An Explicit Shimura Tower of Function Fields over a Number Field: An Application of Takeuchi's List

0.12 MB·

by  Takehiro Hasegawa

An Explicit Shimura Tower of Function Fields over a
Number Field: An Application of Takeuchi’s List
Takehiro HASEGAWA
∗†
Shiga University, Otsu, Shiga 520-0862,Japan
January 27,2017
7
1
0
2 Abstract

n Elkies [E1] proposeda procedurefor constructing explicit towers of curves, and gave two
a towersofShimuracurvesasrelevantexamples.Inthispaper,wepresentanewexplicittowerof
J Shimuracurvesconstructedbyusingthisprocedure.

6
2
Keywords: Functionfield; Toweroffunctionfields; Belyimap

]
T MSC2010: Primary11R58; secondary11G32, 14H05, 14H57
N
. 1 Introduction
h
t
a
m According to a result obtained by Tsfasman, Vladut and Zink, excellent Goppa codes can be
obtainedfrommodulartowers(morespecifically,towersofmodularcurves)([TVZ]). However,the
[
constructionofsuchcodesrequiresexplicitmodulartowers. In1998,Elkiesintroducedaprocedure

1 fordefiningsuchtowers(seePropositionin[E1]),andtherebyconstructedseveralclassical(elliptic)
v
modular towers, two Shimura towers and some Drinfeld modular towers ([E1, E3]). In 2012,
1
Hasegawa, Inuzuka and Suzuki presenteda number of classicalmodular towers constructed by
5
5 the same procedure ([HIS, H]), and Garcia, Stichtenoth, Bassa and Beelen recently constructed
7 several Drinfeld modular towers ([BBGS, BB]). On the other hand, although Elkies suggested
0 new Shimura towers, he did not explicitly construct any (see Section 5 in [E2]). To the author’s
.
1 knowledge,onlytwoexplicitShimuratowershavebeenconstructedthusfar.
0
7 LetKbeatotallyrealnumberfield,andletAbeaquaternionalgebraoverK,namely,acentral
1 simplealgebraoverKofdimension4. Throughoutthispaper,weassumethatAisramifiedatall
:
v butoneoftheinfinite placesofK. LetD(A)denoteadiscriminantofA,thatis,theproductofall
i finiteplacespofKsuchthatAisramifiedatp. ForanidealIofKcoprimetoD(A),Shimuracurves
X
(I)analogoustoclassicalmodularcurvesX (N)arewelldefined(seeSection2below).
0 0
r X
a
Note that the prime number ℓ = 3 is totally ramified in Q(√3)/Q. Our main theorem is as
follows.
Maintheorem. LetKbethefieldQ(√3),whichistotallyreal,andletp denotetheplaceofKlyingabove
3
3. Then,theShimuracurve (p2)isrational,thatis, thegenusof (p2)isequalto0. Foreachnatural
X0 3 X0 3
numbern>1,thecurve (pn)isdefinedbyn 1coordinatesx ,x ,...,x whicharerelatedbythen 2
X0 3 − 1 2 n−1 −
equations
2xj+1−1 3 =2(5−3√3)x3j +4 (j=1,2,...,n 2), (1)
− xj+1+2! x3 2(5 3√3) −
j − −
or, equivalently, the curve (pn) is isomorphic to the locus of (x ,x ,...,x ) in (P1)n 1 satisfying the
above equations (1). In othXe0r w3ords, the Shimura tower {X0(pn3)1}n>21 is definn−e1d recursiv−ely by the affine
equation
y 1 3 (5 3√3)x3+4
2 − =2 − .
− y+2! x3 2(5 3√3)
− −
∗TheauthorwaspartiallysupportedbyJSPSKAKENHIGrantNumber15K17508
†[email protected]
1
Shimuracurves (I)alwayshaveAtkin-Lehnerinvolutionsω. Also,unlikeclassicalmodular
0 I
X
curves, Shimura curves have no cusps. Thus, although the covering maps between Shimura
curvescannotbecomputedbyusingq-expansions,theycanbedeterminedfromtheirramification
behavior. In fact, Elkies illustrated this with two examples(see Third variationin [E1], and also
seeSection2). Inthispaper,we showour maintheorembasedonthemethod devisedbyElkies
(seeSection3).
