数理解析研究所講究録1097 巻 1999 年 151-160

151

Moduli and Modularity of$(\mathrm{Q}, F)$-abelian varieties of$\mathrm{G}\mathrm{L}_{2^{- \mathrm{t}}}\mathrm{y}\mathrm{p}\mathrm{e}$中央大学理工 百瀬文之 (Fumiyuki Momose)
早稲田大学理工志村真帆呂 (Mahoro Shimura)

1

Introduction

Definition ($\mathrm{G}\mathrm{L}_{2}$-type)$A/\mathrm{Q}:\mathrm{Q}$-simple abelian variety of(1)(2) $dimA=[K*. \mathrm{Q}]$ .

$K=\mathrm{E}\mathrm{n}\mathrm{d}_{\mathrm{Q}}^{0}(A):=(\mathrm{E}\mathrm{n}\mathrm{d}_{\mathrm{Q}}(A))\otimes \mathrm{Z}\mathrm{Q}$$T_{l}(A)$: Tate-module ofRemark 1 If:$\rho_{l}$$A/\mathrm{Q}$is$\mathrm{A},$

1.1
(1)

.

-type, it has the following representation.

$G_{\mathrm{Q}}:= \mathrm{G}\mathrm{a}1(\overline{\mathrm{Q}}/\mathrm{Q})-arrow \mathrm{A}\mathrm{u}\mathrm{t}KV_{l}(A)=\prod\lambda|l\mathrm{A}\mathrm{u}\mathrm{t}K\lambda V_{\lambda}(A)$$V_{\lambda}(A):=V \iota(A)\bigotimes_{K\otimes \mathrm{Q}\iota}K_{\lambda},$$\mathrm{A}\mathrm{u}\mathrm{t}_{K_{\lambda}\lambda(}VA)\cong \mathrm{G}\mathrm{L}_{2}(K_{\lambda})$Classification of$A/\mathrm{Q}$: non-CM, where$A\sim B^{r}/\overline{\mathrm{Q}}$’$’/\overline{\mathrm{Q}}\sim$$F:=Z(\mathrm{E}\mathrm{n}\mathrm{d}^{0}(A))$-type,: a number field,$V_{l}(A):=T_{l}(A)\otimes_{\mathrm{Z}_{\mathrm{I}}}\mathrm{Q}_{l}$$\mathrm{G}\mathrm{L}_{2}$$\mathrm{G}\mathrm{L}_{2}$$\mathrm{G}\mathrm{L}_{2}$

.

.

-type

means isogenous over$\overline{\mathrm{Q}}$.

: totally real field.

$F:\dim B=[F:\mathrm{Q}]$$\mathrm{E}\mathrm{n}\mathrm{d}^{0}(B)=\{$$D/F$ :

totally indefinite division quaternion algebra over $F$,$\dim(B)=2[F:\mathrm{Q}]$

If the following diagram commutes, we call $B,$$\exists\varphi_{\sigma}$$(\mathrm{Q}, F)$-abelian variety of: $\sigma B-arrow B$ : $D$ -isogeny$\sigma_{B}-\varphi_{\sigma}arrow B$$\sigma a\downarrow$$\downarrow a$$\sigma Brightarrow\varphi_{\sigma}B$$\mathrm{G}\mathrm{L}_{2}$-type.

152(2)$A/\overline{\mathrm{Q}}$

: CM
$E$

$A\sim E^{\dim}(A)/\overline{\mathrm{Q}},$: CM elliptic curve.

Conjecture 1 (Taniyama-Shimura, Ribet-Serre)-type.
-simple abelian variety ofis isogenou8 over to some -simple factor of $J_{1}(N)$ .
$A/\mathrm{Q}:\mathrm{Q}$$\mathrm{G}\mathrm{L}_{2}$$\Rightarrow A$$\mathrm{Q}$$\mathrm{Q}$Remark 2 It is known that Conj. 1
(1)
(2)

$A$
$A$

is a CM-abelian variety.
is an elliptic curve andtrue for the following cases.

$27\{\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{d}(A)$Moduli of

2$i\mathit{8}$$\mathrm{G}\mathrm{L}_{2}$, (Wiles, Taylor, Diamond, Fontaine).

-type

We fix a totally indefinite quaternion division algebra $D/F$ .
$d:=\{$

1

if $D=F$disc $(D/F)$

if $D\neq F$.

