数理解析研究所講究録
第 1826 巻 2013 年 1-6

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ON SPECIAL VALUES OF TENSOR PRODUCT L-FUNCTIONSOF AN INNER FORM OF GSP(4) AND GL(2)KAZUKI MORIMOTO(GRADUATE SCHOOL OF SCIENCE, OSAKA CITY UNIVERSITY)

ABSTRACT. We $co$ nsider the Rankin-Slebrg integral which represents degree 8tensor product -functions for quaternion unitary groups and. Using thisintegral representation, we prove the algebriacity of special values,$L$$GL_{2}$

1. SET UP
Let $F$ be a number field and $E$ a quadratic extension. For eachthe similitude unitary group $G_{n}=GU(n, n)$ :$n\in \mathbb{N}_{j}$we define$G_{n}(F)=\{g\in GL(2n, E)|^{t}g^{\sigma}J_{n}g=\lambda_{n}(g)J_{n}, \lambda_{n}(g)\in F^{\cross}\}$where$\sigma$is non-trivial element in Gal $(E/F)$ and$J_{n}=[Matrix].$

Let $E\subset D$ be a quaternion algebra over $F$ . For $x\in D$ , wc mean the canonicalinvolution by . For a matrix $A=(a_{ij})$ with cntries in $D$ , we denote thc matrixbyLet us define the quatcrnion similitude unitary groupby$\overline{x}$$\overline{A}.$$(\overline{a_{ij}})$$H_{D}$$H_{D}(F)=\{g\in GL(2, D)|t_{\overline{g}}[Matrix] g=\lambda(g)[Matrix] 1\lambda(g)\in F^{\cross}\}.$

When $D\simeq M_{2}(F)$ , we have an isomorphism$H_{D}(F)\simeq GSp(4, F)=G_{2}(F)\cap GL(4, F)$ .

We note that we can take$\epsilon\in F^{\cross}$such that$D\simeq\{[Matrix]|a, b\in E\}.$

Thus we may suppose that $D\subset Mat_{2\cross 2}(E)$ , so that we can consider
as asubgroup of $GL(4, E)$ . In fact,
can be embedded int $oG_{2}$ , and we fix it Lct
us define a subgroup $H$ ofby
$H_{D}$$H_{D}$$G_{1}\cross G_{2}$$H=\{(g_{1}, h_{2})\in G_{1}\cross H_{D}|\lambda_{1}(g_{1})=\lambda_{2}(h_{2})\},$

and we regard $H$ as a subgroup of$G_{3}$

by the following embedding$H\ni([Matrix], [Matrix])\hookrightarrow[Matrix]\in G_{3}.$22. GLOBAL INTEGRAL
Let $P=MN$ denote the Siegel parabolic subgroup of$G_{3}$

where$M(F)=\{(\begin{array}{ll}g 00 \lambda\cdot(tg^{\sigma})^{-1}\end{array})|g\in GL_{3}(E), \lambda\in F^{\cross}\},$$N(F)=\{(\begin{array}{ll}1_{3} X0 1_{3}\end{array})|tX^{\sigma}=X\in Mat_{3\cross 3}(E)\}.$

Let be a character of
bycharacterof$v$$\mathbb{A}_{E}^{\cross}/E^{\cross}$$v\otimes\tau$

and$\tau$

a character of$\mathbb{A}_{F}^{\cross}/F^{\cross}$Then we define a$P(\mathbb{A}_{F})$$(\nu\otimes\tau)[(\begin{array}{ll}g 00 \lambda\cdot(tg^{\sigma})^{-1}\end{array})(\begin{array}{ll}1_{3} X0 1_{3}\end{array})]=v(\det g)\cdot\tau(\lambda)$.

