The group structure of the normalizer of $Γ_0(N)$

0.15 MB·English

by  Francesc Bars

THE GROUP STRUCTURE OF THE NORMALIZER OF Γ (N)
0
7
0 FRANCESCBARS
0
2

n
a
J Abstract. We determine the group structure of the normalizer of Γ0(N) in
3 SL2(R)moduloΓ0(N). Theseresultscorrect the Atkin-Lehner statement [1,
Theorem 8].
2

] 1. Introduction
T
N ThemodularcurvesX (N)containdeeparithmeticalinformation. Thesecurves
0
. are the Riemann surfaces obtained by completting with the cusps the upper half
h
plane modulo the modular subgroup
t
a
a b
m Γ (N)= SL (Z)c Z .
0 { Nc d ∈ 2 | ∈ }
[ (cid:18) (cid:19)

1 It is clear that the elements in the normalizer of Γ0(N) in SL2(R) induce auto-
morphisms of X (N) and moreover one obtains in that way all automorphisms of
v 0
6 X0(N) for N = 37 and 63 [3]. This is one reason coming from the modular world
6
3 thatshowstheinterestincomputingthegroupstructureofthisnormalizermodulo
6 Γ (N).
0
1
Morris Newman obtains a result for this normalizer in terms of matrices [5],[6],
0
see also the work of Atkin-Lehner and Newman [4]. Moreover, Atkin-Lehner state
7
0 without proof the group structure of this normalizer modulo Γ0(N) [1, Theorem
/ 8]. In this paper we correctthis statementandwe obtainthe rightstructure ofthe
h
normalizermoduloΓ (N). Theresultsareageneralizationofsomeresultsobtained
t 0
a in [2].
m
: 2. The Normalizer of Γ (N) in SL (R)
v 0 2
i
X Denote by Norm(Γ0(N) the normalizer of Γ0(N) in SL2(R).
r Theorem 1 (Newman). Let N =σ2q with σ,q N and q square-free. Let ǫ be the
a ∈
a b
gcdof all integers of the form a d where a,d are integers such that
− Nc d ∈
(cid:18) (cid:19)
Γ (N). Denote by v := v(N) :=gcd(σ,ǫ). Then M Norm(Γ (N)) if and only if
0 0
∈
M is of the form
r∆ u
√δ vδ∆
sN l∆
(cid:18) vδ∆ (cid:19)
with r,u,s,l Z and δ q, ∆ σ. Moreover v =2µ3w with µ=min(3,[1v (N)]) and
∈ | |v 2 2
w = min(1,[1v (N)]) where v (N) is the valuation at the prime p of the integer
2 3 pi i
N.
Date:Firstversion: December22th, 2006.
MSC:20H05(19B37,11G18).
1
2 FRANCESCBARS
This theorem is proved by Morris Newman in [5] [6], see also [2, p.12-14].
Observe that if gcd(δ∆,6)=1 we have gcd(δ∆2, N )=1 because the determi-
δ∆2
nant is one .
3. The group structure of Norm(Γ (N))/Γ (N)
0 0
InthissectionweobtainsomepartialresultsonthegroupstructureofNorm(Γ (N)).
0
Let us first introduce some particular elements of SL (R).
2
Definition 1. Let N be fixed. For every divisor m of N with gcd(m,N/m)=1
′ ′ ′
the Atkin-Lehner involution wm′ is defined as follows,
1 ma b
wm′ = √m′ (cid:18) N′c m′d (cid:19)∈SL2(R)
with a,b,c,d Z.
∈
1 1
