# Period of the Ikeda-Miyawaki lift

Tomoyoshi Ibukiyama  Tomoyoshi Ibukiyama, Department of Mathematics, Graduate
School of Science, Osaka University, Machikaneyama 1-1, Toyonaka, Osaka,
560-0043 Japan<EMAIL_ADDRESS>,  Hidenori Katsurada
Hidenori Katsurada, Department of Mathematics, Hokkaido University, Kita 10,
Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan and Muroran Institute of
Technology, Mizumoto 27-1, Muroran, Hokkaido, 050-8585 Japan
<EMAIL_ADDRESS>and  Hisashi Kojima  Hisashi Kojima Department
of Mathematics, Graduate School of Science and Engineering, Saitama
University, Shimo-Okubo 255, Sakura-ku, Saitama City, 338-8570, Japan
<EMAIL_ADDRESS>
###### Abstract.

In this paper, first we give a weak version of Ikeda’s conjecture on the
period of the Ikeda-Miyawaki lift. Next, we confirm this conjecture rigorously
in some cases.

###### Key words and phrases:

Ikeda-Miyawaki lift, Period

###### 2020 Mathematics Subject Classification:

11F46, 11F67

The first author was partially supported by JSPS KAKENHI Grant Number JP
JP19K03424, JP20H00115, JP23K03031. The second author was partially supported
by JSPS KAKENHI Grant Number JP2103152

##  1\. Introduction

In this paper, we investigate Ikeda’s conjecture on the period of the Ikeda-
Miyawaki lift. Let $k,n$ be positive integers such that $k\equiv n+1\mod 2$
and $k>n$. For a Hecke eigenform $h$ in the Kohnen plus subspace of weight
$k+1/2$ for $\varGamma_{0}(4)$, and a primitive form $g$ of weight $k+n+1$ for
$SL_{2}(\mathbb{Z})$, Ikeda [17] constructed a cusp form $\mathcal{F}_{h,g}$
of weight $k+n+1$ for $Sp_{2n+1}(\mathbb{Z})$ with the following property:

If $\mathcal{F}_{h,g}$ is not identically $0$, it is a Hecke eigenform and its
standard $L$-function $L(s,\mathcal{F}_{h,g},\mathrm{St})$ is expressed as

$L(s,\mathcal{F}_{h,g},\mathrm{St})=L(s,g,\mathrm{St})\prod_{i=1}^{n}L(s+k+n-i,f),$

where $f$ is the primitive form of weight $2k$ for $SL_{2}(\mathbb{Z})$
corresponding to $h$ via the Shimura correspondence, and $L(s,f)$ is Hecke’s
$L$-function of $f$. The existence of such a Hecke eigenform was conjectured
by Miyawaki [29]. Therefore we call $\mathcal{F}_{h,g}$ the Ikeda-Miyawaki
lift of $h$ and $g$. To ensure the non-vanishing of $\mathcal{F}_{h,g}$, Ikeda
[17] proposed the following conjecture concerning the period (or the Petersson
norm ) of $\mathcal{F}_{h,g}$:

$2^{(2n-1)k+4n}\dfrac{\displaystyle\langle{\mathcal{F}}_{h,g},{\mathcal{F}}_{h,g}\rangle}{\displaystyle\langle
g,g\rangle\langle h,h\rangle}=\displaystyle\Lambda(k+n,\mathrm{St}(g)\otimes
f)\prod_{i=1}^{n}\tilde{\xi}(2i)\prod_{i=1}^{n-1}\tilde{\Lambda}(2i+1,f,\mathrm{St})$

(cf. Conjecture 3.1). As for the definitions of several $L$-values in
question, see (2.1), (2.4), (2.5). Ikeda also confirmed the conjecture
approximately in some cases. But, as far as we know, there is no examples that
confirms this conjecture rigorously.

In this paper, first we deduce a weak version of the above conjecture. More
precisely, put

$C_{h,g}=\dfrac{\displaystyle\langle g,g\rangle\langle
h,h\rangle}{\displaystyle\langle{\mathcal{F}}_{h,g},{\mathcal{F}}_{h,g}\rangle}\displaystyle\Lambda(k+n,\mathrm{St}(g)\otimes
f)\prod_{i=1}^{n}\tilde{\xi}(2i)\prod_{i=1}^{n-1}\tilde{\Lambda}(2i+1,f,\mathrm{St}).$

Then we prove that $C_{h,g}$ belongs to $\mathbb{Q}(f)\mathbb{Q}(g)$, where
$\mathbb{Q}(f)$ and $\mathbb{Q}(g)$ are the Hecke fields of $\mathbb{Q}(f)$
and $\mathbb{Q}(g)$, respectively (cf. Theorem 3.6). As a result we prove the
algebraicity of
$\dfrac{\displaystyle\langle{\mathcal{F}}_{h,g},{\mathcal{F}}_{h,g}\rangle}{\displaystyle\langle
g,\ g\rangle^{2}\langle f,\ f\rangle^{n-1}\langle h,h\rangle\Omega_{-}(f)}$,
where $\Omega_{-}(f)$ is Manin-Shimura period for $f$, which will be defined
in Section 1 (cf. Theorem 3.8). Our proof relies on the algebraicities of
critical values of the product of the Hecke $L$-functions and the triple
product $L$-functions of elliptic cusp forms and the standard $L$-functions of
Siegel cusp forms (cf. [7],[30],[31]).

Next, we confirm Ikeda’s conjecture rigorously in some cases (cf. Theorems 6.4
and 6.8). The idea is to compute the special values of $L$-functions stated
above rigorously. We have an algorithm for computing the special values of the
triple product $L$-functions of elliptic cusp forms (cf. [13, Theorem 4.9]).
We also have an algorithm for computing the special vales of the product of
the Hecke $L$-functions elliptic cusp forms (cf. Proposition 5.1). One of key
ingredients in the proof of main examples is to compute the standard
$L$-functions of Siegel cusp forms for $Sp_{3}(\mathbb{Z})$ using the pullback
formula of the Siegel Eisenstein series. This method has been implicitly used
in [13, Section 5]. However, there we used the pull-back formula without
differential operator. In this paper, we use the pullback formula with
differential operator (cf. Theorem 4.8), which allows us to compute the values
in question more efficiently (cf. Remark 4.9).

This paper is organized as follows. In Section 2, we introduce several
$L$-functions and review their algebraicity. In Section 3, we review the
Ikeda-Miyawaki lift and Ikeda’s conjecture on its period, and prove a weaker
version of Ikeda’s conjecture. In Section 4, we review the pullback of the
Siegel-Eisenstein series. Moreover, we give an algorithm for computing the
special value of the standard $L$-function $L(s,F,\mathrm{,}{St})$ at $s=k-3$
when $F$ is a Hecke eigenform of weight $k+2$ for $Sp_{3}(\mathbb{Z})$. In
Section 5, we give an algorithm for computing the special value of the product
of Hecke $L$-functions of primitive forms for $SL_{2}(\mathbb{Z})$. Finally,
in Section 6, we confirm Ikeda’s conjecture in some cases using the results in
Sections 5, 6, and [13, Theorem 4.9].

We thank Cris Poor and David Yuen for valuable comments.

Notation. We denote by ${\mathbb{Z}},$ the ring of rational integers, and by
${\mathbb{Q}},{\mathbb{R}},$ and ${\mathbb{C}}$ the fields of rational
numbers, real numbers, and complex numbers, respectively. Moreover for a prime
number $p,$ we denote by ${\mathbb{Q}}_{p}$ and ${\mathbb{Z}}_{p}$ the field
of $p$-adic numbers and the ring of $p$-adic integers, respectively. We also
denote by $\mathrm{ord}_{p}$ the normalized additive valuation on
${\mathbb{Q}}_{p}$ and write ${\bf e}(z)=e^{2\pi iz}$ for $z\in{\bf C}$.

For a commutative ring $R$, we denote by $M_{mn}(R)$ the set of
$(m,n)$-matrices with entries in $R.$ In particular put $M_{n}(R)=M_{nn}(R).$
For an $(m,n)$-matrix $X$ and an $(m,m)$-matrix $A$, we write
$A[X]={}^{t}XAX,$ where ${}^{t}X$ denotes the transpose of $X$. Put
$\operatorname{GL}_{m}(R)=\\{A\in M_{m}(R)\ |\ \det A\in R^{\times}\\}$, where
$\det A$ denotes the determinant of the square matrix $A$, and $R^{\times}$
denotes the unit group of $R.$ Let $S_{n}(R)$ denote the set of symmetric
matrices of degree $n$ with entries in $R.$ Furthermore, for an integral
domain $R$ of characteristic different from $2,$ let ${\mathcal{H}}_{n}(R)$
denote the set of half-integral matrices of degree $n$ over $R,$ that is,
${\mathcal{H}}_{n}(R)$ is the set of symmetric matrices of degree $n$ whose
$(i,j)$-component belongs to $R$ or $\frac{1}{2}R$ according as $i=j$ or not.
For a subset $S$ of $M_{n}(R)$ we denote by $S^{\mathrm{nd}}$ the subset of
$S$ consisting of non-degenerate matrices. In particular, if $S$ is a subset
of $S_{n}({\mathbb{R}})$ with ${\mathbb{R}}$ the field of real numbers, we
denote by $S_{>0}$ (resp. $S_{\geq 0}$) the subset of $S$ consisting of
positive definite (resp. semi-positive definite) matrices. Let $R^{\prime}$ be
a subring of $R.$ Two symmetric matrices $A$ and $A^{\prime}$ with entries in
$R$ are called equivalent over $R^{\prime}$ and we write
$A\stackrel{{\scriptstyle\sim}}{{{}_{R^{\prime}}}}A^{\prime}$ if there is an
element $X$ of $\operatorname{GL}_{n}(R^{\prime})$ such that
$A^{\prime}=A[X].$ We also write $A\sim A^{\prime}$ if there is no fear of
confusion. For square matrices $X$ and $Y$ we write $X\bot
Y=\left(\begin{array}[]{cc}X&0\\\ 0&Y\end{array}\right).$ For an $m\times n$
matrix, $B=(b_{ij})$ and sequences ${\bf i}=(i_{1},\ldots,i_{r})$ and ${\bf
j}=(j_{1},\ldots,j_{r})$ of integers such that $1\leq i_{1},\ldots,i_{r}\leq
m,1\leq j_{1},\ldots,j_{r}\leq n$, we put

$B\begin{pmatrix}{\bf i}\\\ {\bf j}\end{pmatrix}=(b_{i_{k},j_{l}})_{1\leq
k,l\leq r}.$

Let $T=(t_{ij})_{1\leq i,j\leq m}$ be a symmetric matrix of variables. We say
that $P(T)$ is a polynomial in $T$ if it is a polynomial in $t_{ij}\ (1\leq
i\leq j\leq m)$.

##  2\. Several L-values

Put $J_{n}=\left(\begin{array}[]{cc}0_{n}&-1_{n}\\\ 1_{n}&\ \
0_{n}\end{array}\right),$ where $1_{n}$ denotes the unit matrix of degree $n.$
For a subring $K$ of ${\mathbb{R}}$ put

$\operatorname{GSp}_{n}^{+}(K)=\\{M\in\operatorname{GL}_{2n}(K)\ |\
J_{n}[M]=\mu(M)J_{n}\text{ for some }\mu(M)>0\\},$

and

$\operatorname{Sp}_{n}(K)=\\{M\in\operatorname{GL}_{2n}(K)\ |\
J_{n}[M]=J_{n}\\}.$

Define the standard subgroup $\varGamma_{0}^{(n)}(N)$ by

$\varGamma_{0}^{(n)}(N)=\\{\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)\in\operatorname{Sp}_{n}({\mathbb{Z}})\ |\ C\equiv 0\mod
N\\}.$

If $n=1,$ we drop the superscript $(n).$ Let ${\mathbb{H}}_{n}$ be Siegel’s
upper half-space of degree $n$. For a function $F$ on ${\mathbb{H}}_{n},$ an
integer $k$ and $g=\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)\in\operatorname{GSp}_{n}^{+}({\mathbb{R}}),$ put
$j(g,Z)=\det(CZ+D),$ and

$\left(F|_{k}g\right)(Z)=(\det(g))^{k/2}j(g,Z)^{-k}F\left((AZ+B)(CZ+D)^{-1}\right).$

Let $k$ be a positive integer. We denote by
$M_{k}(\operatorname{Sp}_{n}(\mathbb{Z}))$ or
$M_{k}^{\infty}(\operatorname{Sp}_{n}(\mathbb{Z}))$ the space of holomorphic
or $C^{\infty}$-modular forms, respectively, of weight $k$ for
$\operatorname{Sp}_{n}(\mathbb{Z}).$ We denote by
$S_{k}(\operatorname{Sp}_{n}(\mathbb{Z}))$ the vector subspace of
$M_{k}(\operatorname{Sp}_{n}(\mathbb{Z}))$ consisting of cusp forms. A
holomorphic modular form $F(Z)$ for $\operatorname{Sp}_{n}(\mathbb{Z})$ has
the following Fourier expansion:

$F(Z)=\sum_{A\in\mathcal{H}_{n}(\mathbb{Z})_{\geq 0}}c_{F}(A){\bf
e}(\mathrm{tr}(AZ)),$

and, in particular, a cusp form has the following Fourier expansion:

$F(Z)=\sum_{A\in\mathcal{H}_{n}(\mathbb{Z})_{>0}}c_{F}(A){\bf
e}(\mathrm{tr}(AZ)),$

where $\mathrm{tr}$ denotes the trace of a matrix. Let $F$ and $G$ be elements
of $M_{k}(\operatorname{Sp}_{n}(\mathbb{Z}))$ and assume that either one of
$F$ and $G$ is a cusp form. We then define the Petersson scalar product
$\langle F,G\rangle$ by

$\langle
F,G\rangle=\int_{\operatorname{Sp}_{n}(\mathbb{Z})\backslash\mathbb{H}_{n}}F(Z)\overline{G(Z)}(\det
Y)^{k-n-1}dXdY\qquad(Z=X+\sqrt{-1}Y).$

Let $\tilde{\bf
L}_{n}={\mathcal{R}}_{{\mathbb{Q}}}(\operatorname{GSp}_{n}^{+}({\mathbb{Q}}),\operatorname{Sp}_{n}({\mathbb{Z}}))$,
${\bf
L}_{n}^{\circ}={\mathcal{R}}_{{\mathbb{Q}}}(\operatorname{Sp}_{n}({\mathbb{Q}}),\operatorname{Sp}_{n}({\mathbb{Z}}))$
be the Hecke rings over ${\mathbb{Q}}$ for the Hecke pairs
$(\operatorname{GSp}_{n}^{+}({\mathbb{Q}}),\operatorname{Sp}_{n}({\mathbb{Z}}))$,
$(\operatorname{Sp}_{n}({\mathbb{Q}}),\operatorname{Sp}_{n}({\mathbb{Z}}))$,
respectively. The Hecke ring ${\bf L}_{n}^{\circ}$ is a subring of $\tilde{\bf
L}_{n}.$ Let $T=\operatorname{Sp}_{n}({{\bf Z}})M\operatorname{Sp}_{n}({{\bf
Z}})\in\operatorname{GSp}_{n}^{+}({\mathbb{Q}})$ be an element of $\tilde{\bf
L}_{n}.$ Write $T$ as a disjoint union
$T=\bigcup_{g}\operatorname{Sp}_{n}({\mathbb{Z}})g$. For $k\in{\bf N}$ and for
$F\in M_{k}(\operatorname{Sp}_{n}({\mathbb{Z}}))$ we define the Hecke operator
$F|_{k}T$ as

$F|_{k}T=(\det g)^{k/2-(n+1)/2}\sum_{g}F|_{k}g.$

We define the elements $T(p)$ and $T_{j}(p^{2})$ of $\tilde{\bf L}_{n}$ in a
standard way (cf. [2]): $T(p)=\operatorname{Sp}_{n}({\bf Z})(1_{n}\bot
p1_{n})\operatorname{Sp}_{n}({\bf Z})$ and
$T_{j}(p^{2}){=}\operatorname{Sp}_{n}({\bf Z})(1_{n-j}\bot p1_{j}\bot
p^{2}1_{n-j}\bot p1_{j})\operatorname{Sp}_{n}({\bf Z}).$ For any subring of
$R\subseteq\tilde{\bf L}_{n}$, we say that a modular form $F\in
M_{k}(\operatorname{Sp}_{n}({\mathbb{Z}}))$ is a Hecke eigenform with respect
to $R$ if $F$ is a common eigenfunction of all $T\in R.$ If $R=\tilde{\bf
L}_{n},$ we simply say that $F$ is a Hecke eigenform.