We consider the following three families of isogeny classes of abelian varieties.

(A) { $(A,$(B) { $(B,$(C)$\iota)|A/\mathrm{Q}:\mathrm{Q}$-simple abelian variety of$\iota)|B/\overline{\mathrm{Q}}:(\mathrm{Q},$$F)$-abelian variety of$\mathrm{G}\mathrm{L}_{2}$$\mathrm{G}\mathrm{L}_{2}$-type,-type,.

square free$(\tau \mathfrak{n},d)=1$

Notation: Dirichlet character,$A_{1}(A)\sim A_{2}\Leftrightarrow\exists\chi$$A_{2}\sim A_{1,\chi}/\mathrm{Q}$by ).
: twist ofwith F-structure,(: coarse moduli space of(C),:-module,-modules $(a\in O_{D})_{f}$ason: canonical polarization.
$A_{1}$$A1,\chi$$M(d, \mathrm{m})/\mathrm{Q}$$\iota$$\mathcal{O}_{D}arrow \mathrm{E}\mathrm{n}\mathrm{d}(B)$$V\subset B[\mathrm{m}],$$V\cong O_{D}/\mathrm{m}O_{D;}$$\mathcal{O}_{D}$$\mathrm{L}\mathrm{i}\mathrm{e}B$$\mathrm{T}\mathrm{r}\mathrm{d}_{D/F(a)}=\mathrm{T}\mathrm{r}(a)$$O_{F}$$\mathcal{E}$$W=W(d,\mathrm{m})\subset \mathrm{A}\mathrm{u}\mathrm{t}_{\mathrm{Q}}M(d,\mathrm{m})$$a,$$\mathfrak{p}$$\chi$$(B, \iota, V, \epsilon)_{f}$: prime of$O_{F},$$\mathfrak{P}$;: prime of$O_{D}$.

: $F^{c}-,$$\mathrm{E}\mathrm{n}\mathrm{d}0(A)=\mathrm{M}\mathrm{a}\mathrm{t}(D)$$\iota:F(-\succ \mathrm{E}\mathrm{n}\mathrm{d}^{0}(B)=D$$\mathfrak{m}\subseteq \mathcal{O}\mathrm{I}\mathrm{I}F(M(d,\mathrm{m})/W)(\mathrm{Q})^{0}/(c_{)}\sim$(A)$\iota$

}

$/D$

}

-isog.

$/\sim$
$(A)$

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The following list is generators of $W$.
$W[a]$$\{$ $W[\mathfrak{p}]$$W[\mathfrak{p}]$: $(B, *)\mapsto(B/B[a], *^{J}),$ $(a,d\mathrm{m})=1,$ $(a\sim\alpha’inc+(p)\Leftrightarrow W[\alpha]=W[a’])$: $(B, *)-(B/B[\mathfrak{P}], *)/,$,: $(B, V, *)\mapsto(B/V[\mathfrak{p}], (V+B[\mathfrak{p}])/V[\mathfrak{p}], *’),$.
$\mathfrak{p}|d,$$\mathfrak{P}\supset \mathfrak{p}$$\mathfrak{p}|d$(class of $(B,$ $*)$ ) $=x\in(M(d,\mathrm{m})/W)(\mathrm{Q})^{0}$ ,$B$: simple abelian variety,$\forall \mathfrak{p}|\mathrm{m},$$\exists\sigma\in \mathrm{G}_{\mathrm{Q}},$.

$\exists\gamma_{a}\in Ws.t$$\sim;x\sim y\Leftarrow\Rightarrow\exists \mathfrak{n}(\neq 0)$$(C)$$(C\rangle$$\exists z\in(M(d,\mathrm{m}\mathfrak{n})/W)(\mathrm{Q})$$\exists y=\pi w_{\mathfrak{n}}(z)$$\mathfrak{p}|deg\gamma_{\sigma},$: ideal of$\mathcal{O}_{F},$$(\mathfrak{n},d\mathrm{m})=1$$w_{\mathfrak{n}}$

:

.

,

,

.

Here we consider $W$ as a subgroup of Aut$\pi$$\sigma(B, *)\cong\gamma_{a}(B, *)$$M(d,\mathrm{m}\mathfrak{n})$: $(B, V+V’, *)\mapsto(B/V’, (V+B[\mathfrak{n}])/V’, *’),$.