. Then let $I(s, v\otimes\tau)$ denote the norLet
denote the modulus character of.
ofmalized degenerate principal series representation
Here we employ the normalized induction so thatis unitarizable when${\rm Re}(s)=0$ . Then for a holomorphic section
ofwe have the Siegel
Eisenstein series defined by$P(\mathbb{A}_{F})$$\delta_{P}$$G(\mathbb{A}_{F})$$Ind_{P(A_{F})}^{G(A_{F})}((\nu\otimes\tau)\cdot\delta_{P}^{s})$$I(s, \nu\otimes\tau)$$f^{(s)}$$I(s, \nu\otimes\tau)$$E(g, f^{(s)})= \sum_{\gamma\in P(F)\backslash G(F)}f^{(s)}(\gamma g)$.

This series is absolutely convergent in the right half plane[5] .
Let be an irreducible cuspidal representation ofacter ofsuch that${\rm Re}(s)> \frac{1}{2}$(Langlands$)$$GL_{2}(\mathbb{A}_{F})$$\sigma$and let$\chi$

be a char-$\mathbb{A}_{E}^{\cross}/E^{\cross}$(2.0.1)where$\chi|_{A_{F}^{\cross}}=\omega_{\sigma}$$\omega_{\sigma}$denotes the central character of . Since we have the isomorphism$\sigma$$G_{1}(F)\simeq (GL(2, F)\cross E^{\cross})/\{(a, a^{-1})|a\in F^{\cross}\},$

we can regard
as the irreducible cuspidal automorphic representation of
be the space of automorphic forms forand we denote it by . Let.
Let
be an irreducible cuspidal automorphic representation of.
study
globalThen wea
integral definedLetdenote the central character of$\sigma\otimes\chi$$G_{1}(\mathbb{A}_{F})$$\pi$$V_{\pi}$$\pi.$$(\Pi, V_{\Pi})$$H_{D}(\mathbb{A}_{F})$$\Pi$$\omega\Pi$by(2.0.2)$Z(f^{(s)}, \phi, \Phi)=\int_{Z(A_{F})H(F)\backslash H(A_{F})}E(f^{(s)}, h)\Psi(g_{1})\Phi(h_{2})dh$, where $Z=Z_{G}\cap H,$
denotes thefor $f^{(s)}\in I(s, \nu\otimes\tau),$and, and $h=(g_{1}, h_{2})\in H$ . Here in order for the integral (2.0.2) to becenter ofwell-defined, we assume that$\Psi\in V_{\pi}$$\Phi\in V_{\Pi}$$Z_{G_{3}}$$G_{3}$$\omega_{\Pi}\cdot\omega_{\sigma}\cdot\tau^{2}\cdot(v|_{A_{F}^{\cross}})^{3}=1.$

Proposition 2.1. For $Re(s)\gg 0$ , we have$Z(f^{(s)}, \Psi, \Phi)=\int_{S(A_{F})\backslash H(A_{F})}f^{(s)}(\eta h)W_{\Psi}(g_{1})B_{\Phi}(h_{2})dh$

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where
is the Bessel model ofwith respect to a non-split torus andis,
the Whittaker model ofandis defined as follows: Let us define the Bessel
subgroups $R$ ofby$\Phi$$B_{\Phi}$$\Psi$$W_{\Psi}$$S$$H_{D}$$R(F)=\{(\begin{array}{llll}a^{\sigma} 0 0 00 a 0 00 0 a^{\sigma} 00 0 0 a\end{array})(\begin{array}{llll}1 0 \epsilon b c0 1 c^{\sigma} b0 0 1 00 0 0 1\end{array})\in G_{2}(F)|a\in E^{\cross},$

Then a subgroup

$S$of

$H$

$b\in F,$$c\in E\}.$is defined by$S=\{(\varphi(r), r)|r\in R\}$

where we denote$\varphi[(\begin{array}{llll}a^{\sigma} 0 0 00 a 0 00 0 a^{\sigma} 00 0 0 a\end{array})(\begin{array}{llll}1 0 \epsilon b c0 1 c^{\sigma} b0 0 1 00 0 0 1\end{array})]=(\begin{array}{ll}a 00 a\end{array})(\begin{array}{ll}1 -b0 1\end{array}).$Remark. Our integral representation is a generalization to the similitude quaternion unitary case of Saha’s interpretation [11] of Furusawa’s integral [2]. Note that
we unfold the Rankin-Selberg integral involving the Siegel Eisenstein series on
directly without recourse to the Klingen Eisenstein series on. Thus even whenGSp(4), our local integral is totally different from Saha’s.
$G_{3}$$G_{2}$$H_{D}\simeq$

In order for our investigation to be non-vacuous, we assume that$\Pi$

has a Bessel model of non-split type.