######  Remark 2.1.

Let $k\geq n+1$. Then, by the multiplicity one property for
$S_{k}(\operatorname{Sp}_{n}({\mathbb{Z}}))$ (cf. [3],[8]), a Hecke eigenform
with respect to ${\bf L}_{n}^{\circ}$ is a Hecke eigenform also with respect
to $\tilde{\bf L}_{n}$.

In this case, we denote by ${\mathbb{Q}}(F)$ the field generated over
${\mathbb{Q}}$ by all the Hecke eigenvalues of $T\in\tilde{\bf L}_{n},$ and
call it the Hecke field of $F.$ In this section we review several L-values of
modular forms that appear in this article. Let

$f(z)=\sum_{m=1}^{\infty}c_{f}(m){\bf e}(mz)$

be a primitive form in $S_{k}(\operatorname{SL}_{2}({\mathbb{Z}})).$ For each
prime $p$ let $\alpha_{p}=\alpha_{f,p}$ be a complex number such that
$\alpha_{p}+\alpha_{p}^{-1}=p^{-k/2+1/2}c_{f}(p).$ We define

$\displaystyle
L(s,f)=\prod_{p}\left((1-\alpha_{p}p^{k/2-1/2-s})(1-\alpha_{p}^{-1}p^{k/2-1/2-s})\right)^{-1}.$

We write $\Gamma_{\mathbb{C}}(s)=2(2\pi)^{-s}\Gamma(s)$ and write
$\Gamma_{\mathbb{R}}(s)=\pi^{-s/2}\Gamma(s/2)$ as usual. Put

(2.1) $\widetilde{\xi}(s)=\Gamma_{\mathbb{C}}(s)\zeta(s)$

with $\zeta(s)$ is Riemann’s zeta function. For an integer $l$ satisfying
$1\leq l\leq k-1$, put

$\Lambda(l,f)=\Gamma_{\mathbb{C}}(l)L(l,f).$

Then there exist two (positive) real numbers ${\Omega}_{+}(f)$ and
${\Omega}_{-}(f)$ such that we have (cf. [34])

(2.2)
${\dfrac{\Lambda(l,f,\chi)}{\tau(\chi){\Omega}_{j(l,\chi)}(f)}}\in{\mathbb{Q}}(f)$

for any integer $1\leq l\leq k-1$ and a primitive character $\chi$, where
$j=j(l,\chi)=+$ or $-$ according as $\chi(-1)(-1)^{l}=1$ or $-1$, and
$\tau(\chi)$ is the Gauss sum of $\chi$. We note that ${\Omega}_{+}(f)$ and
${\Omega}_{-}(f)$ are determined by $f$ only up to constant multiples of
${\mathbb{Q}}(f)^{\times}.$ Fixing ${\Omega}_{\pm}(f)$, we define an algebraic
part of $L(l,f,\chi)$ by

$L_{alg}(l,f,\chi)=\frac{\Lambda(l,f,\chi)}{\tau(\chi){\Omega}_{j}(f)}.$

If $\chi$ is the principal character, we simply denote it by $L_{alg}(l,f)$.
For two positive integers $l_{1},l_{2}\leq k-1$ and Dirichlet characters
$\chi_{1},\chi_{2}$ such that $(\chi_{1}(-1)\chi_{2}(-1))=^{l_{1}+l_{2}+1}$,
we define $L_{alg}(l_{1},l_{2};f;\chi_{1},\chi_{2})$ as

(2.3)
$L_{alg}(l_{1},l_{2};f;\chi_{1},\chi_{2})=\dfrac{\Gamma_{\mathbb{C}}(l_{1})\Gamma_{\mathbb{C}}(l_{2})L(l_{1},f,\chi_{1})L(l_{2},f,\chi_{2})}{\sqrt{-1}^{l_{1}+l_{2}+1}\langle
f,f\rangle\tau((\chi_{1}\chi_{2})_{0})},$

where $(\chi_{1}\chi_{2})_{0}$ is the primitive character associated with
$\chi_{1}\chi_{2}$. Then it is known that
$L_{alg}(l_{1}.l_{2},f;\chi_{1},\chi_{2})$ belongs to
${\mathbb{Q}}(f)(\chi_{1},\chi_{2})$, where
${\mathbb{Q}}(f)(\chi_{1},\chi_{2})$ is the field generated over
$\mathbb{Q}(f)$ by all the values of $\chi_{1}$ and $\chi_{2}$ (cf. [33],
[36]). If $\chi_{1}$ and $\chi_{2}$ are principal characters, we simply denote
it by $L_{alg}(l_{1},l_{2};f)$. Let $F$ be a Hecke eigenform in
$S_{k}(\operatorname{Sp}_{n}({\mathbb{Z}}))$ with respect to ${\bf
L}_{n}^{\circ},$ and let the $p$-Satake parameters of $F$ be
$\beta_{1}(p),\cdots,\beta_{n}(p)$. We then define the standard $L$-function
by

$L(s,F,\mathrm{St})=\prod_{p}\\{(1-p^{-s})\prod_{i=1}^{n}(1-\beta_{i}(p)p^{-s})(1-\beta_{i}(p)^{-1}p^{-s})\\}^{-1}.$

We put

(2.4)
$\widetilde{\Lambda}(s,F,\mathrm{St})=\Gamma_{\mathbb{C}}(s)\left(\prod_{i=1}^{n}\Gamma_{\mathbb{C}}(s+k-i)\right)L(s,F,\mathrm{St}).$

Let $\rho(n)=3$ if $n\equiv 1\mod 4$ and $n\geq 5$, and $\rho(n)=1$ otherwise.
For a positive integer $m$ put

$L_{alg}(m,F,\mathrm{St})={\dfrac{\displaystyle\widetilde{\Lambda}(m,F,\mathrm{St})}{\displaystyle\langle
F,F\rangle}}.$

It is known that $L_{alg}(m,F,\mathrm{St})$ belongs to ${\mathbb{Q}}(F)$ if
$\rho(n)\leq m\leq k-n$ and $m\equiv n\mod 2$ and all the Fourier coefficients
of $F$ belong to ${\mathbb{Q}}(F)$ (cf. [5], [30]). When $F$ is a Ikeda-
Miyawaki lift, which we shall explain in Remark 3.7, the algebraicity holds
also for $m=1$. For general $F$, it is expected that
$L_{alg}(1,F,\mathrm{St})\in{\mathbb{Q}}(F)$ even when $n\equiv 1\mod 4$ and
$n\geq 5,$ but this has not been proved in general.

Let $f_{i}(z)$ be a primitive Hecke eigenform in
$S_{k_{i}}(\operatorname{SL}_{2}({\mathbb{Z}}))$ for $i=1,2,3.$ Then we define
the triple product $L$-function $\displaystyle L(s,f_{1}\otimes f_{2}\otimes
f_{3})$ as

$\displaystyle L(s,f_{1}\otimes f_{2}\otimes
f_{3})=\prod_{p}\prod_{i,j,l\in\\{1,-1\\}}\left(1-\alpha_{f_{1},p}^{i}\alpha_{f_{2},p}^{j}\alpha_{f_{3},p}^{l}\,p^{(k_{1}+k_{2}+k_{3}-3)/2-s}\right)^{-1}.$

This satisfies a functional equation for $s\rightarrow k_{1}+k_{2}+k_{3}-2-s$.
Note that $L(s,f_{1}\otimes f_{2}\otimes f_{3})$ is symmetric in the $f_{i}$,
so that we may assume that $k_{1}\geq k_{2}\geq k_{3}$ without loss of
generality. In addition, we always assume that $k_{2}+k_{3}>k_{1}$ hereafter
in this article; this case is called the balanced case. Then there is an
integer $l$ satisfying $(k_{1}+k_{2}+k_{3})/2-1\leq l\leq k_{2}+k_{3}-2$, and
for such $l$, we put

$\displaystyle L_{alg}(l,f_{1}\otimes f_{2}\otimes f_{3})=$
$\displaystyle{\dfrac{L(l,f_{1}\otimes f_{2}\otimes
f_{3})\Gamma_{\mathbb{C}}(l)\Gamma_{\mathbb{C}}(l-k_{1}+1)\Gamma_{\mathbb{C}}(l-k_{2}+1)\Gamma_{\mathbb{C}}(l-k_{3}+1)}{\langle
f_{1},f_{1}\rangle\langle f_{2},f_{2}\rangle\langle f_{3},f_{3}\rangle}}.$

Then $L_{alg}(l,f_{1}\otimes f_{2}\otimes f_{3})$ belongs to
${\mathbb{Q}}(f_{1}){\mathbb{Q}}(f_{2}){\mathbb{Q}}(f_{3})$ ([11], [31], [32])
and is also symmetric in the $f_{i}$. An algorithm to compute
$L_{alg}(l,f_{1}\otimes f_{2}\otimes f_{3})$ has been given in [13, Section
4].

Finally, we define $\displaystyle L(s,\mathrm{St}(f_{2})\otimes f_{1})$ as

$\displaystyle L(s,\mathrm{St}(f_{2})\otimes
f_{1})=\prod_{p}\left(\prod_{i\in\\{1,-1\\}}\prod_{j\in\\{1,-1,0\\}}(1-\alpha_{f_{1},p}^{i}\alpha_{f_{2},p}^{2j}\,p^{k_{1}/2-1/2-s})\right)^{-1},$

and put

(2.5) $\displaystyle\Lambda(s,\mathrm{St}(f_{2})\otimes
f_{1})=\Gamma_{\mathbb{C}}(s)\Gamma_{\mathbb{C}}(s-k_{1}+k_{2})\Gamma_{\mathbb{C}}(s+k_{2}-1)L(s,\mathrm{St}(f_{2})\otimes
f_{1}).$

Moreover we fix the period ${\Omega}_{\pm}(f_{1})$ satisfying (2.2) and for a
positive integer $l$ with $k_{1}/2\leq l\leq k_{2}-1$, put

$L_{alg}(l,\mathrm{St}(f_{2})\otimes
f_{1})={\dfrac{\displaystyle\Lambda(l,\mathrm{St}(f_{2})\otimes
f_{1})}{\displaystyle\langle
f_{2},f_{2}\rangle^{2}{\Omega}_{j^{\prime}}(f_{1})}},$

where $j^{\prime}=+$ or $-$ according as $l$ is odd or even. (We note that
$L_{alg}(l,\mathrm{St}(f_{2})\otimes f_{1})$ would more properly be denoted as
$L_{alg}(l,\mathrm{St}(f_{2})\otimes f_{1};\Omega_{j^{\prime}}(f_{1}))$ since
it does depend upon the choice of ${\Omega}_{\pm}(f_{1})$, but we use the
above short notation.) From the Euler product, we note that

$L(s+k_{2}-1,f_{2}\otimes f_{2}\otimes f_{1})=L(s,\mathrm{St}(f_{2})\otimes
f_{1})L(s,f_{1})$

and we have

(2.6) $\displaystyle L_{alg}(l+k_{2}-1,f_{2}\otimes f_{2}\otimes f_{1})$
$\displaystyle=L_{alg}(l,\mathrm{St}(f_{2})\otimes
f_{1})L_{alg}(l,f_{1})\frac{\Omega_{+}(f_{1})\Omega_{-}(f_{1})}{\langle
f_{1},f_{1}\rangle}.$

We also note that $\Lambda(l,f_{1})/\Omega_{j}(f_{1})$ belongs to
${\mathbb{Q}}(f_{1})^{\times}$ for $k_{1}/2+1\leq l\leq k_{2}-1,$ since we
assumed $k_{1}/2+1\leq k_{2}\leq k_{1}$ and the Euler product of $L(s,f_{1})$
converges in this range. Hence $L_{alg}(l,\mathrm{St}(f_{2})\otimes f_{1})$
belongs to ${\mathbb{Q}}(f_{1}){\mathbb{Q}}(f_{2})$ and non-zero for
$k_{1}/2+1\leq l\leq k_{2}-1$.