Beelen, Garcia and Stichtenoth derived a normal form for equations of recursive Kummer
towersoffunctionfields([BGS]). Intheirnotation,theequationofourmaintheoremiswrittenas
8(5 3√3)(X+1)3+4(X 2)3
Y3 = − − − ,
4(X+1)3+(5 3√3)(X 2)3
− −
wherethevariablesX,YaredefinedbyX=(2x 2)/(x+2),Y=(2y 2)/(y+2).
− −
Let ℓ , 2,3 be a prime number, and let pℓ denote a place of Q(√3) lying above ℓ. Since the
coefficientsoftheaboveequationareintheringofintegersofQ(√3),themodulopℓ reductionof
ourtowercanbedefined,andmoreover,thereductionisasymptoticallyoptimaloveranextension
fieldofFℓ =Z/(ℓ).
Wulftangestudiedoneofthe(modified)ShimuratowersgivenbyElkiesandprovedthatthis
towerisoptimaloverF (seeSection4in[W]). Moreover,BranderattemptedtoconstructaGoppa
26
codefromthesametower([B]). Inthisregard,aGoppacodecanbedefinedbyusingthetowerin
ourmaintheorem.
LetIbeanidealofKcoprimetoD(A). ForanAtkin-Lehnerinvolutionω andamap j: (I)
I 0
X →
P1 (e.g.,seeMaintheoremI(3.2)in[Smr]),theimageofamorphism
C
(I) P1 P1, z (j(z),j(ω(z)))
X0 → C× C 7→ I
is a closed subvariety of dimension 1, and in the open affine set (A1 A1) , this variety is
C× C \{∞}
described by a modular polynomial Φ (x,y). In 1998, Elkies computed a polynomial Φ (x,y) for
I I
N(I)= 3,8,9([E1,E3]),whereNdenotesthenormwithrespecttoK/Q,andfurthermore,in2005,
Voight determineda polynomial ΦI(x,y)for N(I) = 17([V1]). Polynomials Φp3(x,y)and Φp23(x,y)
forourcurvescanbecomputedbyusingourtheorem.
This paper are organized as follows. In Section 2, two examples of explicit Shimura towers
definedbyElkiesin[E1]areintroduced. InSection3,anewexplicitShimuratowerisgiven.
2 Explicit Shimura towers proposed by Elkies
In this section, we introduce two examples by Elkies (see Third variation in [E1]). The reason
revisiting the examples of Elkies in this section is as follows: In [E1], Elkies left a detailed com-
putationalprocessforclassicalmodulartowers(seeExample,FirstandSecondvariationsin[E1]),
butheomittedthemostofacomputationalprocessforShimuratowers. Therefore,animportant
relationship between Shimura towers and arithmetic groups remains unclear. Hence, we shall
writeadetailedcomputationalprocessforShimuratowers,andwerevealarelationshipbetween
Shimuratowersandarithmeticgroups(seeRemarkinthissection). Moreover,thissectionisalso
arecalloftheprocedurebyElkiesforconstructingexplicitShimuratowersinthenextsection.
Takeuchigaveacompletelistofarithmetictrianglegroups∆ofsignature(0;e ,e ,e )(seeTable
1 2 3
(1)in [T]). In fact, there exist 85 such groups. In this and the next sections, we will use three of
thesegroups.
WeconsiderthefirstexamplegivenbyElkies. LetKbethenumberfieldQ(√3),whichistotally
real. Theprimenumbersℓ = 3,2aretotallyramifiedinK/Q. Letp (resp. p )denotetheplaceof
3 2
K lying above 3 (resp. 2), thatis, p = (√3) (resp. p = (5 3√3)). Also, let∆ be the arithmetic
3 2
−
trianglegroupofsignature(0;2,4,12),andletAbethequaternionalgebraassociatedwith∆,which
isramified at p and at exactlyone infinite place. Since the discriminant D(A)is equal to p , the
3 3
2
Shimura curves (pn) are defined. Elkies subsequently constructs the explicit Shimura tower
X0 2
{X0(pn2)}n>1asfollows:
LetHbetheupperhalf-plane. TheShimuracurve (1)=∆ Hisrational. Infact,
X \
3
1 1 1 3 11
2g(∆ H) 2=Area(∆ H) 1 = + + = 2, andtherefore g(∆ H)=0
\ − \ −Xj=1 − eJ! 6 −(cid:18)2 4 12(cid:19) − \
([Smr], and also see p.207 in [T]). We can choose a coordinate J on (1) which takes the values
X
1,0, attheellipticpointsP ,P ,P oforder2,4,12,respectively,thatis, J(P ) =1,J(P ) =0,and
2 4 12 2 4
∞
J(P )= .
