Heegner Points on Modular Curve X

Heegner Points on Modular Curve X_0(11)

[2004.10.09]modular曲線X_0(11)のHeegner点

X₀(11)のHeegner点を計算するプログラム(pari/GP)

■genus 1のmodular曲線X₀(11)にQ-isomorphicなconductor 11の楕円曲線
     E: y²+y = x³-x²-10x-20
のHeegner点を求める。

■楕円曲線Eのねじれ点群と有理点群を求めておく。
pari/gpとmwrank3により、Eのねじれ点群とrankは、
     E(Q)_tors = Z/5Z,
     rank E(Q) = 0
であることが分かる。
よって、EのMordel-Weil群は、
     E(Q) = Z/5Z = { O, [5,5], [16,-61], [16,60], [5,-6]}
である。

[pari/gpによる計算]

gp>  e=ellinit([0,-1,1,-10,-20])
time = 188 ms.
%1 = [0, -1, 1, -10, -20, -4, -20, -79, -21, 496, 20008, -161051, -122023936/161051, [4.346308158205394421969490790, -1.673154079102697210984745395 + 1.320848922269075665602515977*I, -1.673154079102697210984745395 - 1.320848922269075665602515977*I]~, 1.269209304279553421688794616, 0.6346046521397767108443973083 + 1.458816616938495229330889613*I, -1.318701713826012913940217413 + 0.E-29*I, -0.6593508569130064569701087069 - 3.990938775365904887253619872*I, 1.851543623455959317708006712]
gp>  ellglobalred(e)
time = 1 ms.
%2 = [11, [1, 0, 0, 0], 5]
gp>  elltors(e,1)
time = 34 ms.
%3 = [5, [5], [[5, 5]]]
gp>  for(i=1,5,print(ellpow(e,[5,5],i)))
[5, 5]
[16, -61]
[16, 60]
[5, -6]
[0]
time = 11 ms.

[mwrank3による計算]

bash-2.05a$ mwrank3
Program mwrank: uses 2-descent (via 2-isogeny if possible) to
determine the rank of an elliptic curve E over Q, and list a
set of points which generate E(Q) modulo 2E(Q).
and finally search for further points on the curve.
For more details see the file mwrank.doc.
For details of algorithms see the author's book.

Please acknowledge use of this program in published work, 
and send problems to John.Cremona@nottingham.ac.uk.

Version compiled on Feb 11 2003 at 17:40:15 by GCC 3.2.1
using base arithmetic option LiDIA_ALL (LiDIA bigints and multiprecision floating point)
Using LiDIA multiprecision floating point with 15 decimal places.
Enter curve: [0,-1,1,-10,-20]

Curve [0,-1,1,-10,-20] :        No points of order 2
Basic pair: I=496, J=40016
disc=-1113184512
2-adic index bound = 2
By Lemma 5.1(a), 2-adic index = 1
2-adic index = 1
One (I,J) pair
Looking for quartics with I = 496, J = 40016
Looking for Type 3 quartics:
Trying positive a from 1 up to 3 (square a first...)
(1,0,-94,484,-695)      --trivial
(1,0,-28,44,-24)        --trivial
Trying positive a from 1 up to 3 (...then non-square a)
Trying negative a from -1 down to -9
(-2,-1,11,21,-13)       --trivial
(-2,2,-4,32,-28)        --trivial
(-6,2,2,14,-8)  --trivial
(-7,-12,2,16,1) --trivial
Finished looking for Type 3 quartics.
Mordell rank contribution from B=im(eps) = 0
Selmer  rank contribution from B=im(eps) = 0
Sha     rank contribution from B=im(eps) = 0
Mordell rank contribution from A=ker(eps) = 0
Selmer  rank contribution from A=ker(eps) = 0
Sha     rank contribution from A=ker(eps) = 0
Rank = 0
After descent, rank of points found is 0