Denote by Sv′ = 0 v1′ with v′ ∈N\{0}. Atkin-Lehner claimed in [1] the
(cid:18) (cid:19)
following:
Claim 2 (Atkin-Lehner). [1,Theorem8] The quotient Norm(Γ (N))/Γ (N) is the
0 0
direct product of the following groups:
(1) w for every prime q, q 5 q N.
{ qυq(N)} ≥ |
(2) (a) If υ (N)=0, 1
3
{ }
(b) If υ (N)=1, w
3 3
{ }
(c) If υ (N) = 2, w ,S ; satisfying w2 = S3 = (w S )3 = 1 (factor of
3 { 9 3} 9 3 9 3
order 12)
(d) Ifυ (N) 3; w ,S ;wherew2 =S3 =1andw S w
3 ≥ { 3υ3(N) 3} 3υ3(N) 3 3υ3(N) 3 3υ3(N)
commute with S (factor group with 18 elements)
3
(3) Let be λ=υ (N) and µ=min(3,[λ]) and denote by υ =2µ the we have:
2 2 ′′
(a) If λ=0 ; 1
{ }
(b) If λ=1; w
2
(c) Ifλ=2µ;{{w2}υ2(N),Sυ′′}withtherelationsw22υ2(N) =Sυυ′′′′ =(w2υ2(N)Sυ′′)3 =
1, where they have orders 6,24, and 96 for υ = 2,4,8 respectively.
(One needs to warn that for v = 8 the relations do not define totally
this factor group).
(d) If λ > 2µ; { w2υ2(N),Sυ′′}; w22υ2(N) = Sυυ′′′′ = 1. Moreover, Sυ′′ com-
mutes with w2υ2(N)Sυ′′w2υ2(N) (factor group of order 2υ′′2).
Let us give some partial results first.
Proposition 3. Suppose that v(N) = 1 (thus 4 ∤ N and 9 ∤ N). Then the Atkin-
Lehner involutions generate Norm(Γ (N)/Γ (N) and the group structure is
0 0
π(N)
= Z/2Z
∼
i=1
Y
where π(N) is the number of prime numbers N.
≤
Proof. Thisisclassicallyknown. Werecallonlythatwmm′ =wmwm′ for(m,m′)=
1 and easily wmwm′ =wm′wm; the the result follows by a straightforwardcompu-
tation from Theorem 1, see also [2, p.14]. (cid:3)
NORMALIZER OF Γ0(N) 3
When v(N)>1 itis clearthat some elementSv′ appearsin the groupstructure
of Norm(Γ (N))/Γ (N) from Theorem 1.
0 0
Lemma 4. If 4N the involution S Norm(Γ (N)) commutes with the Atkin-
2 0
| ∈
Lehner involutions wm with gcd(m,2)=1 and with the other Sv′.
Proof. By the hypothesis the following matrix belongs to Γ (N)
0
2mk2+2Nt+mkNt (2+2m)(2m+2mk+Nt)
w S w S = 2m 4m .
m 2 m 2 Nt(2m+2mk+Nt) m+Nt+ Nt + kNt + Nt2 !
2m m 2 4m
(cid:3)
Proposition 5. Let N = 2v2(N) ipnii, with pi different odd primes and assume
that v (N) 3, v (N) 1. Then Atkin-Lehner’s Claim 2 is true.
2 3
≤ ≤ Q
For the proof we need two lemmas.
Lemma 6. Let u˜ Norm(Γ (N)) and write it as:
0
∈
1 ∆2δr u
u˜= 2 ,
√δ∆2 sN l∆2δ
(cid:18) 2 (cid:19)
following the notation of Theorem 1. Then:
r u′
w∆2δu˜= s′2N′ v2′ !, if gcd(δ,2)=1,
1 2r u′′
w∆22δu˜= √2 s′2′N′′ 2v2′′ !, if gcd(δ,2)=2.
Proof. This is an easy calculation. (cid:3)
We study now the different elements of the type
r u′
a(r′,u′,s′,v′)= s′2N′ v2′ !,
1 2r u′′
b(r′′,u′′,s′′,v′′)= √2 s′2′N′′ 2v2′′ !.
Observe that b(,,,) only appears when N 0(mod 8).
≡