We note that by the result of Li [6], any irreducible cuspidal automorphic representation ofhas a Bessel model of this type if $D$ does not split. Moreover
if $D\simeq Mat_{2\cross 2}(F)$ , i.e.,
GSp(4), has a Whittaker model or a Bessel modelof some type. If is associated to a holomorphic cusp form, it is non-generic, andPitale-Schmidt [8] shows that it does not have a Bessel model of split type. Thus
such automorphic representations satisfy the above assumption.
The uniqueness of Bessel model is expected for any irreducible admissible representations of $H_{D}(F_{v})$ . However as far as the author knows, there is no reference
which proves the uniqueness in general. For example, for unramified representations of GSp $(4, F_{v})$ , Sugano [12] proves the uniqueness. Then by the uniqueness ofBessel model and Whittaker model, we obtain$H_{D}(\mathbb{A})$$H_{D}\simeq$

$\Pi$

$\Pi$

.

$Z(s)= \prod_{v\not\in S}Z_{v}(W_{\Psi,v}, B_{\Phi,v}, f_{v}^{(s)})\cdot Z_{S}(W_{\Psi,S}, B_{\Phi,S}, f_{S}^{(s)})$

Here

$S$

is a finite set of places such that any place$v\not\in S$(1) 2 does not divide(2)
is unramified quadratic extension or(3)are unramified.
(4) $D(F_{v})\simeq Mat_{2\cross 2}(F_{v})$ .

is finite and satisfies$v$$E_{v}/F_{v}$$\Pi_{v},$ $\pi_{v},$$\nu_{v},$$E_{v}\simeq F_{v}\oplus F_{v}$$\tau_{v}$Then Furusawa and Ichino computed unramified local integrals explicitly.

4Proposition 2.2 (Furusawa-Ichino, Appendix in [7]). Suppose, we havemalized spherical vectors
and$W_{v},$$v\not\in S.$

For nor-$f_{v}^{(s)}$$B_{v}$$Z_{v}(s)= \prod_{i=1}^{3}L(6s+i, \nu|_{F_{v}^{x}}\cdot\epsilon_{E_{v}/F_{v}}^{i+3})^{-1}\cdot L(3s+\frac{1}{2}, \Pi x\sigma\cross(\nu|_{F^{\cross}})^{2}\cross\tau)$

where we normalize the measure on $H(F_{v})$ suitably, andcharacter ofcorresponding tovia local class field theory.

$\epsilon_{E_{v}/F_{v}}$is the quadratic$E_{v}$$F_{v}^{x}$

3. MAIN THEOREM
Assume that
$H_{D}(\mathbb{R})\simeq$

GSp$(4, \mathbb{R})$and$F=\mathbb{Q}.$We possibly have
. We suppose that the central characters ofand are trivial.
Suppose that the archimedean componentof is the holomorphic discrete
series of PGSp
with Harish-Chandra parameter $\ell(e_{1}+e_{2})$ with even integer
where we define$D\simeq Mat_{2\cross 2}(\mathbb{Q})$

$\Pi$

$\pi$

$\Pi$

$II_{\infty}$$(4, \mathbb{R})$$\ell$$e_{i}((t_{1} t_{2} t_{1}^{-1} t_{2}^{-1}))=t_{i} t_{i}\in \mathbb{G}_{m}.$Suppose that is a cuspidal automorphic representation associated to a new formof weight . Then we consider an automorphic form
as the automorphic
form onby extending it trivially, i.e.
$\sigma$$\Psi\in V_{\sigma}$$\ell$$G_{1}(\mathbb{A})$$\Psi(ag)=\Psi(g)$for$a\in \mathbb{A}_{E}^{\cross}$and$g\in GL(2, A_{\mathbb{Q}})$.