The rank and full Mordell-Weil basis have been determined unconditionally.
Regulator = 1

 (1.6 seconds)

■虚２次体K=Q(sqrt{-7})の類数は1である。
Kの整数環O_K=Z[(-1+sqrt{-7})/2]は、M₀(11)={((a b);(c d)) : a,b,c,d \in Z, 11|c }の整数環O=Z+Z((-4 -2); (11 5))に同型である。
τ=(-9+sqrt{-7})/22は、Oの不動点(fixed point)である。実際に、
     ((1 0);(0 1))τ=τ/1=τ,
     ((-4 -2);(11 5))τ=(-4τ-2)/(11τ+5)=(-9+sqrt{-7})/22=τ
となる。

τは判別式-7の２次方程式t²+9t+2=0の根であり、 11τは判別式-7の２次方程式s²+9s+22=0の根であるので、τ=(-9+sqrt{-7})22はX₀(11)のHeegner点である。

modular形式fのFourier係数a_n(E)は、E mod pの有理点から、以下の等式より計算できる。
     f=Σ_n=1^∞{a_nqⁿ} = qΠ_n=1^∞{(1-q¹¹ⁿ)²(1-qⁿ)²} = q-2q²-q³+2q⁴+q⁵+2q⁶-2q⁷-2q⁹-2q¹⁰........

ここで、q=e^2πiτとして、複素数体Cの点
     z = Σ_n=1¹⁰⁰⁰{(a_n/n)qⁿ}
をWeierstrass uniformisationでE(C)に写すと、EのQ(sqrt{-7})-有理点P((1-sqrt{-7})/2,-2-2sqrt{-7})を得る。

[pari/gpによる計算]

gp> read("x0_11.gp")
time = 67 ms.
gp>  tau=(-9+sqrt(-7))/22
time = 0 ms.
%1 = -9/22 + 0.12026142323020866320461889789269365571410269014011137183492429360004858287410380126183604029429743684613929901538624832253379681311941885383760866163091398739623412407856017692271316340295851185004445*I
gp>  q0=exp(2*Pi*I*tau)
time = 3 ms.
%2 = -0.39515111959617535063168342980195099288850717247298087491745329848478165253015833673093490270575974222138283855294957723309849798825224051360340125290343127651615544542994157742440587236904592845502550 - 0.25394820464883811117899444081730311657083474108403541216181985350498192575075778519563644745515743301261214180804701868781844515191378635355035959480821003763687835930828970804742530295453990240947869*I
gp>  f2(q0,1000)
time = 486 ms.
%3 = -0.50768372171182136867551784670181892208779689673224346718685476849216333544511108907614492236860504292465811851132851497491268013139073759059708560422829969941684297923544538412038379127552892113856282 - 0.40562904451604516529142753598929424354673787306147774542102120458224309134668996496697769442177074243769344356157311724198746365650385503909741351082777062649049008879082011123267657101004751995716840*I
gp>  p=ellztopoint(e,%3)
time = 59 ms.
%4 = [0.49999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 - 1.3228756555322952952508078768196302128551295915412250901841672296005344116151418138801964432372718053075322891692487315478717649443136073922136952779400538613585753648641619461498447974325436303504890*I, -1.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 - 5.2915026221291811810032315072785208514205183661649003607366689184021376464605672555207857729490872212301291566769949261914870597772544295688547811117602154454343014594566477845993791897301745214019561*I]
gp>  contfrac(real(p[1]))
time = 4 ms.
%5 = [0, 2]
gp> contfrac(imag(p[1])/sqrt(7))
time = 1 ms.
%6 = [-1, 2]
gp>  contfrac(real(p[2]))
time = 0 ms.
%7 = [-2]
gp>  contfrac(imag(p[2])/sqrt(7))
time = 0 ms.
%8 = [-2]