##  3\. Ikeda-Miyawaki lift

Throughout this section, fix positive integers $k,n$ such that $k\equiv
n+1\mod 2$ and $k>n$. We denote by $S_{k+1/2}^{+}(\varGamma_{0}(4))$ the
Kohnen plus subspace of weight $k+1/2$ for $\varGamma_{0}(4)$. For the detail
of the Kohnen plus subspace, see [25]. For two elements $h_{1}$ and $h_{2}$ of
$S_{k+1/2}^{+}(\varGamma_{0}(4))$, we define the Petersson norm $\langle
h_{1},\ h_{2}\rangle$ as

$\langle h_{1},\
h_{2}\rangle=\frac{1}{6}\int_{\varGamma_{0}(4)\backslash\mathbb{H}_{1}}h_{1}(z)\overline{h_{2}(z)}y^{k-3/2}dxdy\quad(z=x+\sqrt{-1}y).$

Let $h$ be a Hecke eigenform in $S_{k+1/2}^{+}(\varGamma_{0}(4))$ with Fourier
expansion

$h(z)=\sum_{m\in{{\bf N}},\ (-1)^{k}m\equiv 0,1\mod 4}c_{h}(m){\bf e}(mz)$

and let $g$ be a primitive form in
$S_{k+n+1}(\operatorname{SL}_{2}({\mathbb{Z}}))$ with Fourier expansion

$g(z)=\sum_{m=1}^{\infty}c_{g}(m){\bf e}(mz).$

Moreover let

$f(z)=\sum_{m=1}^{\infty}c_{f}(m){\bf e}(mz)$

be the primitive form in $S_{2k}(\operatorname{SL}_{2}({\mathbb{Z}}))$
corresponding to $h$ under the Shimura correspondence. Let
$\alpha_{p}\in{\mathbb{C}}$ be taken such that
$\alpha_{p}+\alpha_{p}^{-1}=p^{-k+1/2}c_{f}(p).$ For
$T\in{{\mathcal{H}}_{2n+2}(\mathbb{Z})}_{>0},$ define $c_{I_{2n+2}(h)}(T)$ as

$c_{I_{2n+2}(h)}(T)=c_{h}(|{\mathfrak{d}}_{T}|)\,{\textfrak{f}}_{T}^{k-1/2}\prod_{p\mid{\mathfrak{f}}_{T}}\alpha_{p}^{-\mathrm{ord}_{p}({\mathfrak{f}}_{T})}F_{p}(T,p^{-n-3/2}\alpha_{p}),$

where ${\mathfrak{d}}_{T}$ is the discriminant of
${\mathbb{Q}}(\sqrt{(-1)^{n+1}\det(2T)}),\
{\mathfrak{f}}_{T}=\sqrt{(-1)^{n+1}\det(2T)/{\mathfrak{d}}_{T}}$, and
$F_{p}(T,X)$ is a polynomial in $X$ with coefficients in ${\mathbb{Q}},$ which
will be defined in Section 4. Define a Fourier series $I_{2n+2}(h)(Z)$ for
$Z\in{\mathbb{H}}_{2n+2}$:

$I_{2n+2}(h)(Z)=\sum_{T\in{{\mathcal{H}}_{2n+2}(\mathbb{Z})}_{>0}}c_{I_{2n+2}(h)}(T){\bf
e}(\mathrm{tr}(TZ)).$

Then $I_{2n+2}(h)$ is a Hecke eigenform in
$S_{k+n+1}(\operatorname{Sp}_{2n+2}({\mathbb{Z}}))$, see [16]. A proof that
$I_{2n+2}(h)$ is a Hecke eigenform for the entire Hecke algebra ${\tilde{\bf\
L}}_{2n+2}$, not just the even part ${\bf\ L}^{\circ}_{2n+2}$, may be found in
[23]. We call $I_{2n+2}(h)$ the Duke-Imamoḡlu-Ikeda lift of $h$ (or of $f$) to
$S_{k+n+1}(\operatorname{Sp}_{2n+2}({\mathbb{Z}})).$ For
$z\in{\mathbb{H}}_{2n+1}$ and $w=x+iy\in{\mathbb{H}}_{1},$ define

$\displaystyle{\mathcal{F}}_{h,g}(z)=\int_{\operatorname{SL}_{2}({\mathbb{Z}})\backslash{\mathbb{H}}_{1}}I_{2n+2}(h)(\left(\begin{array}[]{cc}z&0\\\
0&w\end{array}\right))\overline{g(w)}y^{k+n-1}dw.$

Ikeda [17] showed the following:

If $\displaystyle{\mathcal{F}}_{h,g}(z)$ is not identically zero in
$S_{k+n+1}(\operatorname{Sp}_{2n+1}({\mathbb{Z}}))$, then it is a Hecke
eigenform with respect to ${\bf L}_{2n+1}^{\circ}$ and its standard
$L$-function is given by

(3.1) $\displaystyle
L(s,{\mathcal{F}}_{h,g},\mathrm{St})=L(s,g,\mathrm{St})\prod_{i=1}^{2n}L(s+k+n-i,f).$

This is a part of Ikeda’s results: the case $r=1$ in [17]. The existence of
this type of Hecke eigenform was conjectured by Miyawaki [29]; therefore, we
call ${\mathcal{F}}_{h,g}$ the Ikeda-Miyawaki lift of $h$ and $g$ when
${\mathcal{F}}_{h,g}$ is not identically zero. By Remark 2.1, it is a Hecke
eigenform also with respect to $\tilde{\bf L}_{2n+1}$.

Ikeda proposed the following conjecture relating the Petersson norm of
${\mathcal{F}}_{h,g}$ to those of $f$ and $g$.

######  Conjecture 3.1.

(Ikeda [17]) Let $k>n$ be positive integers with $k+n$ odd. Let $h\in
S_{k+1/2}^{+}(\varGamma_{0}(4))$ be a Hecke eigenform corresponding to the
primitive eigenform $f\in S_{2k}(\operatorname{SL}_{2}({\mathbb{Z}}))$ under
the Shimura correspondence. Let $g\in
S_{k+n+1}(\operatorname{SL}_{2}({\mathbb{Z}}))$ be a primitive eigenform. Then
we have

$2^{(2n-1)k+4n}\dfrac{\displaystyle\langle{\mathcal{F}}_{h,g},{\mathcal{F}}_{h,g}\rangle}{\displaystyle\langle
g,g\rangle\langle h,h\rangle}=\displaystyle\Lambda(k+n,\mathrm{St}(g)\otimes
f)\prod_{i=1}^{n}\tilde{\xi}(2i)\prod_{i=1}^{n-1}\tilde{\Lambda}(2i+1,f,\mathrm{St}).$

The expression in the above conjecture appears to be different from the one in
[17] but it is the same since we have
$\widetilde{\Lambda}(1,f,St)=2^{2k}\langle f,f\rangle$ (see [36]). We consider
the case that ${\mathcal{F}}_{h,g}$ is not identically zero. Then, the above
conjecture can be rewritten as follows:

######  Conjecture 3.2.

In additions to the notation and the assumption as in Conjecture 3.1, assume
that the Ikeda-Miyawaki lift ${\mathcal{F}}_{h,g}$ is not identically zero.
Put

$C_{h,g}=\dfrac{\displaystyle\langle g,g\rangle\langle
h,h\rangle}{\displaystyle\langle{\mathcal{F}}_{h,g},{\mathcal{F}}_{h,g}\rangle}\displaystyle\Lambda(k+n,\mathrm{St}(g)\otimes
f)\prod_{i=1}^{n}\tilde{\xi}(2i)\prod_{i=1}^{n-1}\tilde{\Lambda}(2i+1,f,\mathrm{St}).$

Then

$C_{h,g}=2^{(2n-1)k+4n}.$

Now we modify ${\mathcal{F}}_{h,g}$ and put

$\widetilde{\mathcal{F}}_{h,g}=\langle g,g\rangle^{-1}{\mathcal{F}}_{h,g}.$

Then, we note that we have

(3.2) $\displaystyle C_{h,g}$ $\displaystyle=\dfrac{\displaystyle\langle
h,h\rangle}{\displaystyle\langle
g,g\rangle\langle\widetilde{\mathcal{F}}_{h,g},\
\widetilde{\mathcal{F}}_{h,g}\rangle}$
$\displaystyle\times\Lambda(k+n,\mathrm{St}(g)\otimes
f)\prod_{i=1}^{n}\tilde{\xi}(2i)\prod_{i=1}^{n-1}\tilde{\Lambda}(2i+1,f,\mathrm{St}).$

Let $\\{g_{i}\\}_{i=1}^{d_{2}}$ be a basis of
$S_{k+n+1}(\operatorname{SL}_{2}({\mathbb{Z}}))$ consisting of Hecke
eigenforms, then from the definition of the Ikeda-Miyawaki lift, we easily
have:

$I_{2n+2}(h)(\left(\begin{array}[]{cc}z&0\\\
0&w\end{array}\right))=\sum_{i=1}^{d_{2}}\widetilde{\mathcal{F}}_{h,g_{i}}(z)g_{i}(w).$

We obtain

$I_{2n+2}(h)(\left(\begin{array}[]{cc}z&0\\\
0&w\end{array}\right))=\sum_{A>0,m>0}c_{2n+1,h}(A,m){\bf
e}(\mathrm{tr}(Az)){\bf e}(mw),$

where

$c_{2n+1,h}(A,m)=\sum_{r}c_{I_{2n+2}(h)}(\left(\begin{array}[]{cc}A&r/2\\\
{}^{t}r/2&m\end{array}\right)).$

By replacing the Hecke eigenform $h$ by a constant multiple of $h$, we can
assume that every Fourier coefficient $c_{h}(m)$ belongs to ${\mathbb{Q}}(f)$
for any $m>0.$ If we assume so, then by definition, $c_{I_{2n+2}(h)}(T)$
belongs to ${\mathbb{Q}}(f)$ for any $T\in{{\mathcal{H}}_{2n+2}}_{>0}$ and,
therefore, so does $c_{2n+1,h}(A,m)$ for any $A\in{{\mathcal{H}}_{2n+1}}_{>0}$
and $m>0.$ Hence, the $A$-th Fourier coefficient
$c_{\widetilde{\mathcal{F}}_{h,g}}(A)$ of $\widetilde{\mathcal{F}}_{h,g}$
belongs to ${\mathbb{Q}}(f){\mathbb{Q}}(g)$ for any
$A\in{{\mathcal{H}}_{2n+1}}_{>0}.$

The following theorem is a refinement of [13, Corollary to Theorem 3.1].

######  Theorem 3.3.

We assume that $k>n$ and $n$ is odd. We fix a real number ${\Omega}_{\pm}(f)$
which satisfies (2.2) of Section 2. Then

$\displaystyle|c_{\widetilde{\mathcal{F}}_{h,g}}(A)|^{2}L_{alg}(l,\widetilde{\mathcal{F}}_{h,g},\mathrm{St})=(-1)^{n}C_{h,g}{\dfrac{|c_{{\widetilde{\mathcal{F}}}_{h,g}}(A)|^{2}\,{\Omega}_{-}(f)}{\langle
h,h\rangle}}$ $\displaystyle\times$
$\displaystyle{\dfrac{L_{alg}(k+n,f)L_{alg}(l,g,\mathrm{St})\prod_{i=1}^{n}L_{alg}(l+k+n-2i+1,l+k+n-2i;f)}{L_{alg}(2k+2n,g\otimes
g\otimes
f)\prod_{i=1}^{n-1}L_{alg}(2i+1,f,\mathrm{St})\prod_{i=1}^{n}\widetilde{\xi}(2i)}}$

for any positive definite half-integral matrix $A$ of degree $2n+1$ and an odd
integer $l$ with $1\leq l\leq k-n$.

###### Proof.

The assertion directly follows from (2.3),(2.6), (3.1) and (3.2) remarking
that

$\displaystyle L_{alg}(l+k+n-2i+1,l+k+n-2i;f)$
$\displaystyle=-\frac{\Gamma_{\mathbb{C}}(l+k+n-2i+1)\Gamma_{\mathbb{C}}(l+k+n-2i)L(l+k+n-2i+1,f)L(l+k+n-2i,f)}{\langle
f,\ f\rangle}$

for any $1\leq i\leq n$. ∎

######  Remark 3.4.

Theorem 3.3 has been proved in [13, Corollary to Theorem 3.1] assuming
Conjecture 3.2. We also note that the definition of
$L_{alg}(l+k+n-2i+1,l+k+n-2i;f)$ is a little different from that in [13].

Rewriting Theorem 3.3, we have the following result.

######  Theorem 3.5.

Let the notation be as in Theorem 3.3. Then

$\displaystyle|c_{\widetilde{\mathcal{F}}_{h,g}}(A)|^{2}L_{alg}(l,\widetilde{\mathcal{F}}_{h,g},\mathrm{St})$
$\displaystyle=(-1)^{n+[(n+1)/2]}2^{k-1}|D|^{k}C_{h,g}{\dfrac{|c_{{\widetilde{\mathcal{F}}}_{h,g}}(A)|^{2}}{|c_{h}(|D|)|^{2}}}L_{alg}(k+n,k;f;1,\chi_{D})$
$\displaystyle\times$
$\displaystyle{\dfrac{L_{alg}(l,g,\mathrm{St})\prod_{i=1}^{n}L_{alg}(l+k+n-2i+1,l+k+n-2i;f)}{L_{alg}(2k+2n,g\otimes
g\otimes
f)\prod_{i=1}^{n-1}L_{alg}(2i+1,f,\mathrm{St})\prod_{i=1}^{n}\widetilde{\xi}(2i)}}$

for any positive definite half-integral matrix $A$ of degree $2n+1$, a
fundamental discriminant $D$ such that $c_{h}(|D|)\not=0$ and an odd integer
$l$ with $1\leq l\leq k-n$. In particular, if $c_{h}(1)\not=0$, then

(3.3)
$\displaystyle|c_{\widetilde{\mathcal{F}}_{h,g}}(A)|^{2}L_{alg}(l,\widetilde{\mathcal{F}}_{h,g},\mathrm{St})$
$\displaystyle=(-1)^{n+[(n+1)/2]}2^{k-1}C_{h,g}{\dfrac{|c_{{\widetilde{\mathcal{F}}}_{h,g}}(A)|^{2}}{|c_{h}(1)|^{2}}}L_{alg}(k+n,k;f)$
$\displaystyle\times$
$\displaystyle{\dfrac{L_{alg}(l,g,\mathrm{St})\prod_{i=1}^{n}L_{alg}(l+k+n-2i+1,l+k+n-2i;f)}{L_{alg}(2k+2n,g\otimes
g\otimes
f)\prod_{i=1}^{n-1}L_{alg}(2i+1,f,\mathrm{St})\prod_{i=1}^{n}\widetilde{\xi}(2i)}}.$

###### Proof.

By [26], we have

$\dfrac{|c_{h}(|D|)|^{2}}{\langle h,\
h\rangle}=\frac{2^{k-1}|D|^{k-1}\Gamma_{\mathbb{C}}(k)L(k,f,\chi_{D})}{|D|^{1/2}\langle
f,\ f\rangle}.$

Moreover we have

$\displaystyle
L_{alg}(k+n,k;f,1;\chi_{D})=\frac{\Gamma_{\mathbb{C}}(k+n)\Gamma_{\mathbb{C}}(k)L(k+n,f)L(f,k,\chi_{D})}{\sqrt{-1}^{2k+n+1}s(D)|D|^{1/2}\langle
f,\ f\rangle},$

where $s(D)=\begin{cases}1&\text{ if }k\text{ is even}\\\ \sqrt{-1}&\text{ if
}k\text{ is odd}.\end{cases}$. Thus the assertion follows from Theorem 3.3. ∎

######  Theorem 3.6.

Let the notation be as in Theorem 3.5. Then, $C_{h,g}$ belongs to
$\mathbb{Q}(f)\mathbb{Q}(g)$.

###### Proof.

We can take a positive definite half-integral matrix $A$ of degree $2n+1$, a
fundamental discriminant $D$ and an odd integer $l$ with $n+1\leq l\leq k-n$
such that $c_{\widetilde{\mathcal{F}}_{h,g}}(A)\not=0$ and $c_{h}(|D|)\not=0$.
Then, by construction,
$\dfrac{|c_{\widetilde{\mathcal{F}}_{h,g}}(A)|^{2}}{|c_{h}(|D|)|^{2}}$ is non-
zero and belongs to $\mathbb{Q}(f)\mathbb{Q}(g)$. Moreover, for any odd
integer $3\leq l\leq k-n-1$, the value
$|c_{\widetilde{\mathcal{F}}_{h,g}}(A)|^{2}L_{alg}(l,\widetilde{\mathcal{F}}_{h,g},\mathrm{St})$
belongs to ${\mathbb{Q}}(f){\mathbb{Q}}(g)$. We also note that the value
$L_{alg}(k+n,k;f;1,\chi_{D})L_{alg}(l,g,\mathrm{St})\prod_{i=1}^{n}L_{alg}(l+k+n-2i+1,l+k+n-2i;f)$
is non-zero and belongs to ${\mathbb{Q}}(f){\mathbb{Q}}(g)$. This proves the
assertion. ∎

######  Remark 3.7.

As stated before, $L_{alg}(1,g,\mathrm{St})$ is a rational number and
$\mathbb{Q}(\widetilde{\mathcal{F}}_{h,g})=\mathbb{Q}(f)\mathbb{Q}(g)$.
Therefore, by Theorems 3.5 and 3.6,
$L_{alg}(1,\widetilde{\mathcal{F}}_{h,g},\mathrm{St})$ is algebraic and
belongs to $\mathbb{Q}(\widetilde{\mathcal{F}}_{h,g})$.