12
∞
Ingeneral,foreachnumbern,theShimuracurve (pn)canbeidentifiedwiththeset
X0 2
(P,Q) (1) (1) apointPispn-isogenoustoapointQ
(cid:26) ∈X ×X (cid:12)(cid:12) 2 (cid:27)
(cid:12)
(cid:12)
(e.g., [E1]), and this curve always has(cid:12)the Atkin-Lehner involution ω(n): (P,Q) (Q,P). The
←→
coveringmapπ: (pn) (1)isdefinedbytheprojectionπ((P,Q))=P,anditsdegreeequals
X0 2 →X
1
N(pn) 1+ =N(p )n 1(N(p )+1)=3 2n 1,
2 N(p)! 2 − 2 · −
Yppn
| 2
where N denotes the norm with respect to K/Q. The map π is branched only above the elliptic
pointsP ,P ,P of (1)andalwaysunramifiedabovetheotherpoints. Theramificationindexat
2 4 12
X
apoint(P,Q)of (pn)isequaltothedenominatoroftheirreduciblefraction
X0 2
ord(Q)/ord(P),
whereordrepresentstheorderofapoint,andtheorderofanon-ellipticpointisequalto1.
We can determine the ramification of the first map π: (p ) (1) and the involution ω(1)
0 2
X → X
of the curve (p ) asfollows: The point P is totally ramified, and the point lying above P is
0 2 12 12
X
(P ,P ). ThepointslyingaboveP (resp. P )are(P ,P ),(P ,P )(resp. (P ,P ),(P , )),andthus
12 4 4 2 4 2 4 12 2 4 2
∗
the ramification indices are equal to 2,1 (resp. 1,2), respectively. Here, denotes a non-elliptic
∗
point. Then, the involution ω(1) must interchange the points (P ,P ),(P ,P ) and the points
4 12 12 4
(P ,P ),(P ,P ),respectively. SincetheHurwitzgenusformulayields
2 4 4 12
2g( (p )) 2=3( 2)+(3 1)+(2 1)+(2 1)= 2, andtherefore g( (p ))=0,
0 2 0 2
X − − − − − − X
thecurve (p )isagainrational,and J isapolynomialofdegree3withatriplepole,suchthat J
0 2
X
andJ 1havedoublezeros. Then,wecanchoosearationalcoordinateton (p )suchthat
0 2
− X
J=t(4t 3)2 so J 1=(t 1)(4t 1)2 ,
− − − −
(cid:16) (cid:17)
andthusω(1)interchangesthepointst=0,t= andthepointst=1,t=3/4,respectively. Hence,
∞
3
ω(1)(t)= . (2)
4t
X0((cid:15)(cid:15)p32) t=∞3 t=3/4❍❍❍❍❍2❍❍❍❍ ①①①1①①①①①①t=0 t=1❋❋❋❋❋❋1❋❋❋ ✈✈✈✈2✈✈✈✈✈t=1/4
(1) J= J=0 J=1.
X ∞
Next, we can determine the ramification of the second map π: (p2) (p ) (1) and
X0 2 → X0 2 → X
theinvolution ω(2) ofthecurve (p2)asfollows: ThepointP istotallyramified, andthe point
X0 2 12
lyingaboveP is(P ,P ). Thepoints lyingaboveP (resp. P )are(P , ), (P ,P ), (P ,P )(resp.
12 12 2 4 2 4 4 4 4 4
∗
(P ,P ),(P ,P ),(P , ),(P , )),andthustheramificationindicesareequalto4,1,1(resp. 1,1,2,2),
2 12 2 2 2 2
∗ ∗
respectively. Then, the involution ω(2) interchanges the points (P ,P ),(P ,P ) and the points
4 4 4 4
3
(P ,P ),(P ,P ), respectively, and fixes the point (P ,P ). By the Hurwitz genus formula, the
2 12 12 2 2 2
curve (p2)isrational. Thus,wecanchoosearationalcoordinatexon (p2)suchthat
X0 2 X0 2
x2+3 x4(x2+3)
t= and J = , (3)
4 4 !
