\supset\Gamma_{1}\cap\Gamma_{2}$.

###### Proof.

Suppose $\gamma\in\Gamma_{1}\cap\Gamma_{2}$. Now by definition this means
$K_{1}$ and $K_{2}$ have to be equal for a non zero $\gamma$ to exist in the
intersection. Hence their compositum is $K=K_{1}=K_{2}$. This implies that
$\gamma(\rho_{f})=\rho_{f}\otimes\chi_{\gamma}$ and
$\gamma(\rho_{g})=\rho_{g}\otimes\chi_{\gamma}$. Because of the equation 21,
we also directly have that, for the Yoshida lift $F=Y(f\otimes g)$, we have
$\gamma(\rho_{F})=\rho_{F}\otimes\chi_{\gamma}$. ∎

In our subsequent example of Siegel modular form with extra twist, we show
that it is not necessarily true that we have
$\Gamma=\Gamma_{1}\cap\Gamma_{2}$. In fact, all the examples we have
$\Gamma_{Y}\supsetneq\Gamma_{1}\cap\Gamma_{2}$.

In this section we give an explicit examples of Siegel modular forms with
extra twists. We find examples of Siegel modular forms with extra twists by
taking the Yoshida lifts of two classical modular forms with appropriate,
weight, character and having extra twists themselves.

###  5.2. Proof of Theorem 2

In this subsection, we prove that the group of extra twists for Siegel modular
forms can be very large.

###### Proof.

Let $f,g\in S_{k}(N,\chi)$ be two elliptic modular forms with extra twists
different from complex conjugation. Let $K=K_{f}\cdot K_{g}$ be the compositum
of the Hecke fields $K_{f},K_{g}$ of $f$ and $g$. There exists at least one
$\gamma\in Aut(K)$ such that

(3)
$\displaystyle\gamma(\rho_{f})=\rho_{f}\otimes\chi_{\gamma},\gamma(\rho_{g})=\rho_{g}\otimes\chi_{\gamma}.$

Recall that we assume the necessary condition for the existence of Yoshida
lift of $(f,g)$ is satisfied and let $F=Y(f\otimes g)$ be the Yoshida lift for
the pair $(f,g)$ as defined in § 5.1. We claim that $F$ is a Siegel modular
form with extra twist by same $\gamma$. In other words, we need to show that
there exists a character $\chi_{\gamma}$ such that

(4) $\displaystyle\gamma(\rho_{F})=\rho_{F}\otimes\chi_{\gamma}.$

This is an equality of two $4$ dimensional Galois representation. By Brauer-
Nesbitt theorem, this is only true if all the coefficients in the
characteristic polynomial are equal. The characteristic polynomial of the left
hand side in equation 4 is
$(x^{2}-\gamma(a_{p})x+p^{k-1})(x^{2}-\gamma(a_{p}^{\prime})+p^{k-1})$ and
that of right hand side is
$(x^{2}-a_{p}\chi_{\gamma}(p)x+p^{k-1})(x^{2}-a_{p}^{\prime}\chi_{\gamma}(p)x+p^{k-1})$.
The above equality follows from equation 3. ∎

We expect that the endomorphism algebra of the conjectural motive associated
to $Y(f\otimes g)$ is a direct sum of the endomorphism algebras for $f$ and
$g$.

###  5.3. Examples of the phenomenon described above

In this section, we find explicit examples of Siegel modular forms with extra
twists different complex conjugations. All these examples show that the Hecke
field of Yoshida lift is smaller that the compositum of the individual
elliptic modular forms.

  1. (1)