$V’\cong O_{D}/\mathfrak{n}\mathcal{O}_{D}$.

$M(d,\mathrm{m}\mathfrak{n})arrow M(d,\mathrm{m})_{f}$$(B, V+V’, *)\vdasharrow(B,V, *)$Theorem 1$(\mathrm{A})rightarrow 1:1(\mathrm{B})rightarrow 1:1(\mathrm{C})$.

Remark 3$(\mathrm{A})rightarrow 11(\mathrm{B})$$(\mathrm{B})rightarrow 1:1(\mathrm{C})$2.1

is Ribet-Pyle’s descent.
is a generalization of the theory of Elkies’s local tree.

Trees

Let be the tree attached to lattices of. More precisely, let : $\sigma Barrow B$ be a
-isogeny, then we have the following commutative diagram and . (We note that
corresponds to$G$$V_{\mathfrak{p}}(B)$$\varphi_{\sigma}$

$D$

$i$

$V_{\mathfrak{p}}(B)$$B[\mathfrak{p}^{\infty}].)$$B[\mathfrak{p}^{\infty}]-arrow\sigma B[\mathfrak{p}^{\infty}]$$\backslash C)\nearrow \mathcal{O}_{D}$.

-cyclic$B[\mathfrak{p}^{\infty}]/B[\mathfrak{p}^{i}]\cong B[\mathfrak{p}^{\infty}]$-cyclic map gives the distance between
acts on the lattices,
acts on $G$ .

Hence, thisBecause(1)$\mathfrak{p}\{d$$G_{\mathrm{Q}}$,$\mathcal{O}_{D}$$G_{\mathrm{Q}}$$V_{\mathfrak{p}}(B)$and$\sigma_{V_{\mathfrak{p}}(B)}$

.

154

(a)

$G_{\mathrm{Q}}\cap G$

is fixed point free (in this case$\bullet$(b)$G_{\mathrm{Q}}\cap G$(2)$\mathfrak{p}|d$: the attached object is minimal.

has fixed points (in this case.

),$\mathfrak{p}|\mathrm{m}$$\mathfrak{p}(\mathrm{m})$,:fixed point, and the attached object is minimal.

,$B[\mathfrak{p}^{\infty}]$

has a maximal$\mathcal{O}_{D}$-submodule$V=B[\mathfrak{P}]\subset B[\mathfrak{p}],$$\mathfrak{P}|\mathfrak{p},$$\mathfrak{P}^{2}=\mathfrak{p}O_{D}$.

$B[\mathfrak{p}^{\infty}]arrow B[\mathfrak{p}^{\infty}]/B[\mathfrak{P}]-arrow(B[\mathfrak{p}^{\infty}]/B[\mathfrak{P}])/(B[\mathfrak{p}^{\infty}]/B[\mathfrak{P}])[\mathfrak{P}]=[B[\mathfrak{p}^{\infty}]/[B[\mathfrak{p}]=[B[\mathfrak{p}^{\infty}]$

Hence, in this case$G$

has exactly two vertices.

.

155

(a)

$G_{\mathrm{Q}}\cap G$

is fixed point free (in this case.

$\mathfrak{p}|\delta$–(b)$G_{\mathrm{Q}}\cap G$),$\bullet$: the attached object is minimal.

has fixed points (in this case.

$\mathfrak{p}(\delta)$–,$\bullet$:fixed point, and the attached object is minimal.

$(B)arrow(C)$we choose locally minimal model for each .
Global condition is $(B, *)\mapsto w[a](B, *)$ .
$\mathfrak{p}$If there are more than two minimal points, we identify these by $(B, *)arrowarrow w_{\mathfrak{p}}(B, *)$ and$\sim$$(C)$.

$(C)’$Here: $(M(d, \mathrm{m})/W)(\mathrm{Q})0=(,\prod_{W<W}(M(d,\mathrm{m})/W’)(\mathrm{Q})00)/W$,$x\in(M(d,\mathrm{m})/W)(\mathrm{Q})^{00}\Leftrightarrow G_{\mathrm{Q}}$Invariant

2.2$M(d,\mathrm{m})_{x}$the fiber of$x$transitively.

$\delta$$(\mathrm{A})rightarrow(\mathrm{B})rightarrow(\mathrm{C})’$.

$A/\mathrm{Q}rightarrow B/\overline{\mathrm{Q}}rightarrow x/\mathrm{Q}$$A/\mathrm{Q}$acts on.