We note that $L_{alg}(k+n,\mathrm{St}(g)\otimes f)$ belongs to
$\mathbb{Q}(f)\mathbb{Q}(g)$. Therefore we deduce the following algebraicity
for the ratio of the periods.

######  Theorem 3.8.

The ratio $\dfrac{\langle\widetilde{\mathcal{F}}_{h,g},\
\widetilde{\mathcal{F}}_{h,g}\rangle}{\langle f,\ f\rangle^{n-1}\langle h,\
h\rangle\Omega_{-}(f)}$ belongs to $\mathbb{Q}(f)\mathbb{Q}(g)$, and so does
$\dfrac{\langle{\mathcal{F}}_{h,g},\ {\mathcal{F}}_{h,g}\rangle}{\langle g,\
g\rangle^{2}\langle f,\ f\rangle^{n-1}\langle h,\ h\rangle\Omega_{-}(f)}$.

##  4\. Pullback of Siegel Eisenstein series

To prove our main result, we give an algorithm for computing the standard
$L$-value for a cuspidal Hecke eigenform for
$\operatorname{Sp}_{3}(\mathbb{Z})$ (cf. Theorem 4.8). This type of algorithm
was given in the case $F$ is a cuspidal Hecke eigenform for
$\operatorname{Sp}_{m}(\mathbb{Z})$ with $m\leq 2$ (cf. [10], [14], [21],
[22]). To give such an algorithm, first we express a certain modular form
arising from the pullback of Siegel Eisenstein series as a linear combination
of Hecke eigenforms with their standard $L$-values as coefficients (cf.
Theorem 4.3). We have carried it out in [23], and here we give its refinement.
For a non-negative integer $m$, put

$\Gamma_{m}(s)=\pi^{m(m-1)/4}\prod_{i=1}^{m}\Gamma(s-{i-1\over 2}).$

Let $n,l$ be positive integers such that $l$ is even. Define the Eisenstein
series $E_{n,l}(Z,s)$ by

$E_{n,l}(Z,s)=\bigl{(}\det\mathrm{Im}(Z)\bigr{)}^{s}\sum_{\gamma\in\operatorname{Sp}_{n}(\mathbb{Z})_{\infty}\backslash\operatorname{Sp}_{n}(\mathbb{Z})}j(\gamma,Z)^{-l}|j(\gamma,Z)|^{-2s},$

where
$\operatorname{Sp}_{n}(\mathbb{Z})_{\infty}=\Big{\\{}\begin{pmatrix}A&B\\\
0_{n}&D\end{pmatrix}\in\operatorname{Sp}_{n}(\mathbb{Z})\Big{\\}}$. Then
$E_{n,l}(Z,s)$ converges absolutely as a function of $s$ if the real part of
$s$ is large enough. It has a meromorphic continuation to the whole $s$-plane,
and it belong to $M_{l}^{\infty}(\operatorname{Sp}_{n}(\mathbb{Z}))$. Moreover
it is holomorphic and finite at $s=0$, which will be denoted by $E_{n,l}(Z)$.
In particular, if $E_{n,l}(Z)$ belongs to
$M_{l}(\operatorname{Sp}_{n}(\mathbb{Z})),$ we put

$E_{n,l}^{*}(Z)=Z(n,l)E_{n,l}(Z),$

where

$Z(n,l)=\zeta(1-l)\prod_{i=1}^{[n/2]}\zeta(1+2i-2l).$

From now on, for an element $A\in\mathcal{H}_{n}(\mathbb{Z})_{\geq 0}$, we
denote by $c_{n.l}(A)$ the $A$-th Fourier coefficient $c_{E_{n,l}^{*}}(A)$ of
$E_{n,l}^{*}(Z)$. To determine the Fourier coefficient of $E_{n,l}^{*}(Z)$, we
define a polynomial attached to local Siegel series. For a prime number $p$,
let $\mathbb{Q}_{p}$ be the field of $p$-adic numbers and $\mathbb{Z}_{p}$ the
ring of $p$-adic integers. For an element
$B\in\mathcal{H}_{n}(\mathbb{Z}_{p})$, we define the Siegel series
$b_{p}(B,s)$ as

$b_{p}(B,s)=\sum_{R\in\mathrm{Sym}_{n}(\mathbb{Q}_{p})/\mathrm{Sym}_{n}(\mathbb{Z}_{p})}{\bf
e}_{p}(\mathrm{tr}(BR))\nu_{p}(R)^{-s},$

where ${\bf e}_{p}$ is the additive character of $\mathbb{Z}_{p}$ such that
${\bf e}_{p}(m)={\bf e}(m)$ for $m\in\mathbb{Z}[p^{-1}]$, and
$\nu_{p}(R)=[R\mathbb{Z}_{p}^{n}+\mathbb{Z}_{p}^{n}:\mathbb{Z}_{p}^{n}]$. We
define $\chi_{p}(a)$ for $a\in{\mathbb{Q}}^{\times}_{p}$ as follows:

$\chi_{p}(a):=\left\\{\begin{array}[]{cl}+1&\text{ if }\
{\mathbb{Q}}_{p}(\sqrt{a})={\mathbb{Q}}_{p},\\\ -1&\text{ if }\
{\mathbb{Q}}_{p}(\sqrt{a})/{\mathbb{Q}}_{p}\text{ is quadratic unramified},\\\
0&\text{ if }\ {\mathbb{Q}}_{p}(\sqrt{a})/{\mathbb{Q}}_{p}\text{ is quadratic
ramified}.\end{array}\right.$

For an element $B\in\mathcal{H}_{n}(\mathbb{Z}_{p})^{\mathrm{nd}}$ with $n$
even, we define $\xi_{p}(B)$ by

$\xi_{p}(B):=\chi_{p}((-1)^{n/2}\det B).$

For a nondegenerate half-integral matrix $B$ of size $n$ over
${\mathbb{Z}}_{p}$, define a polynomial $\gamma_{p}(B,X)$ in $X$ by

$\gamma_{p}(B,X):=\left\\{\begin{array}[]{ll}(1-X)\prod_{i=1}^{n/2}(1-p^{2i}X^{2})(1-p^{n/2}\xi_{p}(B)X)^{-1}&\text{
if }n\text{ is \ even},\\\ (1-X)\prod_{i=1}^{(n-1)/2}(1-p^{2i}X^{2})&\text{ if
}\ n\text{ is odd}.\end{array}\right.$

Then it is well known that there exists a unique polynomial $F_{p}(B,X)$ in
$X$ over ${\mathbb{Z}}$ with constant term $1$ such that

$b_{p}(B,s)=\gamma_{p}(B,p^{-s})F_{p}(B,p^{-s})$

(e.g. [20]).

For $B\in\mathcal{H}_{m}(\mathbb{Z})_{>0}$ with $m$ even, let
${\mathfrak{d}}_{B}$ be the discriminant of ${\mathbb{Q}}(\sqrt{(-1)^{m/2}\det
B})/{\mathbb{Q}}$, and $\chi_{B}=({\frac{{\mathfrak{d}}_{B}}{*}})$ the
Kronecker character corresponding to ${\mathbb{Q}}(\sqrt{(-1)^{m/2}\det
B})/{\mathbb{Q}}$. We note that we have $\chi_{B}(p)=\xi_{p}(B)$ for any prime
$p.$ We also note that

$(-1)^{m/2}\det(2B)=\mathfrak{d}_{B}\mathfrak{f}_{B}^{2}$

with $\mathfrak{f}_{B}\in\mathbb{Z}_{>0}$. For any
$T\in\mathcal{H}_{n}(\mathbb{Z}_{p})$ which is not-necessarily non-degenerate,
we define a polynomial $F_{p}^{*}(T,X)$ as follows: For an element
$T\in\mathcal{H}_{n}(\mathbb{Z}_{p})$ of rank $r\geq 1,$ there exists an
element $\widetilde{T}\in\mathcal{H}_{r}(\mathbb{Z}_{p})^{\mathrm{nd}}$ such
that $T\sim_{\mathbb{Z}_{p}}\widetilde{T}\bot O_{n-r}.$ We note that
$F_{p}(\widetilde{T},X)$ depends only on $T$ and does not depend on the choice
of $\widetilde{T}.$ Then we put $F_{p}^{\ast}(T,X)=F_{p}(\widetilde{T},X).$
For an element $T\in\mathcal{H}_{n}(\mathbb{Z})_{\geq 0}$ of rank $r\geq 1,$
there exists an element $\widetilde{T}\in\mathcal{H}_{r}(\mathbb{Z})_{>0}$
such that $T\sim_{\mathbb{Z}}\widetilde{T}\bot O_{n-r}.$ If $r$ is even,
$\chi_{\widetilde{T}}$ depend only on $T$ and does not depend on the choice of
$\widetilde{T}$. Then we write $\chi_{T}^{\ast}=\chi_{\widetilde{T}}$. The
following proposition is due to [4, Proposition 6.3].

######  Proposition 4.1.

Let $n$ and $l$ be positive integers such that $l\geq n+1$. Then
$E_{2n,l}^{*}(Z)$ is holomorphic and belongs to
$M_{l}(\operatorname{Sp}_{2n}(\mathbb{Z}))$ except in the following case:
$l=n+1\equiv 2\text{ mod }4$. In the case that $E_{2n,l}^{*}(Z)$ is
holomorphic we have the following assertion: For
$B\in\mathcal{H}_{2n}(\mathbb{Z})_{\geq 0}$ of rank $m,$ we have

$\displaystyle c_{2n,l}(B)$
$\displaystyle=2^{[(m+1)/2]}\prod_{p\mid\det(2\widetilde{B})}F_{p}^{\ast}(B,p^{l-m-1})$
$\displaystyle\times\begin{cases}\prod_{i=m/2+1}^{n}\zeta(1+2i-2l)L(1+m/2-l,\chi_{B}^{\ast})&\text{
if }m\text{ is even },\\\ \prod_{i=(m+1)/2}^{n}\zeta(1+2i-2l)&\text{ if }\
m\text{ is odd}.\end{cases}$

Here we make the convention that $F_{p}^{*}(B,p^{l-m-1})=1$ and
$\L(1+m/2-l,\chi^{*}_{B})=\zeta(1-l)$ if $m=0$.

Let
$\stackrel{{\scriptstyle\\!\\!\\!\\!\\!\\!\circ}}{{{\mathcal{D}}_{m,l}^{\nu}}}$
be the differential operator in [7], which maps
$M_{l}(\operatorname{Sp}_{2m}(\mathbb{Z}))$ to
$M_{l+\nu}(\operatorname{Sp}_{m}(\mathbb{Z}))\otimes
M_{l+\nu}(\operatorname{Sp}_{m}(\mathbb{Z})).$ For a non-negative integer
$\nu\leq k$, we define a function ${\mathfrak{E}}_{2n}^{k,\nu}(Z_{1},Z_{2})$
on ${\mathbb{H}}_{n}\times{\mathbb{H}}_{n}$ as

$\displaystyle{\mathfrak{G}}_{2n}^{k,\nu}(Z_{1},Z_{2})=(2\pi\sqrt{-1})^{-n\nu}\sqrt{-1}^{-k+\nu}\stackrel{{\scriptstyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\circ}}{{{\mathcal{D}}_{n,k-\nu}^{\nu}}}E^{*}_{2n,k-\nu}(Z_{1},Z_{2})$

for $(Z_{1},Z_{2})\in{\mathbb{H}}_{n}\times{\mathbb{H}}_{n}$. From now on,
until Corollary 4.4, let $l$ be an even integer such that $l\geq n+1$ and we
assume that $l$ does not satisfy $l=n+1\equiv 2\mod 4$. Let $T$ be a $2n\times
2n$ symmetric matrix of variables. Then, there exists a polynomial
$P_{n,l}^{\nu}(X)$ in $T$ such that

$\displaystyle\stackrel{{\scriptstyle\\!\\!\\!\\!\\!\\!\circ}}{{{\mathcal{D}}_{n,l}^{\nu}}}\Bigl{(}{\bf
e}\bigl{(}\mathrm{tr}\Bigl{(}\begin{pmatrix}A_{1}&R/2\\\
{}^{t}R/2&A_{2}\end{pmatrix}\begin{pmatrix}Z_{1}&Z_{12}\\\
{}^{t}Z_{12}&Z_{2}\end{pmatrix}\Bigr{)}\Bigr{)}\Bigr{)}$
$\displaystyle=(2\pi\sqrt{-1})^{n\nu}P_{n,l}^{\nu}\Bigl{(}\begin{pmatrix}A_{1}&R/2\\\
{}^{t}R/2&A_{2}\end{pmatrix}\Bigr{)}{\bf
e}(\mathrm{tr}(A_{1}Z_{1}+A_{2}Z_{2}))$

for $\begin{pmatrix}A_{1}&R/2\\\
{}^{t}R/2&A_{2}\end{pmatrix}\in\mathcal{H}_{2n}(\mathbb{Z})_{\geq 0}$ with
$A_{1},A_{2}\in\mathcal{H}_{n}(\mathbb{Z})_{\geq 0}$ and
$\begin{pmatrix}Z_{1}&Z_{12}\\\
{}^{t}Z_{12}&Z_{2}\end{pmatrix}\in{\mathbb{H}}_{2n}$ with
$Z_{1},Z_{2}\in{\mathbb{H}}_{n}$. For
$A_{1},A_{2}\in\mathcal{H}_{n}(\mathbb{Z})_{\geq 0}$, put

(4.1) $\displaystyle c_{{\mathfrak{G}}_{2n}^{k,k-l}}(A_{1},A_{2})$
$\displaystyle=\sum_{R\in
M_{n}(\mathbb{Z})}P_{n,l}^{k-l}\Bigl{(}\begin{pmatrix}A_{1}&R/2\\\
{}^{t}R/2&A_{2}\end{pmatrix}\Bigr{)}c_{2n,l}\Bigl{(}\begin{pmatrix}A_{1}&R/2\\\
{}^{t}R/2&A_{2}\end{pmatrix}\Bigr{)}.$

Then, by definition, ${\mathfrak{G}}_{2n}^{k,k-l}(Z_{1},Z_{2})$ can be
expressed as

$\displaystyle{\mathfrak{G}}_{2n}^{k,k-l}(Z_{1},Z_{2})=\sum_{A_{1},A_{2}\in\mathcal{H}_{n}(\mathbb{Z})_{\geq
0}}c_{{\mathfrak{E}}_{2n}^{k,k-l}}(A_{1},A_{2}){\bf
e}\Big{(}\mathrm{tr}(A_{1}Z_{1}+A_{2}Z_{2})\Big{)}.$

For $A\in\mathcal{H}_{n}(\mathbb{Z})_{\geq 0}$, define a Fourier series
${\mathcal{G}}_{2n}^{k,k-l}(Z_{1})$ on $\mathbb{H}_{n}$ as

${\mathcal{G}}_{2n}^{k,k-l}(Z_{1},A)=\sum_{A_{1}\in\mathcal{H}_{n}(\mathbb{Z})_{\geq
0}}c_{{\mathfrak{E}}_{2n}^{k,k-l}}(A_{1},A){\bf e}(A_{1}Z_{1}).$

Then, by definition, ${\mathfrak{G}}_{2n}^{k,k-l}(Z_{1},Z_{2})$ can be
expressed as

${\mathfrak{G}}_{2n}^{k,k-l}(Z_{1},Z_{2})=\sum_{A\in{\mathcal{H}}_{n}(\mathbb{Z})_{\geq
0}}{\mathcal{G}}_{2n}^{k,k-l}(Z_{1},A){\bf e}(\mathrm{tr}(AZ_{2})).$

The Fourier series ${\mathcal{G}}_{2n}^{k,k-l}(Z_{1},A)$ belongs to
$M_{k}(\operatorname{Sp}_{n}(\mathbb{Z}))$, and in particular, if $l<k$, then
${\mathcal{G}}_{2n}^{k,k-l}(Z_{1},A)$ belongs to
$S_{k}(\operatorname{Sp}_{n}(\mathbb{Z}))$ (cf. [23, Proposition 3,6]. Define
$\Lambda(m,F,\mathrm{St})$ by

$\Lambda(m,F,\mathrm{St})=\frac{\Gamma(m)\prod_{i=1}^{n}\Gamma(2k-n-i)L(m,F,\mathrm{St})}{\pi^{-n(n+1)/2+nk+(n+1)m}}.$

The following lemma is a modification of [7, Theorem 3.1].