andthe involution ω(2) interchangesthe points x = √ 3,x = √ 3and the points x = 1,x = ,
− − − ∞
respectively,andfixesthepointx= 1. Hence,
−
x+3
ω(2)(x)= . (4)
x 1
−
(p2) x= x=0 x= √ 3 x= √ 3
X0(cid:15)(cid:15)22 2∞ 2 −❍❍❍❍❍❍1❍❍❍❍ ✉✉✉✉1✉✉✉✉✉✉ − −
(p ) t= t=3/4 t=0
X0 2 ∞
x=1❋❋❋❋❋❋1❋❋❋ ✇✇✇✇1✇✇✇✇x✇=−1 x= √−❑2❑❑❑❑❑1❑❑❑❑ rrrr1rrrrxrr=−√−2
t=1 t=1/4.
Wenowhavealltheinformationnecessarytodeterminethecurve (pn). Itfollowsfrom(2)
X0 2
(3)(4)that
3 x2+3 x2+3 3
t ω(1)(t)= ω(1) =
· 4 ⇔ 4 · 4 ! 4
y+3 2
(x2+3) ω(2)(y)2+3 =12 (x2+3) +3 =12,
⇔ ⇔  y 1! 
andhencethecurve (pn)isdefined(cid:16)byn 1co(cid:17)ordinatesx ,...,x satis−fyingthen 2relations
X0 2 − 1 n−1 −
(x2+3)(z2 +3)=12 (j=1,...,n 2), z :=(x +3)/(x 1).
j j+1 − j j j−
Sincethistower{X0(pn2)}n>1hascyclicsteps,andsinceitbecomeunramifiedafterafinitenumber
ofsteps,thisisdominatedbya2-classfieldtowerofthecurve (p5)overanyfinitefieldofodd
X0 2
characteristic.
Next, we consider the other example given by Elkies. Let K be the number field Q(cos 2π),
18
whichistotallyreal,andtheprimenumberℓ=3istotallyramifiedinK/Q. Also,letp denotethe
3
placeofKlyingabove3. Furthermore,let∆bethearithmetictrianglegroupofsignature(0;2,3,9),
and let A be the quaternion algebra associated with ∆, which is ramified at exactly two infinite
places. SinceAisnotramifiedatallfiniteplacesofK,thediscriminantD(A)isequalto1,andthus
theShimuracurvesX0(pn3)aredefined. ElkiesconstructstheexplicitShimuratower{X0(pn3)}n>1as
follows:
TheShimuracurve (1)=∆ Hisrational. Infact,
X \
3
1 1 1 2 8
2g(∆ H) 2=Area(∆ H) 1 = + + = 2, andtherefore g(∆ H)=0
\ − \ −Xj=1 − eJ! 18 −(cid:18)2 3 9(cid:19) − \
([Smz], and also see p.207 in [T]). We can choose a coordinate J on (1) which takes the values
X
1,0, attheellipticpointsP ,P ,P oforder2,3,9,respectively. Inotherwords,J(P )=1,J(P )=0,
2 3 9 2 3
∞
andJ(P )= . Foreachnaturalnumbern,theShimuracurve (pn)canbedefined,andthiscurve
9 ∞ X0 3
alwayshastheAtkin-Lehnerinvolutionω(n). Thedegreeofthemapπ: (pn) (1)equals
X0 3 →X
1
N(pn) 1+ =N(p )n 1(N(p )+1)=4 3n 1.
3 N(p)! 3 − 3 · −
Yppn
| 3
4
Wecandeterminetheramificationofthemap (p ) (1)andtheinvolutionω(1)asfollows:
0 3
X →X
Note that this map is branched only above P ,P ,P and unramified above the other points.