: Neben character of order 2-power.
$\epsilon$

Definition(A)$S_{A’}:=\{\mathfrak{p}\subset O_{F}|\mathfrak{p}|D(K/F), \mathfrak{p}(2\}$.

$\delta_{A’}:=$$\prod_{\prime,\mathfrak{p}\in^{s}A}\mathfrak{p}$.

(B)$\delta_{B}.\cdot=s_{B}.\cdot=\mathfrak{p}\epsilon^{g}\mathrm{t}^{\mathfrak{p}\subset}\mathfrak{p}.(^{\sigma}F-_{\mathrm{P}^{\mathrm{a}}}\mathrm{t}_{0}\mathrm{k}\mathrm{e}\mathrm{r}\psi)\mathfrak{p}\mathrm{r}=\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{h}\mathrm{c}\mathrm{a}\mathrm{n}.\mathrm{d}\mathrm{o}\mathrm{d}\mathrm{d}\mathrm{d}\mathrm{e}’ \mathrm{g}\mathrm{r}AO|\sigma\in \mathrm{q}’ D-\mathrm{i}\mathrm{S}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{y}\psi\sigma.B-arrow B\sigma\}\exists c_{\mathrm{f}}\exists \mathrm{e}\mathrm{e}$$\delta_{B’}:=2$(C)-primary part of$\delta_{B}$.

$S_{C’}:=\{\mathfrak{p}\subset O_{F}|\exists\gamma\in W’, \gamma=w[a]w_{\mathrm{b}}w_{\mathrm{c}}, (a,d\mathrm{m})=1, b|d, \mathrm{c}|\iota \mathrm{n}, \mathfrak{p}|\mathrm{b}\mathrm{c}\}$$\delta_{C’}:=\prod_{\mathfrak{p}\in \mathit{8}_{c}\prime}\mathfrak{p}$$\delta_{C^{J’}}:=2$.

-primary part of$\delta_{C’}$

.

.

156
Theorem 2$\delta_{B}=\delta_{C’}$(We denote this value ).
$\delta$$\delta_{A’}=\delta_{B’}=\delta c\prime\prime$.

Invariant

2.3

Definition
andLet$\mathfrak{p}|d\mathrm{m}$$e_{s}=\{$$\mathfrak{p}|p$

. For

$e$

$x\in(M(d,\mathrm{m})/W’)(\mathrm{Q})^{00}$2 if$W’\cap \mathrm{A}\mathrm{u}\mathrm{t}(B, *)/\overline{\mathrm{F}}_{p}\neq(1)$if$W’\cap \mathrm{A}\mathrm{u}\mathrm{t}(B, *)/\overline{\mathrm{F}}_{p}=(1)$1$e=\#\mathrm{A}\mathrm{u}\mathrm{t}(B, *)/\overline{\mathrm{F}}_{p}$3and$e_{s}$,.

.

Modularity(class of $x=(B,$ $*)$ ) $\in(M(d,\mathrm{m})/W’)(\mathrm{Q})^{00}$ .
satisfy the following conditions;
We assume that$\mathfrak{p}|d\mathrm{m},$$\mathfrak{p}|p\neq 2$$\mathfrak{p}|\mathrm{m}$$(B[\mathfrak{p}]/\overline{\mathrm{F}}_{p})^{\mathrm{e}\mathrm{t}}\neq’(0)$$(C1)\{$$\mathfrak{p}|d$.

$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\mathrm{T}\mathrm{a}\mathcal{O}_{F}/\mathfrak{p}\otimes\overline{\mathrm{F}}/\mathrm{p}\mathrm{n}\overline{\mathrm{F}}_{\mathrm{p}}B[\mathfrak{P}]=1,$$(\mathfrak{P}\subset O_{D}, \mathfrak{P}|\mathfrak{p})$.

$\mathfrak{p}|\mathfrak{m}$$(C2)\{$$\mathfrak{p}|d$(1)$B[p]/\overline{\mathrm{F}}_{p})^{\mathrm{e}’}\mathrm{t}=(0)$$\frac{1}{2}e\leq p-1,$,
if $e_{s}=1,4$ \dag er$e\neq p+1$

$e$

.

$\{$

$\frac{1}{2}2^{t}e_{s}e\leq p-1$

or $\frac{1}{2}2^{t}e_{s}e=2(p-1)$

otherwise.