######  Lemma 4.2.

Let $g$ be a Hecke eigenform in $S_{k}(\operatorname{Sp}_{n}(\mathbb{Z}))$.
Then we have

$\displaystyle\langle
g,\mathfrak{E}_{2n}^{k,k-l}(*,-\overline{Z_{2}})\rangle=c(n,k,l)\Lambda(l-n,g,\mathrm{St})g(Z_{2}),$

where $c(n,k,l)=(-1)^{l(n-1)/2-n(n+1)/2}2^{-4kn+(n-1)l+2n^{2}+3n+2}$.

###### Proof.

By [7, Theorem 3.1], we have

$\displaystyle\langle g,\
\mathfrak{E}_{2n}^{k,k-l}(*,-\overline{Z_{2}})\rangle$ $\displaystyle=\langle
g,\
(2\pi\sqrt{-1})^{n(l-k)}Z(2n,l)\stackrel{{\scriptstyle\\!\\!\\!\\!\\!\\!\circ}}{{{\mathcal{D}}_{n,l}^{k-l}}}(E_{2n,l})(*,-\overline{Z_{2}})\rangle$
$\displaystyle=(2\pi)^{n(l-k)}(-1)^{n(l-k)/2}Z(2n,l)$
$\displaystyle\times(-1)^{kl/2}2^{1-nk+n(n+3)/2}\pi^{n(n+1)/2}\dfrac{\widetilde{\Omega}_{k,l}}{\widetilde{Z}(2n,l)}L(l-n,g,\mathrm{St})g(Z_{2}),$

where

$\widetilde{Z}(2n,l)=\zeta(l)\prod_{i=1}^{n}\zeta(2l-2i)$

and

$\widetilde{\Omega}_{k,l}=\dfrac{\prod_{i=1}^{n}\Gamma(k-n/2-(i-1)/2)\prod_{i=1}^{n}\Gamma(k-(n+1)/2-(i-1)/2)}{\prod_{i=1}^{n}\Gamma(l-(i-1)/2)\prod_{i=1}^{n}\Gamma(l-n/2-(i-1)/2)}.$

By the functional equation of the Riemann zeta function, we can check that

$\displaystyle\dfrac{Z(2n,l)}{\widetilde{Z}(2n,l)}$
$\displaystyle=(-1)^{-l/2-n(n+1)/2}\pi^{-l-2ln+n(n+1)}2^{-l-2ln+n(n+1)+n+1}$
$\displaystyle\times\Gamma(l)\prod_{i=1}^{n}\Gamma(2l-2i).$

Moreover, by the duplication formula for the Gamma function, we show that

$\displaystyle\prod_{i=1}^{n}\Gamma(k-n/2-(i-1)/2)\prod_{i=1}^{n}\Gamma(k-(n+1)/2-(i-1)/2)$
$\displaystyle=\prod_{i=1}^{n}2^{-2k+n+i+1}\pi^{1/2}\Gamma(2k-n-i),$

and

$\displaystyle\prod_{i=1}^{n}\Gamma(l-(i-1)/2)\prod_{i=1}^{n}\Gamma(l-n/2-(i-1)/2)$
$\displaystyle\Gamma(l)\Gamma(l-n)^{-1}\prod_{i=1}^{n}(\Gamma(l-(2i-1)/2)\Gamma(l-2i/2))$
$\displaystyle=\Gamma(l)\Gamma(n-l)^{-1}\prod_{i=1}^{n}2^{-2l+2i+1}\pi^{1/2}\Gamma(2l-2i).$

This proves the assertion.

∎

The following theorem is essentially a refinement of a special case of [23,
Theorem 3.7].

######  Theorem 4.3.

Let $\\{F_{i}\\}_{i=1}^{d}$ be an orthogonal basis of
$S_{k}(\operatorname{Sp}_{n}(\mathbb{Z}))$ consisting of Hecke eigenforms, and
$\\{F_{i}\\}_{d+1\leq i\leq e}$ be a basis of the orthogonal complement
$S_{k}(\operatorname{Sp}_{n}(\mathbb{Z}))^{\perp}$ of
$S_{k}(\operatorname{Sp}_{n}(\mathbb{Z}))$ in
$M_{k}(\operatorname{Sp}_{n}(\mathbb{Z}))$ with respect to the Petersson
product. Then, for any $A\in\mathcal{H}_{n}(\mathbb{Z})_{>0}$ we have

$\displaystyle{\mathcal{G}}_{2n}^{k,k-l}(Z,A)$
$\displaystyle=c(n,k,l)\sum_{i=1}^{d}\frac{\Lambda(l-n,F_{i},\mathrm{St})}{\langle
F_{i},F_{i}\rangle}\overline{c_{F_{i}}(A)}F_{i}(Z)$
$\displaystyle+\sum_{i=d+1}^{e}c_{i,A}F_{i}(Z),$

where $c_{i,A}$ is a certain complex number. Moreover we have $c_{i,A}=0$ for
any $d+1\leq i\leq e$ if $l<k$.

######  Corollary 4.4.

Put

$\widetilde{\mathcal{G}}_{2n}^{k,k-l}(Z,A)=\widetilde{c}(n,k,l){\prod_{i=1}^{n}\Gamma(l-n+k-i)\over\prod_{i=1}^{n}\Gamma(2k-n-i)}{\mathcal{G}}_{2n}^{k,k-l}(Z,A),$

where $\widetilde{c}(n,k,l)=(-1)^{l(n-1)/2+n(n+1)/2}2^{3kn-2nl-n(n+1)/2-1}$.
Then we have

$\displaystyle\widetilde{\mathcal{G}}_{2n}^{k,k-l}(Z,A)$
$\displaystyle=\sum_{i=1}^{d}L_{alg}(l-n,F_{i},\mathrm{St})\overline{c_{F_{i}}(A)}F_{i}(Z)$
$\displaystyle+\sum_{i=d+1}^{e}\widetilde{c}_{i}F_{i}(Z),$

where $\widetilde{c}_{i,A}$ is a certain complex number. Moreover we have
$\widetilde{c}_{i,A}=0$ for any $d+1\leq i\leq e$ if $l<k$.

###### Proof.

For any Hecke eigenform $F$ in $S_{k}(Sp_{n}(\mathbb{Z}))$ we have

$\displaystyle
L_{alg}(l-n,F,\mathrm{St})=2^{e(k,l,n)}{\prod_{i=1}^{n}\Gamma(l-n+k-i)\over\prod_{i=1}^{n}\Gamma(2k-n-i)}\frac{\Lambda(l-n,F,\mathrm{St})}{\langle
F,F\rangle},$

where $e(k,l,n)=-kn-(n+1)l+3n^{2}/2+5n/2+1$. This proves the assertion. ∎

Now we give an explicit formula for
$\stackrel{{\scriptstyle\\!\\!\\!\\!\\!\\!\circ}}{{{\mathcal{D}}_{n,k}^{2}}}$.
The formula has been known in [15, p. 289 (the top of (4.2.1))] for any $n$
and $k$, and here we review it. Let

$T=\begin{pmatrix}R&W\\\ {}^{t}W&S\end{pmatrix}$

be a $2n\times 2n$ symmetric matrix of variables, where $R,W,S$ are $n\times
n$ matrices. We define $P_{\alpha}(T)$ for $0\leq\alpha\leq n$ by

$\det\Big{(}\begin{pmatrix}xR&W\\\
{}^{t}W&S\end{pmatrix}\big{)}=\sum_{\alpha=0}^{n}x^{\alpha}P_{\alpha}(T).$

Define a polynomial $Q_{n,k}^{2}(T)$ in $T$ as

$Q_{n,k}^{2}(T)=\sum_{\gamma=0}^{n}\binom{n}{\gamma}^{-1}\binom{2k-n+1}{n-\gamma}P_{\gamma}(T),$

and put

$\mathbb{D}_{n,k}^{2}=Q_{n,k}^{2}(\partial_{Z}),$

where $\partial_{Z}=\begin{pmatrix}{1+\delta_{ij}\over
2}{\partial\over\partial z_{ij}}\end{pmatrix}_{1\leq i,j\leq n}.$ We denote by
$\mathrm{Hol}(\mathbb{H}_{m})$ the $\mathbb{C}$-vector space of holomorphic
functions on $\mathbb{H}_{m}$. We then define a mapping
$\widetilde{\mathcal{D}}_{n,k}^{2}$ from $\mathrm{Hol}(\mathbb{H}_{2n})$ to
$\mathrm{Hol}(\mathbb{H}_{n})\otimes\mathrm{Hol}(\mathbb{H}_{n})$ as

$\widetilde{\mathcal{D}}_{n,k}^{2}(F)=\mathrm{Res}(\mathbb{D}_{n,k}^{2}(F))$

for $F\in\mathrm{Hol}(\mathbb{H}_{2n})$, where $\mathrm{Res}$ is the
restriction of a function on $\mathbb{H}_{2n}$ to
$\mathbb{H}_{n}\times\mathbb{H}_{n}$.

######  Proposition 4.5.

The operator $\widetilde{\mathcal{D}}_{n,k}^{2}$ maps an element $F$ of
$S_{k}(\operatorname{Sp}_{2n}(\mathbb{Z}))$ to
$S_{k+2}(\operatorname{Sp}_{n}(\mathbb{Z}))\otimes
S_{k+2}(\operatorname{Sp}_{n}(\mathbb{Z}))$.

###### Proof.

The assertion is written in [15, 4.2.1 in p. 289], and easily proved by a
simple computation using [15, Proposition 4.1]. ∎

We consider the special case $n=3$. Then we can derive the following formula:

$\displaystyle P_{0}(T)$ $\displaystyle=-(\det W)^{2},$ $\displaystyle
P_{1}(T)$ $\displaystyle=\sum_{i_{1}=1}^{3}\sum_{4<i_{2}<i_{3}\leq 6,1\leq
i_{4}<i_{5}<i_{6}\leq
6\atop\\{i_{4},i_{5},i_{6}\\}\cap\\{i_{1},i_{2},i_{3}\\}=\emptyset}(-1)^{i_{1}+i_{2}+i_{3}}\det\Big{(}T\begin{pmatrix}i_{1}&i_{2}&i_{3}\\\
1&2&3\end{pmatrix}\Big{)}\det\Big{(}T\begin{pmatrix}i_{4}&i_{5}&i_{6}\\\
4&5&6\end{pmatrix}\Big{)},$ $\displaystyle P_{3}(T)$ $\displaystyle=\det R\det
S,$ and $\displaystyle P_{2}(T)$ $\displaystyle=\det
T-P_{0}(T)-P_{1}(T)-P_{3}(T),$

and

$Q_{3,k}^{2}(T)=\dfrac{2(k-1)(2k-3)(k-2)}{3}P_{0}(T)+\dfrac{(k-1)(2k-3)}{3}P_{1}(T)+\dfrac{2(k-1)}{3}P_{2}(T)+P_{3}(T).$

To calculate the critical values of $L(s,F,\mathrm{St})$ for a Hecke eigenform
$F$ in $S_{k+2}(\mathrm{Sp}_{3}(\mathbb{Z}))$, we have to compare the
differential operator $\widetilde{\mathcal{D}}_{3,k}^{2}$ with
$\stackrel{{\scriptstyle\\!\\!\\!\\!\\!\\!\circ}}{{{\mathcal{D}}_{3,k}^{2}}}$.

######  Proposition 4.6.

We have

$\stackrel{{\scriptstyle\\!\\!\\!\\!\\!\\!\circ}}{{{\mathcal{D}}_{3,k}^{2}}}=-2^{-4}3(2k-1)(2k-3)(k-1)\widetilde{\mathcal{D}}_{3,k}^{2}.$

###### Proof.

We note that $\widetilde{\mathcal{D}}_{3,k}^{2}$ is a constant multiple of
$\stackrel{{\scriptstyle\\!\\!\\!\\!\\!\\!\circ}}{{{\mathcal{D}}_{3,k}^{2}}}$
as a mapping from $\mathrm{Hol}(\mathbb{H}_{6})$ to
$\mathrm{Hol}(\mathbb{H}_{3})\otimes\mathrm{Hol}(\mathbb{H}_{3})$. By [7,
(1.21)], we have

$\stackrel{{\scriptstyle\\!\\!\\!\\!\\!\\!\circ}}{{{\mathcal{D}}_{3,k}^{2}}}((\det
W)^{2})=\prod_{\nu=1}^{2}C_{3}(\nu/2)C_{3}(k-1-\nu/2).$

On the other hand, we have

$\displaystyle\widetilde{\mathcal{D}}_{3,k}^{2}((\det W)^{2})$
$\displaystyle=-\dfrac{2(k-1)(2k-3)(k-2)}{3}P_{0}(\partial_{Z})((\det W)^{2})$
$\displaystyle=-\dfrac{2(k-1)(2k-3)(k-2)}{3}C_{3}(1)C_{3}(1/2),$

where $C_{3}(s)=s(s+1/2)(s+1)$. This proves the assertion. ∎

######  Proposition 4.7.

For $A\in\mathcal{H}_{3}(\mathbb{Z})_{>0}$, let
$\widetilde{\mathcal{G}}_{6}^{k+2,2}(Z,A)$ be the element of
$S_{k+2}(\operatorname{Sp}_{3}(\mathbb{Z}))$ defined in Corollary 4.4. Then,
for any $A\in\mathcal{H}_{3}(\mathbb{Z})_{>0}$, we have

(4.2) $\displaystyle
c_{\widetilde{\mathcal{G}}_{6}^{k+2,k}(*,A)}(A_{1})={-3\Gamma(2k-3)\Gamma(2k-4)\over\Gamma(2k)\Gamma(2k-1)}2^{3k+7}(2k-1)(2k-3)(k-1)$
$\displaystyle\times\sum_{R\in
M_{3}(\mathbb{Z})}c_{6,k}\Big{(}\begin{pmatrix}A_{i}&R/2\\\
{}^{t}R/2&A\end{pmatrix}\Big{)}Q_{3,k}^{2}\Big{(}\begin{pmatrix}A_{i}&R/2\\\
{}^{t}R/2&A\end{pmatrix}\Big{)}.$

The following theorem is a key to proving our main results.