Gal(Q(sqrt{-7})/Q)の自明でない元σ:sqrt{-7}→-sqrt{-7}に対して、 P^σ((1+sqrt{-7})/2,-2+2sqrt{-7})もEのQ(sqrt{-7})-有理点であり、R=P+P^σは、σの作用で不変なので、EのQ-有理点である。

実際に、EのQ-有理点Rを計算すると、(16,-61)となる。

[pari/gpによる計算]

gp>  r7=Mod(w,w^2+7)
time = 0 ms.
%4 = Mod(w, w^2 + 7)
gp>  ellisoncurve(e,[(1-r7)/2,-2-2*r7])
time = 1 ms.
%5 = 1
gp>  elladd(e,[(1-r7)/2,-2-2*r7],[(1+r7)/2,-2+2*r7])
t6me = 4 ms.
%6 = [Mod(16, w^2 + 7), Mod(-61, w^2 + 7)]

また、P,2P((-3+sqrt{-7})/2,(-3+sqrt{-7})/2),3P((5-3sqrt{-7})/2,(3+9sqrt{-7})/2),4P((-11-17sqrt{-7})/2,-104-17sqrt{-17})はK整点であるが、5P(-3/4,(-4-11sqrt{-7})/8)はQ(sqrt{-7})-整点ではない。よって、Pは位数無限のQ(sqrt{-7})-有理点である。

[pari/gpによる計算]

gp>  ellpow(e,[(1-r7)/2,-2-2*r7],2)
time = 6 ms.
%19 = [Mod(1/2*w - 3/2, w^2 + 7), Mod(1/2*w - 3/2, w^2 + 7)]
gp>  ellpow(e,[(1-r7)/2,-2-2*r7],3)
time = 12 ms.
%20 = [Mod(-3/2*w + 5/2, w^2 + 7), Mod(9/2*w + 3/2, w^2 + 7)]
gp>  ellpow(e,[(1-r7)/2,-2-2*r7],4)
time = 4 ms.
%21 = [Mod(-17/2*w - 11/2, w^2 + 7), Mod(-17*w - 104, w^2 + 7)]
gp>  ellpow(e,[(1-r7)/2,-2-2*r7],5)
time = 4 ms.
%22 = [Mod(-3/4, w^2 + 7), Mod(-11/8*w - 1/2, w^2 + 7)]
gp>  ellpow(e,[(1-r7)/2,-2-2*r7],6)
time = 3 ms.
%23 = [Mod(2319/5618*w - 1331/5618, w^2 + 7), Mod(234219/148877*w - 280499/148877, w^2 + 7)]

■虚２次体K=Q(sqrt{-6})の類数は2である。
Kの整数環Z[sqrt{-6}]は、M₀(11)の整数環O=Z+Z((-4 -2);(11 4))に同型である。
τ=(-4+sqrt{-6})/11は、Oの不動点である。
τは判別式-24の２次方程式11t²+8t+2=0の根であり、11τも判別式-24の２次方程式s²+8s+22=0の根であるので、τ=(-4+sqrt{-6})/11はX₀(11)のHeegner点である。

ここで、q=e^2πiτとして、複素数体Cの点
z = Σ_n=1¹⁰⁰⁰{a_nqⁿ/n}
をWeierstrass uniformisationでE(C)に写すと、EのQ(sqrt{-6},sqrt{2})-有理点
P((-2-5sqrt{2})/2+(-3-4sqrt{2})sqrt{-6}/2,(-40-29sqrt{2})/2+(-7+sqrt{2})sqrt{-6}/2)
を得る。