Here $t:=v_{2}(p-1)$ .
(2),$B[p]/\overline{\mathrm{F}}_{p})^{\mathrm{e}’\mathrm{t}}\neq(0)$$\frac{1}{2}e\leq p-1,$

if $e_{s}=1,4$ \dag er .

$e\neq p+1$

$e$

$\{$

otherwise.

$\frac{1}{2}e_{s}e\leq p-1$

Proposition 1 If

$x$

and

$\mathfrak{p}|d$satisfy$(C1)\Rightarrow \mathfrak{p}|\delta$

Proposition 2 We assume the condition $(C1)$ for.

$x$and . We set $e=p^{r}e’,$ $p\{e’$ . Then$\mathfrak{p}$

157

(i)

,

$\mathfrak{p}|\mathrm{m}$(mod$\mathrm{N}(\mathfrak{p})\equiv 1$$\{$$\zeta_{p^{r}}\in F_{\mathfrak{p}}$(ii)$\mathfrak{p}|d$

)

.

,
(mod

$\mathrm{N}(\mathfrak{p})\equiv-1$$\{$$v_{\mathfrak{p}}(p)\geq\varphi(p^{r})$

(iii)

$e’$

$B/\overline{\mathrm{F}}_{p}\sim E^{\dim B},$

$E:\mathit{8}upersingular$

$e’$

)

.

elliptic curve,

Theorem 3$x\in(M(d,\mathrm{m})/W’)(\mathrm{Q})00,$

the object of$x$$\exists \mathfrak{p}|d\mathrm{m},$$\mathfrak{p}|p\neq 2$such that satisfies $(C1)$ and $(C2)$ , thenis modular.

Remark 4 Under the above condition, we can show that
The Galois Image on

Normalizer of Split Cartag subgroupif$\mathfrak{p}|\mathfrak{m}$Normalizer of Cartan subgroupif$\mathfrak{p}|d$$A[\mathfrak{p}]\subset\{$andRemark 5 We omit the caseadditional conditions. That is Theorem 4.
$\mathfrak{p}|\mathfrak{m}$$\mathfrak{p}|3$.

. But we can show the modularity with

Notation$W’(\mathfrak{p})<W’$: index 2,$\gamma\in W’(\mathfrak{p}),$$(\deg\gamma,\mathfrak{p})=1,$$\deg\gamma=a\mathrm{b}\mathrm{c}_{f}(a,d\mathrm{m})=1,$$\mathrm{b}|d,$$\gamma=w[a]w_{\mathfrak{h}}w_{\mathrm{c}}$$\mathrm{c}|\mathrm{m}$,.

Theorem 4$x\in(M(d,\mathrm{m})/W’)(\mathrm{Q})^{00},$$\exists \mathfrak{p}|d\mathrm{m},$$(\mathrm{C}0)(M(d,\mathrm{m})/W’(\mathfrak{p}))_{x}(\mathrm{R})=\phi$$(\mathrm{c}1)’(B[\mathfrak{p}]/\overline{\mathrm{F}}_{3})^{e’t}\neq(0)$$(\mathrm{C}2)^{}$$e_{S}=1,$$\Rightarrow the$.

$\mathfrak{p}|p=3$

such that.

$e=2$ (or $2\cross 3$ -power).

object of$x$

is modular.

Remark 6 Moreover, we have a criterion of modularity for the case of the action ofthe
on the tree has more than two fixed point8 byof Skinner- Wiles.
$u\mathit{8}ing$$re\mathit{8}ul\partial$$\mathrm{G}_{\mathrm{Q}}$

158

4

Q-curves

Let $D=F=\mathrm{Q},$ $\dim B=1$ .
(square free), $M(1, N)=X_{0}(N)$ .
We set$\mathrm{m}=N$

Theorem 5 If $x\in(X_{0}(N)/W’)(\mathrm{Q})^{00}$ satisfies the following condition8, then the objectof is modular.
$x$(A) In the case of $p\geq 5$ ,$(C1)$$(C2)$$x/\overline{\mathrm{F}}_{\mathrm{p}}\neq \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}$$e$$e_{s}$

point.