######  Theorem 4.8.

Let $F$ be a Hecke eigenform in $S_{k+2}(\operatorname{Sp}_{3}(\mathbb{Z}))$.
Let $\dim S_{k+2}(\operatorname{Sp}_{3}(\mathbb{Z}))=d$ and that
$F_{1},\ldots,F_{d}$ be a basis of
$S_{k+2}(\operatorname{Sp}_{3}(\mathbb{Z}))$ consisting of Hecke eigenforms
such that $F_{1}=F$. Let $A_{1},\ldots,A_{d}$ and $A$ be elements of
$\mathcal{H}_{3}(\mathbb{Z})_{>0}$ and put $a_{ij}=c_{F_{j}}(A_{i})$ for
$i=1,\ldots,d$ and $j=1,\ldots,d$. Moreover, put
$C(k;A_{i},A)=c_{\widetilde{\mathcal{G}}_{6}^{k+2,k}(*,A)}(A_{1})$. Here
$c_{6,k}(*)$ is the Fourier coefficient of the Siegel Eisenstein series
$E_{6,k}^{*}$ as stated at the beginning of this section. Suppose that
$\det(a_{ij})_{1\leq i,j\leq d}\not=0$. Then, we have

$\displaystyle|c_{F}(A)|^{2}L_{alg}(k-3,F,\mathrm{St})$
$\displaystyle=\frac{c_{F}(A)\begin{vmatrix}C(k;A_{1},A)&a_{1,2}&\ldots&a_{1,d}\\\
\vdots&\vdots&\ddots&\vdots\\\
C(k;A_{d},A)&a_{d,2}&\ldots&a_{d,d}\end{vmatrix}}{\det(a_{ij})_{1\leq i,j\leq
d}}.$

In particular if $\dim S_{k+2}(\operatorname{Sp}_{3}(\mathbb{Z}))=1$, then

$\displaystyle|c_{F}(A)|^{2}L_{alg}(k-3,F,\mathrm{St})=C(k;A,A).$

###### Proof.

By Corollary 4.4 and Proposition 4.6, for any $1\leq i\leq d$, we have

$c_{F}(A)C(k;A_{i},A)=\sum_{j=1}^{d}a_{ij}c_{F}(A)\overline{c_{F_{j}}(A)}L_{alg}(k-3,F_{j},\mathrm{St}).$

Thus the assertion can be derived by Cramer’s formula. ∎

######  Remark 4.9.

In view of Theorem 3.5, to confirm Conjecture 3.2 for the Ikeda-Miyawaki lift
$F$ in $S_{k+2}(\operatorname{Sp}_{3}(\mathbb{Z}))$, we have to compute the
$L$-value $L(m,F,\mathrm{St})$ at least one critical point $m$ such that
$1\leq m\leq k-1$ and $m\equiv 1\mod 2$. Besides $m=k-3$, we can obtain a
formula also for $|c_{F}(A)|^{2}L_{alg}(k-1,F,\mathrm{St})$ using Corollary
4.4. In this case $\widetilde{\mathcal{G}}_{6}^{k+2,0}(Z,A)$ is essentially
obtained by the restriction of $E_{6,k+2}^{*}$ without differential operator.
However, to get it we have to solve a system of linear equations of
$e$-variables with $e=\dim_{\mathbb{C}}M_{k+2}(Sp_{3}(\mathbb{Z}))$. On the
other hand, in the above proposition, we only have to solve a system of linear
equations of $d$-variables with $d=\dim
S_{k+2}(\operatorname{Sp}_{3}(\mathbb{Z}))$. For example, in the case $k=14$,
we have $d=7$ and $e=3$. This implies that the pullback formula with
differential operator allows us to compute the value in question efficiently.

We have an explicit formula for $F_{p}^{*}(T,X)$ for any half-integral matrix
$T$ over ${\mathbb{Z}}_{p}$ (cf. [20], [18],[19]), and an algorithm for
computing it (cf. [27]). Therefore, by Proposition 4.1, we can compute
$c_{6,k}(B)$ for any $B\in\mathcal{H}_{6}(\mathbb{Z})_{\geq 0}$. Therefore, by
using the above theorem together with Proposition 4.7, we can compute
$|c_{F}(A)|^{2}L_{alg}(k-3,F,\mathrm{St})$ with the aid of Mathematica, as
will be shown in Propositions 6.3 and 6.7.

##  5\. Computation of $L_{alg}(l_{1},l_{2};f)$

For our later purpose, we explain how to compute $L_{alg}(l_{1},l_{2};f)$
following [36]. Let $l_{1},l_{2},k$ be positive integers satisfying the
following conditions:

(5.1) $\displaystyle l_{1}-l_{2}\geq 3\text{ is odd},k\text{ is even and
}l_{1}+1<k\leq l_{1}+l_{2}-3.$

Put $\nu=k-l_{1}-1$, and define a function
$G_{\nu}(E^{*}_{l_{1}-l_{2}+1},E^{*}_{l_{1}+l_{2}-k+1})(\tau)$ on
$\mathbb{H}_{1}$ as

$\displaystyle G_{\nu}(E^{*}_{l_{1}-l_{2}+1},E^{*}_{l_{1}+l_{2}-k+1})(\tau)$
$\displaystyle=(2\pi\sqrt{-1})^{-\nu}\sum_{\mu=0}^{\nu}(-1)^{\nu-\mu}\begin{pmatrix}\nu\\\
\mu\end{pmatrix}\frac{\Gamma(l_{1}-l_{2}+1+\nu)\Gamma(l_{1}+l_{2}-k+1+\nu)}{\Gamma(l_{1}-l_{2}+1+\mu)\Gamma(l_{1}+l_{2}-k+1+\nu-\mu)}$
$\displaystyle\times\frac{\partial^{\mu}E^{*}_{l_{1}-l_{2}+1}}{\partial\tau^{\mu}}\frac{\partial^{\nu-\mu}E^{*}_{l_{1}+l_{2}-k+1}}{\partial\tau^{\nu-\mu}}.$

Here, for a positive even integer $l$, let $E_{l}^{*}=E_{1,l}^{*}$ is the
Eisenstein series defined in Section 4. Then,
$G_{\nu}(E^{*}_{l_{1}-l_{2}+1},E^{*}_{l_{1}+l_{2}-k+1})$ belongs to
$S_{k}(SL_{2}(\mathbb{Z}))$ (cf. [36, p. 147]). Let $f_{1},\ldots,f_{d}$ be a
basis of $S_{k}(SL_{2}(\mathbb{Z}))$ consisting of primitive forms. Then, by
[36, (77)], similarly to Corollary 4.4 we have

$\displaystyle
G_{\nu}(E^{*}_{l_{1}-l_{2}+1},E^{*}_{l_{1}+l_{2}-k+1})(\tau)=\gamma(k,l_{1})\sum_{i=1}^{d}L_{alg}(l_{1},l_{2};f)f_{i}(\tau),$

where $\gamma(k,l_{1})=\frac{(-1)^{k/2}\Gamma(k-1)}{2^{k-1}\Gamma(l_{1})}.$
Thus, similarly to Theorem 4.8, we obtain the following formula for
$L(l_{1},l_{2},f)$

######  Proposition 5.1.

Let $l_{1},l_{2},k$ be positive integers satisfying the conditions (5.1), and
$f_{1},\ldots,f_{d}$ be as above. Let $m_{1},\ldots,m_{d}$ be positive
integers, and put $a_{ij}=c_{f_{j}}(m_{i})$. Write

$G_{\nu}(E^{*}_{l_{1}-l_{2}+1},E^{*}_{l_{1}+l_{2}-k+1})(\tau)=\sum_{m=1}^{\infty}C(k,l_{1},l_{2};m){\bf
e}(m\tau).$

Suppose that $\det(a_{ij})_{1\leq i,j\leq d}\not=0$ and put $f=f_{1}$. Then we
have

$\displaystyle\gamma(k,l_{1})L_{alg}(l_{1},l_{2};f)$
$\displaystyle=\frac{\begin{vmatrix}C(k,l_{1},l_{2};m_{1})&a_{1,2}&\ldots&a_{1,d}\\\
\vdots&\vdots&\ddots&\vdots\\\
C(k,l_{1},l_{2};m_{d})&a_{d,2}&\ldots&a_{d,d}\end{vmatrix}}{\det(a_{ij})_{1\leq
i,j\leq d}}.$

######  Remark 5.2.

  * (1)

A result similar to above holds also for $k=l_{1}+1$ (i.e. $\nu=0$).

  * (2)

The above method cannot be applied to the case $l_{1}=l_{2}+1$ as it is
because $E_{1,2}^{*}$ is not a holomorphic modular form. But it is nearly
holomorphic, and a result similar to above can be obtained by modifying the
above method (cf. [33]).

##  6\. Examples

To give examples, for an even integer $l\geq 6$, we define a function
$\delta_{l}(\tau)$ on $\mathbb{H}_{1}$ as

$\delta_{l}(\tau)=\frac{1}{8\pi\sqrt{-1}}\Big{(}(\frac{l}{2}-1)E_{l-2}^{*}(4\tau)\frac{d}{d\tau}\theta(\tau)-\theta(\tau)\frac{d}{d\tau}E_{l-2}^{*}(4\tau)\Big{)},$

where

$\theta(\tau)=1+2\sum_{n=1}^{\infty}{\bf e}(n^{2}\tau).$

Then, $\delta_{l}(\tau)$ belongs to $S_{l+1/2}^{+}(\varGamma_{0}(4))$ (cf.
[26, p. 187]).

###  6.1. The case $k=10$

Let $k=10$ and $n=1$. Then $S_{21/2}^{+}(\varGamma_{0}(4))$ and
$S_{20}(\operatorname{SL}_{2}(\mathbb{Z}))$ are one dimensional, and
$h(\tau):=120\delta_{6}(\tau)$ is a Hecke eigenform in
$S_{21/2}^{+}(\varGamma_{0}(4))$ with the following Fourier expansion:

$h(\tau)=q-56q^{4}+360q^{5}-13680q^{8}\cdots,$

where $q={\bf e}(\tau)$. (cf. [26, p. 178]). The primitive form $f$ in
$S_{20}(\operatorname{SL}_{2}(\mathbb{Z}))$ corresponding to $h$ has the
following Fourier expansion:

$f(\tau)=q+456q^{2}+\cdots.$

Moreover, $S_{12}(\operatorname{SL}_{2}(\mathbb{Z}))$ is also one dimensional,
and it is spanned by the Ramanujan delta function, which is denoted by
$g_{12}$. Moreover, $S_{12}(\operatorname{Sp}_{3}({\mathbb{Z}}))$ is one
dimensional and spanned by $\widetilde{\mathcal{F}}_{h,g_{12}}$. To verify
Ikeda’s conjecture in this case, we provide several propositions.

######  Proposition 6.1.

Let $h,f$ and $g_{12}$ be as above. Let $A=\begin{pmatrix}1&0&1/2\\\
0&1&1/2\\\ 1/2&1/2&1\end{pmatrix}$. Then we have
$c_{\widetilde{\mathcal{F}}_{h,g_{12}}}(A)=-2^{7}\cdot 3^{3}\cdot 5$.

###### Proof.

Since $S_{12}(\operatorname{SL}_{2}(\mathbb{Z}))$ is one dimensional, we have

$I_{4}(h)\Big{(}\begin{pmatrix}z&O\\\
O&w\end{pmatrix}\Big{)}=\widetilde{F}_{h,g_{12}}(z)g_{12}(w).$

Hence we have

$\sum_{r\in M_{3,1}(\mathbb{Z})}c_{I_{4}(h)}\Big{(}\begin{pmatrix}A&r/2\\\
{}^{t}r/2&1\end{pmatrix}\Big{)}=c_{\widetilde{\mathcal{F}}_{h,g_{12}}}(A).$

By a simple computation, we have

$\displaystyle\sum_{r\in
M_{3,1}(\mathbb{Z})}c_{I_{4}(h)}\Big{(}\begin{pmatrix}A&r/2\\\
{}^{t}r/2&1&\end{pmatrix}\Big{)}$
$\displaystyle=c_{I_{4}(h)}\Big{(}\begin{pmatrix}A&\bf 0\\\ {}^{t}\bf
0&1\end{pmatrix}\Big{)}+\sum_{r\in\mathcal{R}_{1}}c_{I_{4}(h)}\Big{(}\begin{pmatrix}A&r/2\\\
{}^{t}r/2&1\end{pmatrix}\Big{)}+\sum_{r\in\mathcal{R}_{2}}c_{I_{4}(h)}\Big{(}\begin{pmatrix}A&r/2\\\
{}^{t}r/2&1\end{pmatrix}\Big{)},$

where

$\mathcal{R}_{1}=\Big{\\{}\pm\begin{pmatrix}0\\\ 1\\\
0\end{pmatrix},\pm\begin{pmatrix}0\\\ 1\\\
1\end{pmatrix},\pm\begin{pmatrix}1\\\ 0\\\
0\end{pmatrix},\pm\begin{pmatrix}1\\\ 0\\\ 1\end{pmatrix}\Big{\\}},\text{ and
}\mathcal{R}_{2}=\Big{\\{}\pm\begin{pmatrix}0\\\ 0\\\
1\end{pmatrix},\pm\begin{pmatrix}1\\\ 1\\\
1\end{pmatrix},\pm\begin{pmatrix}1\\\ -1\\\ 0\end{pmatrix}\Big{\\}}.$

∎

We see that we have $\det\Big{(}2\begin{pmatrix}A&\bf 0\\\ {}^{t}\bf
0&1\end{pmatrix}\Big{)}=\mathfrak{d}_{\big{(}\begin{smallmatrix}A&\bf 0\\\
{}^{t}\bf 0&1\end{smallmatrix}\big{)}}=8$. Hence we have

$c_{I_{4}(h)}\Big{(}\begin{pmatrix}A&\bf 0\\\ {}^{t}\bf
0&1\end{pmatrix}\Big{)}=c_{h}(8)=-13680.$

Similarly, for any $r\in\mathcal{R}_{1}$, we have
$\det\Big{(}2\begin{pmatrix}A&r/2\\\
{}^{t}r/2&1\end{pmatrix}\Big{)}=\mathfrak{d}_{\big{(}\begin{smallmatrix}A&r/2\\\
{}^{t}r/2&1\end{smallmatrix}\big{)}}=5$, and

$c_{I_{4}(h)}\Big{(}\begin{pmatrix}A&r/2\\\
{}^{t}r/2&1\end{pmatrix}\Big{)}=c_{h}(5)=360.$

For any $r\in\mathcal{R}_{2}$, we have
$\mathfrak{d}_{\big{(}\begin{smallmatrix}A&r/2\\\
{}^{t}r/2&1\end{smallmatrix}\big{)}}=1$ and
$\mathfrak{f}_{\big{(}\begin{smallmatrix}A&r/2\\\
{}^{t}r/2&1\end{smallmatrix}\big{)}}=2$. By [20] we have

$F_{2}(\begin{pmatrix}A&r/2\\\ {}^{t}r/2&1\end{pmatrix},X)=1-12X+32X^{2}.$

Hence we have

$\displaystyle c_{I_{4}(h)}\Big{(}\begin{pmatrix}A&r/2\\\
{}^{t}r/2&1\end{pmatrix}\Big{)}=c_{h}(1)2^{19/2}\alpha_{f,2}^{-1}F_{2}(\begin{pmatrix}A&r/2\\\
{}^{t}r/2&1\end{pmatrix},2^{-5/2}\alpha_{f,2})$
$\displaystyle=2^{19/2}(\alpha_{f,2}+\alpha_{f,2}^{-1}-2^{-1/2}\cdot
3)=c_{f}(2)-2^{9}\cdot 3=-1080.$

This proves the assertion.