Let $f,g\in S_{2}^{1}(30,\chi)$ be two newforms with $\chi$ is a Dirichlet
character on $\chi\mathrel{\mathop{\ordinarycolon}}({\bf Z}/{30{\bf
Z}})^{\times}\rightarrow{\bf C}^{\times}$ such that $\chi$ is determined on
its generators by $\chi(7)=-\zeta_{8}^{2}$ and $\chi(11)=-1$. Note we can do
so because this is a four dimensional space. Since the conductor of the
character is $15$, condition $(2)$ of Yoshida lifting is satisfied and we can
consider the Yoshida lifting $Y(f\otimes g)$ and this is an example of an
explicit Siegel modular form with an extra twist. In this example the
classical modular forms could be chosen with the expansion
$f=q+\zeta_{8}q^{2}+(\zeta_{8}^{3}-\zeta_{8}^{2}-1)q^{3}+\zeta_{8}^{2}q^{4}+...$
and
$g=q+\zeta_{8}^{3}q^{2}+(\zeta_{8}^{9}-\zeta_{8}^{6}-1)q^{3}+\zeta_{8}^{6}q^{4}+...$.
For a detailed study of this space, let $K_{f,g}$ be the compositum of the
coefficient fields of $f$ and $g$. It must be said that no matter what $f$ and
$g$ we choose, in any case $K_{Y(f\otimes g)}\subsetneq K_{f,g}$.

Let $\Gamma_{F}$ be the group of extra twists of $F$. Then it is readily
evident that $\Gamma_{F}\supseteq\Gamma_{1}\cap\Gamma_{2}$. For the $f$ and
$g$ we have chosen, $\Gamma_{1}=\Gamma_{2}$. Not also that $K_{f,g}={\bf
Q}(\zeta_{8})$. We see that in this expansion the coefficient of $q^{2}$ in
the sum of $f+g$ is $\zeta_{8}+\zeta_{8}^{3}$. We claim that all of the
coefficients of the sum $f+g$ belong to the field ${\bf
Q}(\zeta_{8}+\zeta_{8}^{3})$.

It is easy to check manually that for any power $1\leq i\leq 8$,
$\zeta_{8}^{i}+\zeta_{8}^{3i}$ can be written in terms of
$\zeta_{8}+\zeta_{8}^{3}$. Hence all the sum of Hecke eigenvalues of $f$ and
$g$ are in ${\bf Q}(\zeta_{8}+\zeta_{8}^{3})$. From the definition of Yoshida
lift 21, we get that the Hecke eigenvalues of $Y(f\otimes g)\in{\bf
Q}(\zeta_{8}+\zeta_{8}^{3})$. Hence this is an explicit example of when field
of Hecke eigenvalues of the Yoshida lift is a proper subset of the compositum
of the field of the Hecke eigenvalues of the concerned classical modular
forms.

  2. (2)

Consider the $8$ dimensional complex vector space $S_{2}^{1}(100,\chi)$ where
the character $\chi$ is defined by $\chi(51)=-1$ and
$\chi(77)=\frac{\mu^{6}-3\mu^{2}}{4}$; here $\mu$ is a root of the polynomial
$p(x)\mathrel{\mathop{\ordinarycolon}}=x^{8}-7x^{4}+16=0$. Such a character
$\chi$ has conductor $20$ and the order of the group of inner twists is $8$.

Let
$\mu_{1}\mathrel{\mathop{\ordinarycolon}}=\frac{7+\sqrt{15}i}{2},\mu_{2}\mathrel{\mathop{\ordinarycolon}}=-(\frac{7+\sqrt{15}i}{2})$
be the root of $p(x)$. For $i=1,2$, we have the Fourier expansions

$f_{i}=q+\mu_{i}q^{2}+\frac{3\mu_{i}^{7}-13\mu_{i}^{3}}{8}q^{3}+....$

The Hecke eigenvalue ring $K_{f_{1}f_{2}}$ is of dimension $8$ over ${\bf Q}$.
The Hecke eigenvalue ring is determined by the coefficients of $1,q$ and
$q^{2}$ itself. Hence by Definition 21, the coefficient ring,
$K_{Y(f_{1}\otimes f_{2})}$ is determined by the sum of Hecke eigenvalues of
$q$ and $q^{2}$ under the two roots. By a small calculation, we see that this
field happens to be ${\bf Q}(\sqrt{15}i)$ which is of extension degree $2$
over ${\bf Q}$. Hence $K_{f_{1}f_{2}}$ is of extension degree $4$ over
$K_{Y(f_{1}\otimes f_{2})}$. In these case also $K_{Y(f_{1}\otimes
f_{2})}\subsetneq K_{f_{1},f_{2}}$.