Theorem 3.1. Suppose that $\ell>6$ . Let
andbe arithmetic automorphic forms in the sense of Harris [4]. Then for an integerm such thatwe have$\Phi\in V_{\Pi}$$\Psi\in V_{\sigma}$$2<m \leq\frac{\ell}{2}-1,$$\frac{L(m,\Pi\cross\sigma)}{\pi^{4m}\langle\Psi\otimes\Phi,\Psi\otimes\Phi\rangle}\in\overline{\mathbb{Q}}$and$( \frac{L(m,\Pi\cross\sigma)}{\pi^{4m}\langle\Psi\otimes\Phi,\Psi\otimes\Phi\rangle})^{\tau}=\frac{L(m,\Pi^{\tau}\cross\sigma^{\tau})}{\pi^{4m}\langle\Psi^{\tau}\otimes\Phi^{\tau},\Psi^{\tau}\otimes\Phi^{\tau}\rangle}$for all$\tau\in$Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$. Here we define$\langle\Psi\otimes\Phi, \Psi\otimes\Phi\rangle=\int_{Z_{H}(A_{Q})H(\mathbb{Q})\backslash H(A_{Q})}|\Psi(g_{1})\Phi(h_{2})|^{2}dh$where we denote $h=(g_{1}, h_{2})\in H(A_{\mathbb{Q}})$ , and $dh$ is the Tamagawa measure on$H(\mathbb{A}_{\mathbb{Q}})$.

We can prove this by a similar way with Garrett-Harris [3]. For a detail of theproof, we refer to [7].

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3.1. Period Relation. Let
be an irreducible cuspidal automorphic representation of GSp
as in Theorem 3.1. Further we assume that is tempered
and non-endoscopic. We suppose that there exists an irreducible cuspidal automorphic representation
ofsuch that for every place such that,$(\Pi, V_{\Pi})$$\Pi$$(4, \mathbb{A}_{\mathbb{Q}})$$(\Pi_{D}, V_{\Pi_{D}})$$v$$H_{D}(\mathbb{A}_{\mathbb{Q}})$$H_{D}(\mathbb{Q}_{v})\simeq GSp(4, \mathbb{Q}_{v})$$\Pi_{v}\simeq\Pi_{D,v}.$

Then$\Pi_{D}$satisfies the condition in Theorem 3.1. Comparing the equations in The, we obtain the following relation.
andorem 3.1 for$\Pi$$\Pi_{D}$Corollary 3.1. For any arithmetic forms$\Phi\in V_{\Pi}$and$\Phi_{D}\in V_{\Pi_{D}}$, we have$\langle\Phi, \Phi\rangle/\langle\Phi_{D}, \Phi_{D}\rangle\in\overline{\mathbb{Q}}$and$(\langle\Phi, \Phi\rangle/\langle\Phi_{D}, \Phi_{D}\rangle)^{\tau}=\langle\Phi^{\tau}, \Phi^{\tau}\rangle/\langle\Phi_{D}^{\tau}, \Phi_{D}^{\mathcal{T}}\rangle$

for any$\tau\in$Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$.

Here we define the pairing$\langle\Phi_{D},$$\Phi_{D}\rangle$

by$\langle\Phi, \Phi\rangle=\int_{Z_{H_{D}}(\mathbb{A}_{Q})H_{D}(\mathbb{Q})\backslash H_{D}(\mathbb{A}_{Q})}|\Phi_{D}(h)|^{2}dh$where $dh$ is the Tamagawa measure on$H_{D}(\mathbb{A}_{\mathbb{Q}})$, and we define$\langle\Phi,$$\Phi\rangle$

similarly.