######  Proposition 6.2.

  * (1)

We have $L_{alg}(22,f\otimes g_{12}\otimes g_{12})=\dfrac{2^{51}\cdot
5}{11\cdot 17}$.

  * (2)

We have $L_{alg}(7,g,\mathrm{St})=\dfrac{2^{15}}{3}$.

  * (3)

We have $L_{alg}(11,10;f)L_{alg}(17,16;f)=\dfrac{2^{34}\cdot 13}{3^{4}\cdot
5\cdot 17^{2}}$.

###### Proof.

The assertions (1) and (2) follow from [13, Table 2] and [9, Table1],
respectively. As explained in Remark 5.2, we cannot compute either
$L_{alg}(11,10;f)$ or $L_{alg}(17,16;f)$ individually using Proposition 5.1.
However, we have

$L_{alg}(11,10;f)L_{alg}(17,16;f)=L_{alg}(17,10;f)L_{alg}(16,11;f).$

Then the value in (3) can be computed using Proposition 5.1. ∎

######  Proposition 6.3.

Let $h,g_{12}$ and $A$ be as above. Then

$|c_{\widetilde{\mathcal{F}}_{h,g_{12}}}(A)|^{2}L_{alg}(7,\widetilde{\mathcal{F}}_{h,g_{12}},\mathrm{St})=\dfrac{2^{37}\cdot
3^{2}\cdot 11\cdot 13}{17}.$

###### Proof.

By Theorem 4.8, we have

$\displaystyle|c_{\widetilde{\mathcal{F}}_{h,g_{12}}}(A)|^{2}L_{alg}(7,\widetilde{\mathcal{F}}_{h,g_{12}},\mathrm{St})$
$\displaystyle=\frac{-3\Gamma(17)\Gamma(16)}{\Gamma(20)\Gamma(19)}2^{37}\cdot
19\cdot 17\cdot 9$ $\displaystyle\times\sum_{R\in
M_{3}(\mathbb{Z})}c_{6,10}\Big{(}\begin{pmatrix}A&R/2\\\
{}^{t}R/2&A\end{pmatrix}\Big{)}Q_{3,10}^{2}\Big{(}\begin{pmatrix}A&R/2\\\
{}^{t}R/2&A\end{pmatrix}\Big{)}.$

Thus the assertion can be verified by a computation with Mathematica. ∎

By Propositions 6.1, 6.2, 6.3 combined with (3.3), we can make sure the
following theorem.

######  Theorem 6.4.

Let $h$ and $g_{12}$ be as above. Then $C_{h,g_{12}}=2^{14}$, that is,
Conjecture 3.2 holds for this case, and so does Conjecture 3.1.

###  6.2. The case $k=14$

Let $k=14$ and $n=1$. Then $S_{28}(\operatorname{SL}_{2}(\mathbb{Z}))$ is two
dimensional and spanned by two primitive forms $f_{\pm}(\tau)$ such that

$f_{\pm}(\tau)=q+(-4140\pm 108\lambda)q^{2}+\cdots.$

The space $S_{29/2}^{+}(\varGamma_{0}(4))$ is also two dimensional and spanned
by $\delta_{6}(\tau)E_{8}^{*}(\tau)$ and $\delta_{8}(\tau)E_{6}^{*}(\tau)$.
Then, by a simple computation, we see that we can take a basis
$\\{h_{+},h_{-}\\}$ of $S_{29/2}^{+}(\varGamma_{0}(4))$ consisiting of Hecke
eigenforms such that

$\displaystyle h_{\pm}(\tau)=q+(-12332\pm 108\lambda)q^{4}+(123360\mp
1080\lambda)q^{5}+(1126824\mp 10152\lambda)q^{8}+\cdots$

with $\lambda=\sqrt{18209}$. We note that $f_{\pm}$ corresponds to $h_{\pm}$
under the Shimura correspondence. Let $g_{16}$ be the unique primitive form in
$S_{16}(\operatorname{SL}_{2}(\mathbb{Z}))$. Let $A$ be as in Section 6.1, and
$A_{1}=\begin{pmatrix}1&0&0\\\ 0&1&1/2\\\ 0&1/2&1\end{pmatrix}$, and
$A_{2}=\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}$. Then,
$S_{16}(\operatorname{Sp}_{3}({\mathbb{Z}}))$ is three dimensional and spanned
by the Hecke eigenforms $f_{1},f_{2},f_{3}$ such that

$c_{f_{1}}(A)=c_{f_{2}}(A)=c_{f_{3}}(A)=1,$
$c_{f_{1}}(A_{1})=3(29+\lambda)/26,c_{f_{2}}(A_{1})=3(29-\lambda)/26,c_{f_{3}}(A_{1})=16,$

and

$c_{f_{1}}(A_{2})=2(2293+19\lambda)/13,c_{f_{2}}(A_{2})=2(2293-19\lambda)/13,c_{f_{3}}(A_{2})=-40$

(cf. [13, Table 6] and Corrections to [13] bellow). We note that

$f_{1}=c\widetilde{\mathcal{F}}_{h_{+},g_{16}},\
f_{2}=c\widetilde{\mathcal{F}}_{h_{-},g_{16}}\text{ with some
}c\in\mathbb{Q}(\lambda)^{\times}.$

######  Proposition 6.5.

Let $h_{+},f_{+}$ and $g_{16}$ be as above. Then we have
$c_{\widetilde{\mathcal{F}}_{h_{+},g_{16}}}(A)=-2^{5}\cdot 567(-107+\lambda)$.

###### Proof.

Similarly to Proposition 6.1, we have

$c_{\widetilde{\mathcal{F}}_{h_{+},g_{16}}}(A)=c_{h_{+}}(8)+8c_{h_{+}}(5)+6(c_{f_{+}}(2)-2^{13}\cdot
3).$

This proves the assertion. ∎

######  Proposition 6.6.

  * (1)

We have

$L_{alg}(30,f_{+}\otimes g_{16}\otimes g_{16})=-\dfrac{2^{61}\cdot
21(81594529+464593\lambda)}{2586133225}.$

  * (2)

We have $L_{alg}(11,g_{16},\mathrm{St})=\dfrac{2^{19}\cdot 3^{2}\cdot
839}{13}$.

  * (3)

We have

$L_{alg}(15,14;f_{+})L_{alg}(25,24;f_{+})=\dfrac{2^{48}\cdot(26136063+188401\lambda)}{27962195625}.$

###### Proof.

(1) On [13, Table 2], only the norm of $L_{alg}(30,f_{+}\otimes g_{16}\otimes
g_{16})$ has been given, and here we give the value itself. For non-negative
integers $i_{1},i_{2},i_{3}$, let $b(i_{1},i_{2},i_{3},28,16,16;2)$ be the
quantity defined in [13, p. 166], and let

${\bf M}(4,f_{+},g_{16},g_{16})=2^{-57}\begin{pmatrix}28\\\
14\end{pmatrix}\begin{pmatrix}14\\\ 2\end{pmatrix}L_{alg}(30,f_{+}\otimes
g_{16}\otimes g_{16})$

as define in [13, p.164]. Let $\Phi(X)$ be the Hecke polynomial of the Hecke
operator $T(2)$ acting on $S_{28}(SL_{2}(\mathbb{Z}))$. Then we have
$\Phi(X)=-195250176+8280X+X^{2}$. Then by [13, Theorem 4.9], we have

$\displaystyle{\bf M}(4,f_{+},g_{16},g_{16})$
$\displaystyle=\Phi^{\prime}(c_{f_{+}}(2))^{-1}\sum_{i_{1}=0}^{1}\sum_{j_{1}~{}0}^{1-i_{1}}a_{j_{1}}c_{f_{+}}(2)^{1-i_{1}-j_{1}}b(i_{1},0,0,4,28,16,16;2)$

where $a_{0}=1$ and $a_{1}=8280$. Thus the assertion (1) can be verified by a
computation with Mathematica.

(2) The assertion (2) follows from [9, Table1]. (There are some errors on that
table, but they are corrected in www.neil-dummigan.staff.shef.ac.jp.)

(3) We have

$L_{alg}(15,14:f)L_{alg}(25,24;f_{+})=L_{alg}(25,14,f_{+})L_{alg}(24,15,f_{+}).$

Then the assertion (3) can be confirmed using Proposition 5.1. ∎

######  Proposition 6.7.

Let $h_{+},g_{16}$ and $A_{0}$ be as above. Then

$\displaystyle|c_{\widetilde{\mathcal{F}}_{h_{+},g_{16}}}(A)|^{2}L_{alg}(11,\widetilde{\mathcal{F}}_{h_{+},g_{16}},\mathrm{St})$
$\displaystyle=\dfrac{2^{49}\cdot
34862967(-222920204581+1281418453\lambda)}{633217975}.$

###### Proof.

We note that

$|c_{\widetilde{\mathcal{F}}_{h_{+},g_{16}}}(A)|^{2}L_{alg}(11,\widetilde{\mathcal{F}}_{h_{+},g_{16}},\mathrm{St})=L(11,f_{1},\mathrm{St}).$

Therefore, by Theorem 4.8 we have

$|c_{\widetilde{\mathcal{F}}_{h_{+},g_{16}}}(A)|^{2}L_{alg}(11,\widetilde{\mathcal{F}}_{h_{+},g_{16}},\mathrm{St})=\frac{\begin{vmatrix}C(14;A,A)&1&1\\\
C(14;A_{1},A)&c_{f_{2}}(A_{1})&c_{f_{3}}(A_{1})\\\
C(14;A_{2},A)&c_{f_{2}}(A_{2})&c_{f_{3}}(A_{2})\end{vmatrix}}{\begin{vmatrix}1&1&1\\\
c_{f_{1}}(A_{1})&c_{f_{2}}(A_{1})&c_{f_{3}}(A_{1})\\\
c_{f_{1}}(A_{2})&c_{f_{2}}(A_{2})&c_{f_{3}}(A_{2})\end{vmatrix}}.$

Here for $A,B\in\mathcal{H}_{3}(\mathbb{Z})_{>0}$, $C(k;B,A)$ is that defined
in Theorem 4.8. Thus the assertion can be verified by a computation with
Mathematica. ∎

By an argument similar to that in Theorem 6.4, we can deduce the following
theorem.

######  Theorem 6.8.

Let $h_{+}$ and $g_{16}$ be as above. Then, $C_{h_{+},g_{16}}=2^{18}$, that
is, Conjecture 3.2 holds true for $h_{+},g_{16}$ and so does Conjecture 3.1.

Corrections to [13]

* p. 165, l. -4: For ‘$P_{r,\nu_{1},\nu_{2},\nu_{3}}(*)$’, read ‘$P_{r-2,\nu_{1},\nu_{2},\nu_{3}}(*)$’.

* p. 166, l. 15: For ‘$c_{3,k_{1},k_{2},k_{3}}(*)$’, read

‘$c_{3,(k_{2}+k_{3}-k_{1}-r)/2,,(k_{1}+k_{3}-k_{2}-r)/2,(k_{1}+k_{2}-k_{3}-r)/2}(*)$’.

* p. 175, Table 6: The $T_{7}$-th Fourier coefficient of $f_{3}$ should be $-40$.

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# Human 3D keypoints via spatial uncertainty modeling

Francis Williams1,4, Or Litany2,3, Avneesh Sud1, Kevin Swersky1, Andrea
Tagliasacchi1,5
1Google Research, 2Stanford University, 3NVIDIA, 4NYU, 5University of Toronto

###### Abstract

We introduce a technique for 3D human keypoint estimation that directly models
the notion of spatial uncertainty of a keypoint. Our technique employs a
principled approach to modelling spatial uncertainty inspired from techniques
in robust statistics. Furthermore, our pipeline requires no 3D ground truth
labels, relying instead on (possibly noisy) 2D image-level keypoints. Our
method achieves near state-of-the-art performance on Human3.6m while being
efficient to evaluate and straightforward to implement.

##  1 Introduction

The ability to identify semantic human keypoints is a classical problem in
computer vision that finds many applications in the real world spanning from
gaming [15], athletics [14], robotics [12], and has now entered our household
in allowing us to control smart devices with our body (e.g. Google Nest Hub
Max and Facebook Portal). In the supervised setting, the identification of 2D
keypoints is typically considered a “solved” problem [10, 1], with most of the
recent research efforts either targeting 3D keypoint estimation [12, 9, 17],
or having moved to the more challenging unsupervised learning setting [5].

While the success of 2D keypoint estimation pipelines is largely due to the
ease of generating 2D annotations, specifying ground truth 3D keypoints from a
single image is ill-posed. Hence, researchers have proposed circumventing this
issue by leveraging statistical body models [12], motion-capture data [17], or
resorting to multi-camera setups [16, 4] where 3D is recovered from 2D
estimates via triangulation. In this paper, we perform several core technical
contributions _to the latter_ , towards the general objective of enabling
high-quality motion capture to “in-the-wild” settings. In particular, existing
multi-camera techniques [4, 6, 11, 13, 14] suffer from two significant
shortcomings: \raisebox{-.75pt}{1}⃝ they assume a training set with a
sufficiently large set of cameras to minimize self-occlusion and enable
accurate extraction of ground truth 3D labels; \raisebox{-.75pt}{2}⃝ they are
not equipped with spatial uncertainty estimates about their prediction.

If we hope to generalize the performance of multi-view setups to computer
vision and be robust to outliers in “in-the-wild” (i.e. outside capture studio
alike [3]), then both these shortcomings ought to be improved:
\raisebox{-.75pt}{1}⃝ our architecture requires only 2D labels which possibly
include noise, and \raisebox{-.75pt}{2}⃝ our model produces an interpretable
notion of keypoint uncertainty. We realize these improvements via a principled
approach to modelling uncertainty: representing a keypoint as a parameterized
probability distribution in 3D space that can be marginalized onto an image
plane to compute a loss in 2D. Our modelling approach is motivated by robust
statistics to enable training with large outliers in the labels. Finally,
while we use explicit 2D labels for our experiments, we remark that our method
could be trained on the output of an ensemble of off-the-shelf 2D keypoint
models, leading to a fully unsupervised 3D human pose pipeline.

