cubic zeta 1ifting (Tomoyoshi IBUKIYAMA) (Department of Math., Graduate School of Sci., Osaka Univ. 1 \Re $\Phi^{\mathrm{J}}$ 1

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1 cubic zeta 1ifting (Tomoyoshi IBUKIYAMA) (Department of Math., Graduate School of Sci., Osaka Univ. 1 \Re $\Phi^{\mathrm{J}}$ 1 W. Kohnen } $SL_{2}(\mathbb{Z})$ \sigma ) $E^{l\downarrow}$ Kim-Shahi 2 1 Symmetric cube zeta 3 $[8],[9],$ [7] W. Kohnen $S_{2k-2}(SL_{2}(\mathbb{Z}))$ $2k-2$ $SL_{2}(\mathbb{Z})$ $S_{k-1/2}(\Gamma_{0}(4))$ $k-1/2$ $k-1/2$

2 138 $e(x)=e^{2\pi ix}$ $H_{1}$ $\theta(\tau)=\sum_{p\in \mathbb{z}}e(p^{2}\tau)(\tau\in H_{1})$ $\gamma=(\begin{array}{ll}a b4c d\end{array})\in\gamma_{0}(4)$ $( \theta(\gamma\tau)/\theta(\tau))^{2}=(\frac{-1}{d})(c\tau+d)$ $\theta(\gamma\tau)/\theta(\tau)$ 1/2 $H_{1}$ $f$ (\mbox{\boldmath $\tau$}) $\gamma\in\gamma_{0}(4)$ $f(\gamma\tau)=f(\tau)(\theta(\gamma\tau)/\theta(\tau))^{2k-1}$ $k-1/2$ $S_{k-1/2}(\Gamma_{0}(4))$ 4 1 new forms Kohnen $f\in S_{k-1/2}(\Gamma_{0}(4))$ $f$ $f( \tau)=\sum_{\mathrm{n}=1}^{\infty}\mathrm{c}(n)e(n\tau)$ $n\equiv 0$ or $(-1)^{k}$ -1 $\mathrm{m}\mathrm{o}\mathrm{d} $S_{k-1/2}(\Gamma_{0}(4))$ (Shimura, Kohnen) 4$ $\mathrm{a}\mathrm{a}$ $S_{k-1/2}^{+}(\Gamma_{0}(4))$ $c(n)=0$ Kohnen $S_{2k-2}(SL_{2}(\mathbb{Z}))\cong S_{k-1/2}^{+}(\Gamma_{0}(4))$. Hecke index 1 $f_{\hat{u}}$ index 1 ( Zagier Skoruppa )

3 138 2 \S 1 2 $GL_{2}(\mathbb{C})$ $\rho_{k,j}(g)=\det(g)^{k}$ Sym(j)(g) 4) $Sym(j)$ { $j$ $H_{2}$ 2 $Sp(2, \mathbb{r})$ 4 $H_{2}$ $F$ $(F _{k,j}[g])(z)=\rho_{k_{\dot{j}}},(cz+d)^{-1}f(gz)$ $g=(\begin{array}{ll}a BC D\end{array})\in Sp(2, \mathbb{r})$ $Sp(2,\mathbb{R})$ $H_{2}$ $F$ $Sp(2,\mathbb{Z})$ $\rho_{k,j}=\det Sym$ (D $\Phi(F)(\tau)=\lim_{\lambdaarrow\infty}F(_{0}^{\mathcal{T}}$ i0\lambda $\Phi(F)=0$ $F$ $\rho_{k,j}$ $S_{k,j}$ (Sp(2, $\mathbb{z}$)) ( $k,$ $j$ ) 3 (g) 1 $\sim$. $Z\in $\theta(z)$ $\det$ H_{2}$ $\theta(z)=\sum_{p\epsilon \mathbb{z}^{2}}e(^{t}pzp)$