2 3 9
The points lying above P are(P ,P ),(P ,P ), and thus the ramificationindices areequalto 1,3,
9 9 9 9 3
respectively. Also,thepointslyingaboveP (resp. P )are(P ,P ),(P , )(resp. (P , ),(P , )),and
3 2 3 9 3 2 2
∗ ∗ ∗
thus the ramification indices are equal to 1,3 (resp. 2,2), respectively. Then, the involution ω(1)
must interchange the points (P ,P ),(P ,P ) and fix the point (P ,P ). Since, from the Hurwitz
3 9 9 3 9 9
genusformula
2g( (p )) 2=4( 2)+(3 1)+(3 1)+(2 1)+(2 1)= 2, andtherefore g( (p ))=0,
0 3 0 3
X − − − − − − − X
thecurve (p )isagainrational. Then,wecanchoosearationalcoordinateton (p )suchthat
0 3 0 3
X X
(t 1)3(9t 1) (3t2+6t 1)2
J = − − so J 1= − ,
− 64t3 − − 64t3 !
andtheinvolutionω(1) interchangesthepointst=1,t=0whilefixingthepointt= . Hence,
∞
ω(1)(t)=1 t. (5)
−
X0((cid:15)(cid:15)p43) t=∞❍❍❍❍❍❍1❍❍❍ ✇✇✇✇3✇✇✇✇✇t=0
(1) J=
X ∞
t=1❊❊❊❊❊❊1❊❊❊ ✇✇✇✇3✇✇✇✇t✇=1/9 t=(−3+2√◆3◆)◆/◆3◆◆2◆◆◆◆◆ ♣♣♣♣♣2♣♣t♣=♣♣(♣−3−2√3)/3
J=0 J=1.
Next, the ramification of the map (p2) (p ) (1) and the involution ω(2) can be
X0 3 → X0 3 → X
determinedasfollows: ThepointslyingaboveP are(P ,P ),(P , ),andtheramificationindices
9 9 3 9
∗
are3,9,respectively. ThepointslyingaboveP (resp. P )are(P ,P ),(P ,P ),(P ,P ),(P , ),(P , ),
3 2 3 9 3 3 3 3 3 3
∗ ∗
(P , )(resp. (P , ),(P , ),(P , ),(P , ),(P , ),(P , )),andthereforetheramificationindicesequal
3 2 2 2 2 2 2
∗ ∗ ∗ ∗ ∗ ∗ ∗
1,1,1,3,3,3 (resp. 2,2,2,2,2,2), respectively. Then, the involution ω(2) interchanges the points
(P ,P ),(P ,P ) and the points (P ,P ),(P ,P ), respectively. By the Hurwitz genus formula, the
3 9 9 3 3 3 3 3
curve (p2)isagainrational. Thus,wecanchoosearationalcoordinatexon (p2)suchthat
X0 3 X0 2
(x3 1)3(9x3 1)
t=x3 and J = − − , (6)
− 64x9 !
and the involution ω(2) interchanges the points x = 1,x = and the points x = ζ ,x = ζ2,
∞ 3 3
respectively. Here,ζ isacubeofunity. Hence,theinvolutionω(2) isgivenby
3
x+2
ω(2)(x)= . (7)
x 1
−
X0((cid:15)(cid:15)p323) x=3∞ x=1●●●●●●1●●●x1=ζ3✇✇✇✇1✇✇✇✇✇x=ζ23
(p ) t= t=1.
X0 3 ∞
Wenowhavealltheinformationnecessarytodeterminethecurve (pn). Itfollowsfrom(5)
X0 3
(6)(7)that
t+ω(1)(t)=1 x3+ω(1)(x3)=1
⇔
y+2 3
x3+ω(2)(y)3 =1 x3+ =1,
⇔ ⇔ y 1!
−
andhencethecurve (pn)isdefinedbyn 1coordinatesx ,...,x satisfyingthen 2relations
X0 3 − 1 n−1 −
x3+z3 =1 (j=1,...,n 2), z :=(x +2)/(x 1).
j j+1 − j j j−
5
Sincethistower{X0(pn3)}n>1hascyclicsteps,andsinceitbecomeunramifiedafterfinitelymany
steps,thisisdominatedbya3-classfieldtowerofthecurve (p4).
X0 3
Remark. ItfollowsfromTable(1)in[T]thattheShimuracurve (p )correspondstothetriangle
0 3
X
group ∆ of signature (0;3,3,9), that is, (p ) = ∆ H. In fact, ∆ has exactly three elliptic points
0 3
X \
(P ,P ), (P ,P ), (P ,P ) of order 3,3,9, respectively. Voight studied Shimura curves of genus at
9 3 3 9 9 9
most2([V2]). Itfollows fromTable 4.3in[V2] thatthe curve (p2)correspondstoagroup Γof
X0 3
signature(0;3,3,3,3),andtheellipticcurve (p3)correspondstoagroupofsignature(1;3,3,3).In
X0 3
fact,Γhasexactlyfourellipticpoints(P ,P ),(P ,P ),(P ,P ),(P ,P )oforder3,3,3,3,respectively.