[pari/gpによる計算]

gp>  tau=(-4+sqrt(-6))/11
time = 0 ms.
%25 = -4/11 + 0.22268088570756164529066218860962649017872249824151546622115386975008730704157409332180540300951274861987223647514674446589626049634888703105688945011247986843899206795706150647705728229735561547486193*I
gp>  z0=f2(exp(2*Pi*I*tau),1000)
time = 381 ms.
%26 = -0.15882879211510579636430227271912781529638240464384567578493037113085848483539422706992217824565132107569758327069092235806887319449934046639144707260405449931839582904124657019634631138115048014466894 - 0.24453812691369989217080875059881915143692338996456510360871230053174983929209723562732673268059423630966634772698079666697599660082422064430139785789787914202528449809320916898198835407804721853282096*I
gp>  p=ellztopoint(e,z0)
time = 49 ms.
%27 = [-4.5355339059327376220042218105242451964241796884423701829416993449768311961552675971259688358191039318375346155772807425623120901396268430316103037427498395785330566648187639818894998762528819551514286 - 10.602437844450276321405711478082326555720142236226527704872266768684172633821172688134373896685954654930396467501548095807168078569752445927685297975233412423355742048900737185143344438310706355028282*I, -40.506096654409878207624486501040622139260242192965747061061856200865620937700552063330619247750802804657700770348228306861410122809835689583339761707949069555491728655948831094959099282266715339878286 - 6.8411632921722460501630479199647475049380109284879648214586170059264283041918025558083618291089922461894549628777429589070360840144331982168785883172361012863731013844445624172987305483471065208589195*I]
gp>  contfrac(real(p[1]))
time = 1 ms.
%28 = [-5, 2, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6]
gp>  contfrac(imag(p[1])/sqrt(6))
time = 1 ms.
%29 = [-5, 1, 2, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22]
gp>  contfrac(real(p[2]))
time = 1 ms.
%30 = [-41, 2, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 4]
gp>  contfrac(imag(p[2])/sqrt(6))
time = 2 ms.
%31 = [-3, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 1, 1, 4]

循環連分数[6^・,1,1^・]=αは、２次方程式
     α = 6+1/(1+1/(1+1/α))
つまり、
     2α²-12α-7 = 0
の正根なので、
     α = (6+5sqrt{2})/2
を得る。よって、循環連分数[-5,2,6^・,1,1^・]=[-5,2,α]を求めると、
     [-5,2,α] = -5+1/(2+1/α) = (-2-5sqrt{2})/2
となる。
また、[22^・,3,5,3^・]=βは、２次方程式
     β = 22+1/(3+1/(5+1/(3+1/β))) = (1138β+357)/(51β+16)
つまり、
     β²-22β-7 = 0
の正根なので、
     β = 11+8sqrt{2}
を得る。よって、
     [-5,1,2,22^・,3,5,3^・] = [-5,1,2,β] = (-13β-4)/(3β+1) = (-3-4sqrt{2})/2
となる。
同様にして、
     [40^・,1,1^・] = (40+29sqrt{2})/2
     [-40,2,40^・,1,1^・] = (-40-29sqrt{2})/2
     [4^・,1^・] = 2+2sqrt{2}
     [-3,4^・,1^・] = (-7-sqrt{2})/2
を得る。

[pari/gpによる計算]

gp>  r2=Mod(w,w^2-2)
time = 1 ms.
%1 = Mod(w, w^2 - 2)
gp>  -5+1/(2+1/((6+5*r2)/2))
time = 17 ms.
%2 = Mod(-5/2*w - 1, w^2 - 2)
gp>  contfracpnqn([22,3,5,3,b])
time = 0 ms.
%3 = 
[1138*b + 357 1138]

[51*b + 16 51]

gp>  b-(1138*b + 357)/(51*b + 16)
time = 1 ms.
%4 = (51*b^2 - 1122*b - 357)/(51*b + 16)
gp>  factor(51*b^2 - 1122*b - 357)
time = 6 ms.
%5 = 
[b^2 - 22*b - 7 1]

gp>  contfracpnqn([-5,1,2,b])
time = 0 ms.
%6 = 
[-13*b - 4 -13]

[3*b + 1 3]

gp>  subst((-13*b-4)/(3*b+1),b,(11+8*r2))
time = 2 ms.
%7 = Mod(-2*w - 3/2, w^2 - 2)
gp>  contfracpnqn([40,1,1,c])
time = 0 ms.
%8 = 
[81*c + 41 81]