4 (-1) 140 $Sp(2, \mathbb{z})$ $\Gamma_{0}(4)=\{\gamma=(\begin{array}{ll}A BC D\end{array})$ $\in Sp(2,\mathbb{Z})$ ; $C\equiv 0\mathrm{m}\mathrm{o}\mathrm{d} 4\}$ $\gamma=(\begin{array}{ll}a BC D\end{array})$ $\psi(\gamma)=(_{\neg\det(d}^{-1})$ $(\theta(\gamma Z)/\theta(Z))^{2}=\psi(\gamma)\det(CZ+D)$ $\theta(\gamma Z)/\theta(Z)$ 1/2 $\chi$ $\Gamma_{0}$ \epsilon $F$ $H_{2}$ $\chi$ $F(\gamma Z)=\chi(\gamma)$ ( $\theta$ (4) $\det Sym(j)$ $\gamma\in\gamma_{0}(4)$ ( $\gamma$z)/ $\theta$(z)) -1Sym $(j)$ ($CZ+$ D)F(Z) $S_{k-1/2,j}(\Gamma_{0}(4), \chi)$ $\psi$ $\chi$ $\chi)$ Hecke $S_{k-1/2,j}($\Gamma 0(4), $\chi$ Haupt type, $\chi=\psi$ Neben type $\ovalbox{\tt\small REJECT}$ 1 Neben type $1_{2}$ -1 2 $-1_{4}$ $\det(-1_{2})=1$ Neben type $\psi^{l})$ $l=0$ 1 $F\in S_{k-1/2,j}($\Gamma 0(4), $F(Z)= \sum a(t)\prec tr(tz)$ $T$ ( $T$ \text{ }\star\backslash \mathrm{f}\mathrm{f}\mathrm{f}\text{ }\acute{1}\overline{\mathrm{t}}f^{ }\mathrm{j}$) $\text{ }\mathrm{i}\mathrm{e}\text{ }\{\mathrm{i}\xi $a(t)$ $j+1\text{ }\grave{\grave{\text{ }} }J\mathrm{s}\text{ ^{}\mathrm{v}}\text{ ${}^{t}\mu\mu \mathrm{m}$od $4_{\text{ }}$ \supset $(\begin{array}{ll}1 11 1\end{array})$ $\text{ }$ $k+lt$ $(\begin{array}{ll}0 $\mathrm{f}\mathrm{h}\ovalbox{\tt\small REJECT}_{\grave{1}}\ovalbox{\tt\small REJECT}^{\backslash }\Phi$ _{ } ^{}-}C_{\text{ } }\mu\in \mathbb{z}^{2}$ 00 0\end{array})$ $(\begin{array}{ll}1 00 0\end{array})$ $(\begin{array}{ll}4a 2b2b 4c\end{array})$ (a, $b,$ $c\in \mathbb{z}$ ) $7\mathrm{F} $ $(-1)^{k+l}T\equiv$ $(\begin{array}{ll}0 00 1\end{array})$, $\text{ _{}\mathrm{n}}^{\mathrm{a}}$ $\#[]\mathrm{h}$

5 141 $a(t)=0$ $F$ $S_{k-1/2,j}^{+}$ $(\Gamma_{0}(4), \psi^{l})$ $\mathrm{a}\mathrm{a}$ $l=0$ $l=1$ (Haupt Neben) (Haupt Neben ) 2 $\psi^{l})$ (Hayashida and Ibukiyama) $S_{k-1/2,j}^{+}$ ( 0(4)) $k+l$ $Sp(2, \mathbb{z})$ index 1 $k+l$ $Sp(2, \mathbb{z})$ index 1 2 Hecke $j$ $S_{k-}^{+}$1/2,j $(\Gamma_{0}(4),\psi)\cong S_{j+\theta}$,2k-6(Sp(2, $\mathbb{z}$)) Spinor Zhuravlev 2 Euler factor 2 (1) $j$ $j$ (2) $j+3$ $\psi$ 1 (3)

6 142 $j=0,$ =3 $\det 3Sym(2k-6)$ $\det Sym(j)$ (Neben) (4) $Sp(2, \mathbb{r})$ $Sp(2)$ Ihara, Langlands $Sp(2)/\{\pm 1_{2}\}\cong SO$ (5) $SO$ (5) $Sp(2, \mathbb{r})$ 2 $SP(2, \mathbb{r})$ $Sp(2, \mathbb{r})$ 2 ( ) (4) (3) 2 $Sp(2)$ $SO$ (5) $Sp(2)$ $SO$ (5) SO $(3,2)$ 5 Spinor (Andrianov ). $F\in S_{k,j}$ (Sp(2, $\mathbb{z}$)) $L(s, F)= \prod_{\mathrm{p}}$ ( $1-\lambda(p)p^{-s}+(\lambda(p)^{2}-\lambda(p^{2})-p-1$ ) $p-2s-\lambda$ (p)p $-3s+p2\mu-4s$ ) $-1$ $\mathrm{a}\mathrm{a}$ $\mu=2k+j-3$ $\delta$ $\lambda(p^{\delta})$ Hecke $T(p^{\delta})=\{g\in M_{4}(\mathbb{Z})_{1}.{}^{t}gJg=p^{\delta}J\}$