3 3 3 3 3 9 9 9
3 A new explicit Shimura tower
In Section 2, we introduced in detail two examples given by Elkies. In this section, using the
methodproposedbyElkies,weconstructanewShimuratower(seeMaintheoreminSection1).
LetKbethenumberfieldQ(√3). Theprimenumbersℓ=3,2aretotallyramifiedinK/Q. Letp
3
(resp. p )denotetheplaceofKlyingabove3(resp. 2). Also,let∆bethearithmetictrianglegroup
2
of signature (0;3,3,6), and let A be the quaternion algebra associated with ∆, which is ramified
atp andatexactlyoneinfiniteplace. SincethediscriminantD(A)equalsp ,theShimuracurves
2 2
X0(pn3)aredefined,andthentheShimuratower{X0(pn3)}n>1isconstructedasfollows:
TheShimuracurve (1)=∆ Hisrational. Infact,
X \
3
1 1 2 2 5
2g(∆ H) 2=Area(∆ H) 1 = + + = 2, andtherefore g(∆ H)=0
\ − \ −Xj=1 − eJ! 6 −(cid:18)3 3 6(cid:19) − \
([Smz], and also see p.207 in [T]). We can choose a coordinate J on (1) which takes the values
X
1,0,∞attheelliptic pointsP3,P′3,P6 of order3,3,6,respectively. Specifically, J(P3) = 1,J(P′3) = 0,
andJ(P )= . Foreachnaturalnumbern,theShimuracurve (pn)canbedefined,andthiscurve
6 ∞ X0 3
hastheAtkin-Lehnerinvolutionω(n). Thedegreeofthemapπ: (pn) (1)isequalto
X0 3 →X
1
N(pn) 1+ =N(p )n 1(N(p )+1)=4 3n 1.
3 N(p)! 3 − 3 · −
Yppn
| 3
We candetermine the ramification of (p ) (1)and the involution ω(1) asfollows: Note
0 3
X → X
thatthismapisbranchedonlyaboveP3,P′3,P6andunramifiedabovetheotherpoints. Thepoints
lyingaboveP6are(P6,P′3),(P6,P3),andthustheramificationindicesareequalto2,2,respectively.
The points lying above P′3 (resp. P3) are (P′3,P6),(P′3,∗) (resp. (P3,P6),(P3,∗)), and therefore the
ramificationindicesareequalto1,3(resp. 1,3),respectively. Then,theinvolutionω(1)interchanges
thepoints(P′3,P6),(P6,P′3)andthepoints(P3,P6),(P6,P3),respectively. SincebytheHurwitzgenus
formula
2g( (p )) 2=4( 2)+(2 1)+(2 1)+(3 1)+(3 1)= 2, andtherefore g( (p ))=0,
0 3 0 3
X − − − − − − − X
thecurve (p )isalsorational. Then,wecanchoosearationalcoordinateton (p )suchthat
0 3 0 3
X X
4(2t+1)3 t(t 4)3
J = so J 1= − ,
(t2+10t 2)2 − −(t2+10t 2)2!
− −
andtheinvolutionω(1) interchangesthepointst = ,t= 5+3√3andt = 0,t = 5 3√3,from
∞ − − −
which
(5 3√3)t 2
ω(1)(t)= − − . (8)
− t+(5 3√3)
−
(p ) t= 5+3√3 t= 5 3√3
X0(cid:15)(cid:15)43 − ▲▲▲▲▲▲2▲▲▲▲▲ rrrrr2rrrrrr− −
(1) J=
X ∞
6
t=∞●●●●●●1●●● ✉✉✉✉3✉✉✉✉t✉=−1/2 t=0●●●●●●1●●● ✇✇✇3✇✇✇✇✇✇t=4
J=0 J=1.