We give two examples only but from LFMFDB it is evident that the group of
extra twists for Yoshida lifts can be very large although it can be controlled
(as expected) from the individual classical elliptic modular forms.

## References

  * [1] Modular functions of one variable. V, Lecture Notes in Mathematics, Vol. 601, Springer-Verlag, Berlin-New York (1977). Edited by J.-P. Serre and D. B. Zagier.
  * [2] A. Andrianov, Introduction to Siegel modular forms and Dirichlet series, Universitext, Springer, New York (2009), ISBN 978-0-387-78752-7.
  * [3] T. Arakawa, _Vector-valued Siegel’s modular forms of degree two and the associated Andrianov $L$-functions_, Manuscripta Math. 44 (1983), no. 1-3, 155–185.
  * [4] S. Arias-de Reyna, L. Dieulefait, and G. Wiese, _Compatible systems of symplectic Galois representations and the inverse Galois problem I. Images of projective representations_ , Trans. Amer. Math. Soc. 369 (2017), no. 2, 887–908.
  * [5] D. Banerjee and E. Ghate, _Adjoint lifts and modular endomorphism algebras_ , Israel J. Math. 195 (2013), no. 2, 507–543.
  * [6] A. F. Brown and E. P. Ghate, _Endomorphism algebras of motives attached to elliptic modular forms_ , Ann. Inst. Fourier (Grenoble) 53 (2003), no. 6, 1615–1676.
  * [7] A. Conti, _Galois level and congruence ideal for $p$-adic families of finite slope Siegel modular forms_, Compos. Math. 155 (2019), no. 4, 776–831.
  * [8] F. Diamond and J. Shurman, A first course in modular forms, Vol. 228 of _Graduate Texts in Mathematics_ , Springer-Verlag, New York (2005), ISBN 0-387-23229-X.
  * [9] E. Ghate, E. González-Jiménez, and J. Quer, _On the Brauer class of modular endomorphism algebras_ , Int. Math. Res. Not. (2005), no. 12, 701–723.
  * [10] L.-K. Hua, _On the automorphisms of the symplectic group over any field_ , Ann. of Math. (2) 49 (1948) 739–759.
  * [11] L. Kaup and B. Kaup, Holomorphic Functions of Several Variables, De Gruyter, Berlin, New York (1983), ISBN 9783110838350.
  * [12] A. Kumar, M. Kumari, and A. Weiss, _On the Lang–Trotter conjecture for Siegel modular forms_ , arXiv preprint arXiv:2201.09278 (2022).
  * [13] F. Momose, _On the $l$-adic representations attached to modular forms_, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 1, 89–109.
  * [14] A. Pitale, Siegel modular forms, Vol. 2240 of _Lecture Notes in Mathematics_ , Springer, Cham (2019), ISBN 978-3-030-15674-9; 978-3-030-15675-6. A classical and representation-theoretic approach.
  * [15] K. A. Ribet, _Galois action on division points of Abelian varieties with real multiplications_ , Amer. J. Math. 98 (1976), no. 3, 751–804.
  * [16] B. Roberts, _Global $L$-packets for ${\rm GSp}(2)$ and theta lifts_, Doc. Math. 6 (2001) 247–314.
  * [17] A. Saha, _On ratios of Petersson norms for Yoshida lifts_ , Forum Math. 27 (2015), no. 4, 2361–2412.
  * [18] P. Scholze, _On torsion in the cohomology of locally symmetric varieties_ , Ann. of Math. (2) 182 (2015), no. 3, 945–1066.
  * [19] J.-P. Serre, _Propriétés galoisiennes des points d’ordre fini des courbes elliptiques_ , Invent. Math. 15 (1972), no. 4, 259–331.
  * [20] ———, Abelian $l$-adic representations and elliptic curves, Vol. 7 of _Research Notes in Mathematics_ , A K Peters, Ltd., Wellesley, MA (1998), ISBN 1-56881-077-6. With the collaboration of Willem Kuyk and John Labute, Revised reprint of the 1968 original.
  * [21] J.-L. Waldspurger, _Quelques propriétés arithmétiques de certaines formes automorphes sur ${\rm GL}(2)$_, Compositio Math. 54 (1985), no. 2, 121–171.