7 143 $J=(\begin{array}{ll}0-1_{2}\mathrm{l}_{2} 0\end{array})$ $\text{ }$ 6 normalization $GSp(+2,\mathbb{R})=\{g=(\begin{array}{ll}A BC D\end{array})\in$ $M_{4}(\mathbb{R}),{}^{t}gJg=n(g)J(n(g)>0)\}$ $g$ $F\in S_{k,j}$ $(2, (Sp \mathbb{z})$ $F _{k,j}[g]=\rho_{k,j}(cz+d)^{-1}f(gz)$ $T(p^{\delta})= \bigcup_{\mathit{9}i}sp(2, \mathbb{z})g_{i}$ (disjoint) $F _{k,j}t(p^{\delta})=p^{\delta(2k+j-3)} \sum_{i}f [g_{i}]$ $F _{k,j}t(p^{\delta})=\lambda(p^{\delta})f$ { Zhuravlev Zhuravlev $\Gamma_{0}(4)\ni\gamma\prec$ $(\gamma, \theta(\gamma Z)/\theta(Z))$ $\tilde{\gamma}_{0}(4)$ $g\in M_{4}(\mathbb{R}),{}^{t}gJg=m^{2}g$ $g_{1}=m^{-1}g=(\begin{array}{ll}a BC D\end{array})$ $H_{2}$ $F _{k-1/2,j}$ [( $\phi$(z))] $g,$ $=Sym(j)(CZ+D)^{-1}\phi(Z)^{-2k+1}F(gZ)$ $K_{1}=((_{0}^{1}00p000p000$2 $p$ $00,p^{1/2}$) $K_{2}=($, $p)$ $p0$) $\tilde{\gamma}_{0}(4)$ $T_{i}(p)= \tilde{\gamma}_{0}(4)k_{i}\tilde{\gamma}_{0}(4)=\bigcup_{j}\tilde{\gamma}_{0}(4)\tilde{g_{j}}$

8 $\text{ }$ \check 144 $F _{k-}$ 1/2,pTi $(p)=p^{i(k+j-7/2)} \sum_{j}f _{k-1/2,\rho}[\tilde{g}_{j}]\psi(\det(d_{j}))$ $D_{j}$ 2 2 $p^{-1}g_{j}$ $S_{k-1/2,j}^{+}($ \Gamma 0(4), $\psi)$ $\psi)$ (cf $p=2$ $S_{k-1/2,j}^{+}($ 0(4) $\rangle$ [?] $)$. C $F\in S_{k-1/2,j}(\Gamma_{0}(4))$ \mathrm{t}\dot{\text{ }^{}\vee}\mathrm{c}$ 1(p) $\mathrm{f}\mathrm{p}\text{ }$ $rightarrow $T_{1}(p)F=\lambda(p)F$, $T_{2}(p)F=\alpha)(p)F$ $L(s, F)$ $=$ $\prod_{p}(1-\lambda(p)\psi(p)p^{-s}+(\mu(p)+p^{2k+2j-5}(1+p^{2}))p^{-2s}-\lambda(p)\psi(p)p^{2k+2j-s}+p^{4k+4j-6})^{-1}$ $a\leq b\leq d\leq c$ $a+c=b$ + $d=\delta$ $T$(pa, c, $p^{b},p$ $p^{d}$ ) $\psi(p^{\delta})t(p^{a+b},p^{a+d},p^{c+d},p^{b+c})$ (1) $k\geq 5$ ( $k\geq 5$ )

9 145.. $j$ $\dim S_{k-1/2,j}^{+}(\Gamma_{0})=\dim S_{j+}$ $(Sp(2, \mathbb{z}))$ 3,2k-6 $k,$ $j$ ( ) $k=3$ $j=$ $0$ ( ) 1 $\det 3Sym(j)$ (2) : $j$ $\sum_{k=0}^{\infty}s_{k-1/2,j}$ (\Gamma 0(4)) $A=$ { (4Z); } I $f$ $f(z)\in\oplus_{k=0}^{\infty}a_{2k}(sp(2,\mathbb{z}))$ $A_{2k}$ $\mathbb{z})$ (Sp(2, ) $2k$ $j=2$ $j=4$ $\mathrm{a}\backslash _{\mathrm{o}}$ 9 $S_{k-1/2,j}^{+}$ $\psi)$ $($ 0(4), ( ) \mathbb{z})$ $S_{k,j}$ (Sp $(2,