$p$1

2$\geq 5(\neq 3)$$1$

4$\geq 5(\neq 3)$$1$

6$\geq 13(\neq 7)$$2$2$\geq 5(\neq 3)$$2$

4$\geq 13(\neq 5)$$2$

6$\geq 13(\neq 7)$(B) In the case of $p=3$ ,$(C0)$$(X_{0}(N)/W’(3))_{x}(\mathrm{R})=\phi$$(C2)/$$e_{s}=1,$

$e=2$

.

.

Example 1(4-1) $N=p\geq 5,$

If$\neq 7$;$x\in(X\mathrm{o}(p)/w_{p})(\mathrm{Q})$is non-cuspidal point and if$x$(mod ) is non-supersingular$p$then : modular.
If $p=7$ and $e\neq 6$ , then modular.

$point_{f}$$x$(4-2) $N=35,39(\Rightarrow X_{0}(N)/w_{N}=\mathrm{P}^{1})$;$x\in(X_{0}(N)/w_{N})(\mathrm{Q})$ : non-cuspidal $point\Rightarrow x$ : modular.
If $N=65_{f}$ then $X_{0}(N)/W_{N}$ is an elliptic curve with positive rank.
$x\in(X_{0}(N)/w_{N})(\mathrm{Q}):$ non-cuspidal $point\Rightarrow x$ : modular.

(4-3) $N=p=3$ ;By using the Fricke’8 explicit defining equation of $X_{0}(3)$ :$j:=j(\tau)=27(\mathcal{T}+1)(9\mathcal{T}+1)^{3}/\tau$$\{$$w_{3}(\mathcal{T})=1/\tau$,.

we obtain the conditions in Theorem. 4 explicitly as follows ( : the valuation at 3).
and $|a|<2$ .
implies is a root of $X^{2}-aX+1=0,$$v_{3}(a)\leq
3$ .
$(C1)$ implies $v_{3}(j)\leq 0$ . It implies $v_{3}(a)\geq 1$ or$v_{3}$$(C\mathrm{O})$$\tau$$a\in \mathrm{Q}$

159

If

is a root of $X^{2}-aX+1=0$ where
thengive.s modular Q-curve.
$\tau$$a$satisfies above conditions;$j(\tau)$5$\mathrm{Q}\mathrm{M}$-abelian surfaces

Let $D/Q$ be a indefinite quaternion division algebra. (i.e. $D\neq F,$ $F=Q,$ $\dim B=2$ ): Shimura curve.

$M(d, m)/\mathrm{Q}$

Theorem 6 For $p|dm,$ $p\neq 2$ ,If $x\in(M(d, m)/W’)(\mathrm{Q})^{00}$ satisfies the following conditions, then the object of is mod$x$

ular.

(1) $p|m$ ; Conditions $(C1)$ and $(C2)$ are$\mathit{8}ame$

as of Q-curves.

(2) $p|d$ ; For $y\in M(d, m)_{x}$ ,$(C1)$$(C2)$$y/\overline{\mathrm{F}}_{p}\neq \mathrm{d}\mathrm{o}\mathrm{u}\mathrm{b}\mathrm{l}\mathrm{e}$$e_{s}$$e$point.

$p$1

2$\geq 3$1

4$\geq 3$1

6$\geq 11(\neq 5)$22$\geq 3$2 4
2

6$\geq 11(\neq 3,7)$$\neq 1+2$

-power, $1+5\cross 2$ -power.

ExamPle 2(5-1) The following diagram is the covering map of $M(6,1)$ reduction mod 3 and its quotient
curves by its involutions. $(M(6,1)/w_{a}=\mathrm{P}^{1},$ $a\neq 1)$$/w_{3}$$\downarrow$$e_{s}=1,$$e=4$$\cross:e=6$,$\bullet$$e_{s}=1,$: $e=4,0$ :$\mathrm{F}_{3}$$e=4$-rational points $(e_{S}=1, e=2)$

160

(5-2)

contained in the following list, then $M(d, 1)/w_{d}=\mathrm{P}^{1}$ and $\forall x\in(M(d, 1)/w_{d})(Q)$is modular.
$Ifdi\mathit{8}$$d=14,21,33,34,35,39,46,51,55,62,69,87,94,95,111,119,159$ .

If is contained in the following list, then $M(d, 1)/w_{d}$ is an elliptic curves with positive$d$rank and $\forall x\in(M(d, 1)/w_{d})(\mathrm{Q})$ is modular.
$d=$57,65, 77, 129, 143.

Remark 7 Even for $m\neq 1$ , We have obtained some examples.