3.2. Remarks on Tbeorem 3.1.
3.2.1. critical point. The critical points in Theorem 3.1 does not cover all criticalpoints on the right half plane ${\rm Re}(s)>0$ . Indeed the critical points forandare not included due to the analytic property of Eisenstein series.
$s= \frac{1}{2}$$\frac{1}{6}$3.2.2. Split case. WhenGSp(4), similar results are proved by many people.
Furusawa [2] discovered an integral representation of this -function and he proved
the algebriacity at the rightmost critical point for Siegel cusp forms and elliptic
cusp form of full level. Pitale-Schmidt [9] extended his result with respect to thelevel of elliptic cusp forms, and Saha [10] extended with respect to both of levels ofSiegel cusp forms and elliptic cusp form. Saha [11] also proved the algebraicity forother critical points combining the pull-back formula and differential operators. On
the other hand, B\"ocherer-Heim [1] showed the algebraicity at all critical points inthe full modular balanced mixed weight case using Heim’s integral representation.
$H_{D}\simeq$$L$

3.2.3. Yoshida’s Conjecture. When the irreducible cuspidal automorphic representation of GSp
is associated to a Siegel cusp form, our result is compatible
with Yoshida’s calculation [13] on Deligne peirod.
$(4, \mathbb{A}_{\mathbb{Q}})$

Acknowledgements. This article is based on a talk delivered by the author atthe 2012 RIMS conference Automorphic forms and automorphic -functions. Theauthor thanks the organizer of the conference, Tomonori Moriyama and Atsushi
Ichino, for the opportunity to speak.
The research of the author was supported in part by Grant-in-Aid for JSPS
Fellows (23-6883) and JSPS Institutional Program for Young Researcher Overseas Visits project: Promoting international $yo$ ung researchers in mathematics andmathematical sciences led by OCAMI.
$L$

6REFERENCES[1] $B6$cherer, S., and B. Heim. “Critical values of -functions on $GSp_{2}\cross G1_{2}.$ ” Mamthmatische
Zeitschnfl 254, no. 3 (2006): 485-503.
(2) and their special values.“ Joumal fur die[2] Furusawa, M. “On -functions for GSp(4)Reine und Angewandte Mathematik 438 (1993): 187-218.
[3] Garrett, P., and M. Harris. “Special values of triple product -functions.” Amencan Joumalof Mathematics 115 (1993): 161-240.
[4] Harris, M. “Arithmetic vector bundles and automorphic forms on Shimura varieties. .” Inventiones Mathathematicae 82 (1985): 151-189.
[5] Langlands, R. On the Functional Equations Satisfied by Eisenstein Senes. Lecture Notes in$L$$L$$\cross GL$

$L$

$I$

Mathematics 544: Springer, 1976.
[6] Li, J.- . “Nonexistence of singular cusp forms.” Compositio Mathematica 83, no. 1 (1992):$S$43-51.
[7] Morimoto, K. “On $L$-functions for quaternion unitary groups of degree 2 and $GL$ (2) (withan Appendix by M. Furusawa and A. Ichino).” preprint[8] Pitale, A., and R. Schmidt. “Bessel models for lowest weight representations of GSp.
Intemational Mathematics Research Notices 2009, no. 7 (2009): 1159-1212.
[9] Pitale, A., and R. Schmidt. “Integral representation for -functions for GSp(4) $GL$ (2).
$(4, \mathbb{R})$$)$$L$$\cross$

Joumal of Number Theory 129 (2009): 1272-1324.
-functions for holomorphic forms on GSp(4) $GL$ (2) and their special values.”
Intemational Mathematics Research Notices 2009, no. 10 (2009): 1773-1837.
[11] Saha, A. “Pullbacks of Eisenstein series from $GU$ (3, 3) and critical -values for GSp(4)$GL$ (2).
Pacific Joumal of Mathematics 246, no. 2 (2010): 435-486.
[12] Sugano, T. “On holomorphic cusp forms on quaternion unitary groups of degree 2.” Joumalof the Faculty of Science. Section I A. Mathematics 31, no. 3 (1985): 521-568.
[13] Yoshida, H. “Motives and Siegel modular forms.” Amemcan Joumal of Mathematics 123(2001): 1171-1197.
[10] Saha, A.

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