10 148 ( ) (3) 7Kim-Shahidi lifling $f$ 1 $f= \sum_{n=1}^{\infty}a$ (n)qn $L(s, f)= \prod_{pgood}(1-a(p)p^{-s}+p^{k-1-2s})^{-}1$ $1-a(p)p^{-s}+p^{k-1-2s}=(1-\alpha p^{-\ell})(1-\beta p^{-s})$ symmetric cube zeta $L(s, f, Sym(3))= \prod_{pgood}((1-\alpha^{\theta}p^{-s})(1-\alpha^{2}\beta p^{-s})(1-\alpha\beta^{2}p^{-s})(1-\beta^{3}p^{-s}))^{-1}$ $\mathrm{g}\mathrm{l}(4)$ Kim Shahidi $k=2$ (s, $f,$ $Sym(3)$ ) 3 Spinor zeta l Kim (1) 1 2 $\det Sym(k-2)$ (2) 1 $\Gamma_{0}(p)$ Iwahori subgroup $\mathrm{a}\mathrm{a}$ ( local rep. Steinberg rep. ) level 1 $\mathrm{a}\mathrm{a}$ level 1? Kim 2 $\Gamma_{0}(11)$ Ihara- Langlands 11 Steinberg ( level 11 Iwahori subgroup new form ) 1 ( ) 2 $k=12$ $SL_{2}(\mathbb{Z})$ Ramanujan Delta $\det Sym$ (10),

11 J. 147 $S_{13,10}(Sp(2, \mathbb{z}))$ $\dim S_{13,10}(Sp(2, \mathbb{z}))=2$ Euler 2-factor 3-factor $L(, s, \triangle, Sym(3))$ Euler factors [9] References [1] T. Arakawa, Vector Valued Siegel s Modular Forms of Degree Two and the Associated Andrianov $\mathrm{l}$-functions. Manuscripta Math. 44(1983), [2] M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkh\"auser, 1985, Boston-Basel-Stuttgart. [3] S. Hayashida, Skew-holomorphic Jacobi forms of index 1 and Siegel $\mathrm{w}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}\mathrm{s}_{)}$ modular forms of half integral Number Theory 106(2004), [4] T. Ibukiyama, Construction of half integral weight Siegel modular forms of from automorphic forms of the compact twist $Sp(n, \mathbb{r})$ $\mathrm{s}_{7}1$). $(2)$ J. reine $\mathrm{u}$. angew. Math. 359 (1985), [5] T. Ibukiyama, On Jacobi forms and SIegel modular forms of half integral weights, Comment. Math. Univ. St. Paul, 41 (1992), n0.2, [6] S. Hayashida and T. Ibukiyama, Siegel modular forms of half integral $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}_{\text{ }}$ weight and a lifting preprint. [7] T. Ibukiyama, Vector valued Siegel modular forms of half integral weight, preprint. [8] T. Ibukiyama, Conjecture on Shimura correspondence of Siegel modular forms of degree two, in preparation. [9] T. Ibukiyama, Numerical example of a cubic zeta function coming from a Siegel modular form, in preparation. [10] W. Kohnen, Modular forms of half-integral weight on 248 (1980), n0.3, , Math. Ann.

12 148 [11] N.-P. Skoruppa, Developments in the theory of Jacobi forms, AutOmorphic functions and their applications (Khabarovsk, 1988), , Acad. Sci. USSR, Inst. Appl. Math., Khabarovsk, See also MPIpreprint (1989). [12] G. Shimura, On modular forms of half integral weight, Ann. of Math. 97(1973), [13] R. Tsushima, An explicit dimension formula for the spaces of generalized Siegel modular forms with respect to, Proc. Japan Acad. Ser. $Sp(2, \mathbb{z})$ A Math. Sci. 59(1983), no. 4, [14] R. Tsushima, Dimension Formula for the Spaces of Siegel Cusp Forms of Half Integral Weight and Degree Two, Comm. Math. Univ. St. Pauli Vol. 52 No. 1(2003), [15] R. Tsushima, Dimension Formula for the Spaces of Jacobi Forms of Degree Two, in preparation. [16] V. G. Zhuravlev, Hecke rings for covering of a symplectic group, Math. Sbornik 121 (163) (1983), [17] V. G. Zhuravlev, Euler expansions of theta transforms of Siegel modular forms of half-integral weight and their analytic properties, Math. Sbornik 123 (165) (1984), Tomoyoshi Ibukiyama Department of Mathematics, Graduate School of Science Osaka University Machikaneyama 1-16, Toyonaka, Osaka 56&0043 Japan

Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara Sp(2, R) p

Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara 80 1963 Sp(2, R) p L holomorphic discrete series Eichler Brandt Eichler