[2*c + 1 2]

gp>  c-(81*c+41)/(2*c+1)
time = 0 ms.
%9 = (2*c^2 - 80*c - 41)/(2*c + 1)
gp>  contfracpnqn([-41,2,c])
time = 0 ms.
%10 = 
[-81*c - 41 -81]

[2*c + 1 2]

gp>  subst((-81*c-41)/(2*c+1),c,(40+29*r2)/2)
time = 2 ms.
%11 = Mod(-29/2*w - 20, w^2 - 2)
gp>  contfracpnqn([4,1,d])
time = 0 ms.
%12 = 
[5*d + 4 5]

[d + 1 1]

gp>  d-(5*d + 4)/(d+1)
time = 3 ms.
%13 = (d^2 - 4*d - 4)/(d + 1)
gp>  contfracpnqn([-3,c])
time = 0 ms.
%14 = 
[-3*c + 1 -3]

[c 1]

gp>  subst((-3*c + 1)/c,c,2+2*r2)
time = 1 ms.
%15 = Mod(1/2*w - 7/2, w^2 - 2)

Gal(Q(sqrt{-6},sqrt{2})/Q(sqrt{-6}))の自明でない元σ:sqrt{2}→-sqrt{2}に対して、R=P+P^σはσの作用により不変であるので、Q(sqrt{-6})-有理点である。
実際にRを求めると、(-2+sqrt{-6},5)となる。
同様に、R'(-2-sqrt{-6},5)もQ(sqrt{-6})-有理点であるので、S=R+R'を求めると、Q-有理点(5,-6)を得る。

[pari/gpによる計算]

gp>  rm6=Mod(u,u^2+6)
time = 0 ms.
%18 = Mod(u, u^2 + 6)
gp>  P1=[(-2-5*r2)/2+(-3-4*r2)*rm6/2,(-40-29*r2)/2+(-7+r2)*rm6/2]
time = 5 ms.
%19 = [Mod(Mod(-2*u - 5/2, u^2 + 6)*w + Mod(-3/2*u - 1, u^2 + 6), w^2 - 2), Mod(Mod(1/2*u - 29/2, u^2 + 6)*w + Mod(-7/2*u - 20, u^2 + 6), w^2 - 2)]
gp>  P2=[(-2+5*r2)/2+(-3+4*r2)*rm6/2,(-40+29*r2)/2+(-7-r2)*rm6/2]
time = 1 ms.
%20 = [Mod(Mod(2*u + 5/2, u^2 + 6)*w + Mod(-3/2*u - 1, u^2 + 6), w^2 - 2), Mod(Mod(-1/2*u + 29/2, u^2 + 6)*w + Mod(-7/2*u - 20, u^2 + 6), w^2 - 2)]
gp> ellisoncurve(e,P1)
time = 20 ms.
%21 = 1
gp>  ellisoncurve(e,P2)
time = 2 ms.
%22 = 1
gp>  elladd(e,P1,P2)
time = 7 ms.
%23 = [Mod(Mod(u - 2, u^2 + 6), w^2 - 2), Mod(Mod(5, u^2 + 6), w^2 - 2)]
gp>  ellisoncurve(e,[rm6-2,5])
time = 0 ms.
%24 = 1
gp>  elladd(e,[rm6-2,5],[-rm6-2,5])
time = 0 ms.
%25 = [Mod(5, u^2 + 6), Mod(-6, u^2 + 6)]

[参考文献]

[1]菅真紀子, "θ-合同数と楕円曲線", 1999, p90-102, thesis.
[2]Henri Darmon, "Rational Points on Modular Elliptic Curves", CBMS 101, AMS, 2004, ISBN0-8218-2868-1, p34-44.

Last Update: 2005.06.12

H.Nakao