At this stage, the ramification of (p2) (p ) (1) and the involution ω(2) can be de-
X0 3 → X0 3 → X
termined as follows: The points lying above P are (P , ),(P , ), and the corresponding indices
6 6 6
∗ ∗
are equal to 6,6, respectively. The points lying above P′3 (resp. P3) are (P′3,P3), (P′3,P′3), (P′3,P′3),
(P′3,∗),(P′3,∗),(P′3,∗)(resp. (P3,P′3),(P3,P3),(P3,P3),(P3,∗),(P3,∗),(P3,∗)),andthusthecorrespond-
ing indices are equal to 1,1,1,3,3,3 (resp. 1,1,1,3,3,3), respectively. Then, the involution ω(2)
interchangesthepoints(P3,P′3),(P′3,P3),thepoints(P3,P3),(P3,P3)andthepoints(P′3,P′3),(P′3,P′3),
respectively. Bythe Hurwitzgenus formula, the curve (p2)isrational. Thus, we canchoose a
X0 3
rationalcoordinatexon (p2)suchthat
X0 2
(5+3√3)x3+4
t= , (9)
−x3 2(5+3√3)
−
whereby the involution ω(2) interchanges the points x = 1 √3,x = 1 + √3, the points x =
−
(1 √3)ζ ,x=(1 √3)ζ2andthepointsx=(1+ √3)ζ ,x=(1+ √3)ζ2,respectively. Here,ζ isa
− 3 − 3 3 3 3
cubeofunity. Hence,theinvolutionω(2) isgivenby
x 1
ω(2)(x)= 2 − . (10)
− x+2
(p2) x=1+ √3 x=(1+ √3)ζ x=(1+ √3)ζ2
X0(cid:15)(cid:15)33 ❖❖❖❖❖❖❖1❖❖❖❖❖❖ 1 ♥♥♥3♥♥♥♥1♥♥♥♥♥♥ 3
(p ) t= .
X0 3 ∞
x=1 √3 x=(1 √3)ζ x=(1 √3)ζ2
− ❖❖❖❖❖❖❖1❖❖❖❖❖ 1− ♠♠♠3♠♠♠1♠♠♠♠♠♠ − 3
t=0.
Wenowhavealltheinformationnecessarytodeterminethecurve (pn). Itfollowsby(8)(9)
X0 3
(10)that
(5 3√3)t 2 (5+3√3)x3+4 4
ω(1)(t)= − − ω(1) =
− t+(5 3√3) ⇔ −x3 2(5+3√3)! x3
− −
(5+3√3)ω(2)(y)3+4 4 y 1 3 (5 3√3)x3+4
= 2 − =2 − ,
⇔ −ω(2)(y)3 2(5+3√3) x3 ⇔ − y+2! x3 2(5 3√3)
− − −
andhencethecurve (pn)isdefinedbyn 1coordinatesx ,...,x satisfyingthen 2relations
X0 3 − 1 n−1 −
(5 3√3)x3+4 x 1
z3 =2 − j (j=1,...,n 2), z := 2 j− .
j+1 x3 2(5 3√3) − j − xj+2
j − −
Remark. VoightgavealistofShimuracurvesofgenusatmosttwo([V2]). ItfollowsfromTable
4.2in[V2]thatourShimuracurve (p )correspondstoagroupΓ ofsignature(0;3,3,3,3),and
0 3 1
X
our Shimura curve (p2) corresponds to a group Γ of signature (0;3,3,3,3,3,3). In fact, Γ has
X0 3 2 1
exactlyfour elliptic points (P3,P6), (P′3,P6), (P6,P3), (P6,P′3) of order 3,3,3,3,respectively, and Γ2
hasexactlysixellipticpoints(P3,P3),(P3,P3),(P3,P′3),(P′3,P3),(P′3,P′3),(P′3,P′3)oforder3,3,3,3,3,3,
respectively.
Remark. While the curves given by Elkies are defined over Q, ours is defined over Q(√3). In
general,oursalsohasamodeloverQ(seeProposition5.1.2in[V1]). However,suchamodeldoes
notperhapshaveasimpleequationasinourtheorem.
7
Remark. Takeuchigaveacompletelistofarithmetictrianglegroupsofsignature(0;e ,e ,e ): There
1 2 3
exist85suchgroups,ofwhich76groupshavenocusps(seeCasesII,...,XIXofTable(1)in[T]).
NotethatShimuracurveshavenocusps. ElkiesconstructedtwoShimuratowersfromtwogroups
(0;2,4,12)and (0;2,3,9),respectively ([E1], and also see Section 2). This time, we constructed a
Shimuratowerfromthegroup(0;3,3,6)(seeSection3). Ithinkthatbyusingthesameprocedure
asElkies,explicitShimuratowerscannotbeconstructedfromtheother73groupsanylonger.
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