##  2 Method

Figure 1: Model architecture – We predict keypoints as probability
distributions in 3D parameterized by a mean $\bm{\mu}$ and covariance $\Sigma$
representing the position and spatial uncertainty of a keypoint. As input, our
model accepts multiple images of a subject taken from multiple (known) camera
views. We feed these images through a 2D backbone using the architecture from
Xiao et al. [21] which predicts one heatmap per keypoint within each view. To
predict $\mu$, we triangulate the spatial mean of each heatmap using known
camera parameters. To predict $\Sigma$, we aggregate the bottleneck features
of the backbone network, and use a fully connected network to predict a
decomposition of $\Sigma$ that is guaranteed to be positive definite.

As input, we are given a multi-view setup consisting of $C$ cameras with known
extrinsics and intrinsics $\\{\bm{P}_{c}\in^{3\times 4}\\}$ capturing images
$\\{\mathcal{I}_{c}\in\mathbb{R}^{3\times w\times h}\\}$ where $w$ and $h$ are
the width and height of the image width. We then seek to predict $K$ spatial
distributions (over $\mathbb{R}^{3}$) corresponding to the $K$ keypoints of a
human subject. We assume as a prior that each keypoint follows a multivariate
t-distribution with mean $\bm{\mu}$ and scale matrix $\Sigma$:

$P(\bm{x})\propto[1+\frac{1}{\nu}(\bm{x}-\bm{\mu})\Sigma^{-1}(\bm{x}-\bm{\mu})]^{\frac{-(\nu+3)}{2}}$
(1)

where $\nu$ is a hyperparameter which controls the falloff of the
t-distribution. A large value of $\nu$, leads to a distribution which is more
robust to outliers (due to a faster falloff), but slower to train; we found
via a parameter sweep that setting $\nu{=}5$ led to good results. We remark
that the covariance of our distribution is a constant multiple of the scale
matrix, and therefore refer to $\Sigma$ as the covariance in what follows. Our
choice of parameterization is motivated by three factors:
\raisebox{-.75pt}{1}⃝ it leads to interpretable keypoints, since, similar to a
Gaussian, we can view the mean $\bm{\mu}$ as the most likely spatial location
of a keypoint, and the covariance $\Sigma$ as defining a spatial uncertainty;
\raisebox{-.75pt}{2}⃝ the log likelihood loss of a multivariate-t distribution
is robust to outliers, due to its rapidly decaying density;
\raisebox{-.75pt}{3}⃝ a multivariate t-distribution can be projected with
perspective onto a 2D version of itself, enabling 2D supervision.

Similarly to Algebraic Triangulation proposed by Iskakov et al. [4], our
method consists of a 2D backbone network applied to each view, followed by a
differentiable triangulation step. Unlike Iskakov et al. [4], our method
directly models keypoint uncertainty, outputting a 3D distribution (as opposed
to a point-wise quantity) while requiring only 2D labels which can contain
noise; see Figure 1 for a schematic of our architecture.

#### Predicting keypoint distributions

To output a multivariate t-distribution, we must predict its two parameters
$\bm{\mu}$ and $\Sigma$. To predict $\bm{\mu}$, we follow the Algebraic
triangulation method in [4]. First, a 2D backbone network accepts the $C$
views of a subject as input and produces $C$ collections of $K$ heatmaps
$\\{H_{c,k}\\}_{c\in[C],k\in[K]}$ for each keypoint in each view. We apply a
global softmax activation to each of these heatmaps to convert them to a
probability distribution, and take the spatial mean of the image to generate a
2D keypoint prediction

$\bm{\mu}_{c,k}^{\text{2D}}=\sum_{i,j=0}^{w,h}\left(\begin{matrix}i\\\
j\end{matrix}\right)\operatorname{softmax}(H_{c,k})[i,j]$ (2)

where $H_{c,k}[i,j]$ is the pixel at coordinate $(i,j)$ of the heatmap for
keypoint $k$ in view $c$, and $w,h$ are the width and height of the heatmap
images. Again following Iskakov et al. [4], we use the architecture from Xiao
et al. [21] as our 2D backbone, which is initialized with pretrained weights.

Similarly to Iskakov et al. [4], to output 3D mean estimates, we triangulate
the 2D predictions by solving a total least squares problem [2] which amounts
to minimizing a quadratic energy formed by the intersection of points in
projective space subject to a unit norm constraint.

Similarly to Kumar et al. [7], to predict the covariance $\Sigma$, we use the
bottleneck layer for each image, $z_{1},\ldots z_{C}$,
$z_{i}{\in}\mathbb{R}^{128}$ of the backbone network. Since the uncertainty
about a keypoint should be a function of all views of a person, we aggregate
these bottleneck features and predict a lower triangular matrix
$L{\in}\mathbb{R}^{3\times 3}$ as

$L=\rho\bigg{(}\frac{1}{C}\sum_{i=1}^{C}\phi(z_{i})\bigg{)}$ (3)

where $\rho$ and $\phi$ are both fully connected networks. We apply a shifted
ELU activation ($ELU(x)+1$) to the diagonal entries of $L$ to ensure they are
strictly positive and then compute sigma as

$\Sigma=LL^{T}$ (4)

which is positive definite by construction; note we omit the keypoint index
$k$ for brevity of notation.

#### Supervision

3D keypoint labels are often acquired via some kind of triangulation process
which may introduce bias into the labels, which will be learned downstream by
the network. Thus, we propose to learn the triangulation directly from a
collection of (possibly noisy) 2D keypoint labels in each camera view. To
train our model, we maximize the likelihood of these 2D labels under a
projected version of our 3D keypoint distribution. In contrast to training
directly on 3D data, our method aims to produce 3D predictions whose
projections agree with 2D keypoint labels under perspective projection.
Moreover, training in this way allows our method to naturally consider an
ensemble 2D labels (given, for example, by multiple 2D keypoint models or
multiple labellers), aggregating these results in a way that is robust to
noise and outliers.

#### Projecting keypoint distributions

To supervise on 2D labels, we project our keypoint distributions from 3D onto
the image planes of each. Furthermore, we require that the projections of our
3D distributions have an explicit form so we can evaluate a loss in each
image. We first observe that, similar to a Gaussian, for any point $x$ sampled
from a multivariate t-distribution (1), its image $x^{\prime}=Ax+b$ under an
affine transformation also follows a multivariate t with parameters
$\bm{\mu}^{\prime}=A\bm{\mu}+b$ and $\Sigma^{\prime}=A\Sigma A^{T}$. While
perspective projection is _not_ an affine transformation, we can rely on para-
perspective projection [2], an affine approximation to perspective to map our
distributions from 3D to 2D. We remark that [22] use a similar projection
technique for projecting Gaussians from 3D to 2D. Assuming that $x$ and
$\bm{\mu}$ are expressed in the same coordinate system as the camera (where
the positive $z$-axis is pointing in the “look-at” direction of the camera,
and the origin is the camera center), we can write the para-perspective
operation as

$x^{\prime}=\frac{1}{\|\bm{\mu}\|}\Pi x,\qquad\Pi=\begin{bmatrix}1&0&0\\\
0&1&0\\\ 0&0&0\end{bmatrix}$ (5)

which is equivalent to first projecting the point $x$ onto a plane centered at
$\bm{\mu}$ parallel to the image plane, then scaling the projection of $x$ by
the distance between the mean $\bm{\mu}$ and the camera origin. We remark that
because of the assumed camera orientation the norm $\|\bm{\mu}\|$ is the depth
of the keypoint. Applying the projection (5) to each keypoint and each view,
we get $C$ multivariate t-distributions on each image plane. To train, we
minimize the negative log likelihood of 2D keypoint labels
$k_{1},\ldots,k_{K}$ under these distributions. Taking the negative log of
(1), yields the following loss for each predicted keypoint
$\bm{\mu}_{i},\Sigma_{i}$ and 2D label $k_{i}$ within a given camera view $c$:

$\displaystyle L_{c}(\bm{\mu}_{k},\Sigma_{k})$
$\displaystyle=\log|\Sigma_{k}^{\prime}|+(\nu+2)\log\|k_{i}-\bm{\mu}_{k}^{\prime}\|^{2}_{\Sigma_{k}^{\prime-1}}$
(6)

where we use Mahalanobis norms to short-hand notation, and
$\bm{\mu}^{\prime}_{k}$ and $\Sigma_{k}^{\prime}$ are the para-perspective
projections of $\bm{\mu}_{k}$ and $\Sigma_{k}$ onto the image plane:

$\bm{\mu}^{\prime}_{k}=\Pi\frac{\bm{\mu}_{k}}{\|\bm{\mu}_{k}\|},\qquad\Sigma^{\prime}_{k}=\frac{1}{\|\bm{\mu}\|^{2}}\,\Pi\,\Sigma_{k}\,\Pi^{T}$
(7)

Our final loss function is simply the average of the losses of each keypoint
in each camera:

$\mathcal{L}=\frac{1}{CK}\sum_{c,k=1}^{C,K}L_{c}(\bm{\mu}_{k},\Sigma_{k})$ (8)

Figure 2: Results from the test set of Human3.6M trained on 2 cameras. The
green circles represent the 95% confidence ellipsoid of the predicted keypoint
distribution and the red points are the ground truth labels. The wrist
keypoints have greater spatial uncertainty since the model makes more error in
these areas. Also note the ground truth keypoints in Human3.6M that are used
for training are extremely precise, as they are obtained by tracking a
skeleton with a mo-cap system; this justifies the small spatial uncertainty
predicted by the model.

Method Avg Dir Disc Eat Greet Phone Pose Purch Sit SitD Smoke Photo Wait Walk
WalkD WalkT AT-Conf [4] 22.6 20.4 22.6 20.5 19.7 22.1 19.5 23.0 25.8 33.0 23.0
20.6 21.6 23.7 20.7 21.3 VT-Conf [4] 20.8 19.9 20.0 18.9 18.5 20.5 18.4 22.1
22.5 28.7 21.2 19.4 20.8 22.1 19.7 20.2 Ours 21.6 18.3 21.6 20.0 18.8 21.1
18.9 22.4 25.2 31.1 21.9 20.6 20.7 22.7 19.6 20.1

Table 1: MPJPE on all 4 cameras on Human3.6m

Method Avg Dir Disc Eat Greet Phone Pose Purch Sit SitD Smoke Photo Wait Walk
WalkD WalkT AT-Conf [4] 30.9 24.3 31.2 26.7 27.7 33.2 24.2 30.3 39.0 48.4 31.8
33.0 26.6 26.5 29.7 25.8 VT-Conf [4] 27.8 21.4 27.2 24.3 24.1 29.1 22.2 27.7
35.3 46.4 28.9 28.3 24.1 23.2 26.9 23.2 Ours 30.8 24.0 30.8 26.1 28.1 33.6
23.9 30.1 39.1 49.5 30.9 32.5 26.0 26.5 29.3 25.9

Table 2: MPJPE on a model trained on 2 cameras (and tested on 2 different
cameras) on Human3.6m

Method Avg Dir Disc Eat Greet Phone Pose Purch Sit SitD Smoke Photo Wait Walk
WalkD WalkT AT-Conf [4] 158.4 105.4 173.4 49.9 1,029.9 67.2 55.0 68.6 70.2
505.1 61.6 76.7 75.2 40.6 61.4 50.18 VT-Conf [4] 39.4 34.3 36.3 37.3 43.0 40.9
31.9 41.1 44.3 54.3 40.5 39.5 41.1 32.4 37.6 34.0 Ours 90.7 118.4 108.7 56.5
154.5 69.2 86.3 66.6 75.6 183.7 69.8 96.4 90.7 45.7 79.8 59.7

Table 3: MPJPE on all model trained on 2 antipodal cameras (and tested on 2
antipodal cameras) on Human3.6m.

##  3 Experiments and results

We evaluate our method on the widely used Human3.6M dataset [3]. Human3.6M
consists of video sequences of 7 subjects doing 15 actions taken from 4
cameras. Of these, 2 subjects are reserved for validation and 5 for testing.
We evaluate our technique against the two state of the art methods proposed in
Iskakov et al. [4]. While our method is similar in spirit to algebraic
triangulation (AT) from Iskakov et al. [4], we outperform it in our
experiments. In particular, our method is more robust to camera configurations
where the triangulation problem is _ill-conditioned_ ; see Section 3.2. Our
method performs slightly worse than the volumetric triangulation (VT) from
Iskakov et al. [4], however it is much faster to evaluate, conceptually
simpler, and requires no data preprocessing to compute bounding boxes, thus
making it easier to integrate into practical vision pipelines. Furthermore,
our method provides additional explainability of the results in the form of
explicit geometric uncertainty; see Figure 2 for example predictions from our
pipeline.

###  3.1 Comparison with 4 cameras – Table 1

As a baseline, we first compare our method to Iskakov et al. [4] using the
full 4 cameras available in the dataset. As in Iskakov et al. [4], we report
the MPJPE (Mean Per Joint Position Error), which measures the ground L2
distance between the ground truth keypoints relative to the pelvis. The
results are reported in Table 1. Our method slightly outperforms the algebraic
triangulation method, and performs slightly worse than the volumetric
triangulation method from Iskakov et al. [4].

###  3.2 Comparison with 2 cameras – Table 2 & 3

We now compare our model on the same train/test split as the four camera case,
except trained on a subset of 2 cameras and evaluated on 2 different cameras.
In both the train and evaluation setups the cameras are pointing in the same
direction, thus introducing possible occluded body parts which could increase
error. We find our method outperforms the algebraic method but underperforms
the volumetric method; see Table 2 for quantitative results.

#### Robustness Stress Test

Finally, as a robustness stress test we consider a 2 camera train/test split
as above, but where the cameras are antipodal. This camera configuration leads
to many predictions where the triangulation problem is ill-posed, causing
large outliers. We see that in this case, our method is much more robust than
algebraic triangulation, but much less robust than the volumetric
triangulation; see Table 3 for quantitative results.

##  4 Conclusions and Future Work

In this technical report, we presented a technique for learning 3D keypoints
from image (possibly noisy) image labels. Unlike prior work, our method
requires only 2D supervision and produces outputs which are equipped with an
interpretable notion of spatial uncertainty. Our method achieves near state-
of-the-art results on a standard benchmark while remaining extremely simple to
implement and fast to evaluate. We believe that such a method paves the way
for important future work in the problem of human keypoint detection.

#### Fully Unsupervised Keypoint Detection

We remark that many existing methods exist in the literature for image level
keypoint estimation (_e.g_. [18, 19, 20]). Since our pipeline requires only
noisy 2D annotations, we could leverage an ensemble of 2D predictions from
existing models to generate labels on the fly. Our model, would not only learn
to predict the mean of this ensemble, but also the covariance, providing an
explicit measurement for the consensus between labellers. Furthermore, the
backbone network in our model is, in general, not pretrained on the same
dataset as it is evaluated on (for example, our backbone was pretrained on
COCO [8]). Thus, an ensemble prediction would be fully unsupervised, and could
be used, for example, to construct very large datasets using only a few
calibrated cameras.

#### Temporal Predictions

Spatial uncertainty is a core requirement for many temporal models such as
Kalman Filters. A natural extension of our pipeline, is to aggregate keypoint
predictions over multiple frames to predict the next frame in a way that
minimizes uncertainty. Furthermore, leveraging the ensemble predictions
described in the previous paragraph, our pipeline could be extended to perform
fully unsupervised spatio-temporal keypoint prediction.

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