Lemma7. ForN 4(mod8)alltheelementsofthenormalizeroftypea(r ,u,s,v )
′ ′ ′ ′
≡
belong to the order six group S ,w S2 =w2 =(w S )3 =1 .
{ 2 4| 2 4 4 2 }
Proof. Straightforwardfrom the equalities:
a(r ,u,s,v ) Γ (N) s u 0(mod 2)
′ ′ ′ ′ 0 ′ ′
∈ ⇔ ≡ ≡
a(r′,u′,s′,v′)S2 Γ0(N) r′ v′ u′ 1 s′ 0(mod 2)
∈ ⇔ ≡ ≡ ≡ ≡
a(r′,u′,s′,v′)w4 Γ0(N) r′ v′ 0 u′ s′ 1(mod 2)
∈ ⇔ ≡ ≡ ≡ ≡
a(r ,u,s,v )w S Γ (N) r u s 1 v 0(mod 2)
′ ′ ′ ′ 4 2 0 ′ ′ ′ ′
∈ ⇔ ≡ ≡ ≡ ≡
a(r ,u,s,v )S w Γ (N) v u s 1 r 0(mod 2)
′ ′ ′ ′ 2 4 0 ′ ′ ′ ′
∈ ⇔ ≡ ≡ ≡ ≡
a(r ,u,s,v )S w S Γ (N) r v s 1 u 0(mod 2)
′ ′ ′ ′ 2 4 2 0 ′ ′ ′ ′
∈ ⇔ ≡ ≡ ≡ ≡
(cid:3)
4 FRANCESCBARS
Lemma 8. Let N be a positive integer with υ (N) = 3. Then all the elements
2
of the form a(r ,u,s,v ) and b(r ,u ,s ,v ) correspond to some element of the
′ ′ ′ ′ ′′ ′′ ′′ ′′
following group of 8 elements
S ,w S2 =w2 =1,S w S w =w S w S
{ 2 8| 2 8 2 8 2 8 8 2 8 2}
Proof. If follows from the equalities:
a(r ,u,s,v ) Γ (N) r v 1,u s 0(mod 2)
′ ′ ′ ′ 0 ′ ′ ′ ′
∈ ⇔ ≡ ≡ ≡ ≡
a(r′,u′,s′,v′)S2 Γ0(N) r′ v′ u′ 1,s′ 0(mod 2)
∈ ⇔ ≡ ≡ ≡ ≡
a(r′,u′,s′,v′)w8S2w8 Γ0(N) r′ v′ s′ 1,u′ 0(mod 2)
∈ ⇔ ≡ ≡ ≡ ≡
a(r ,u,s,v )S w S w Γ (N) r v s v 1(mod 2)
′ ′ ′ ′ 2 8 2 8 0 ′ ′ ′ ′
∈ ⇔ ≡ ≡ ≡ ≡
b(r ,u ,s ,v )w Γ (N) r v 0,u s 1(mod 2)
′′ ′′ ′′ ′′ 8 0 ′′ ′′ ′′ ′′
∈ ⇔ ≡ ≡ ≡ ≡
b(r ,u ,s ,v )S w S Γ (N) r v u s 1(mod 2)
′′ ′′ ′′ ′′ 2 8 2 0 ′′ ′′ ′′ ′′
∈ ⇔ ≡ ≡ ≡ ≡
b(r ,u ,s ,v )S w Γ (N) r 0,u s v 1(mod 2)
′′ ′′ ′′ ′′ 2 8 0 ′′ ′′ ′′ ′′
∈ ⇔ ≡ ≡ ≡ ≡
b(r ,u ,s ,v )w S Γ (N) v 0,u s r 1(mod 2)
′′ ′′ ′′ ′′ 8 2 0 ′′ ′′ ′′ ′′
∈ ⇔ ≡ ≡ ≡ ≡
(cid:3)
We can now proof Proposition 5].
Proof. [ of Proposition 5] Let N = 2υ2(N) ipnii, with pi different primes and
assumethat 9∤N. If υ (N) 1 we are done by proposition3. Suppose υ (N)=2
2 2
and let u˜ Norm(Γ (N)). B≤y lemmas 6 andQ7, w u˜ = α, α S ,w S2 = w2 =
∈ 0 δ ∈ { 2 4| 2 4
(w S )3 = 1 and it follows that u˜ = w α. Since w ((δ,2) = 1) commutes with
4 2 δ δ
S and the Atkin-Lehner involutions commute one to each other, we are already
2
done. In the situation 8 N the proof is exactly the same but using lemmas 6 and
8 instead. || (cid:3)
4. Counterexamples to Claim 2.
In the above section we have seen that Atkin-Lehner’s claim is true if v(N) 2
≤
i.e. for v (N) 3 and v (N) 1. Now we obtain counterexamples when v (N)
2 3 2
≤ ≤
and/or v (N) are bigger.
3
Lemma 9. Claim 2 for N =48 is wrong.
Proof. We know by Ogg [7] that X (48) is an hyperelliptic modular curve with
0
hyperelliptic involution not of Atkin-Lehner type. The hyperelliptic involution
always belongs to the center of the automorphism group. We know by [3] that
Aut(X (48))=Norm(Γ (48))/Γ (N). NowifClaim2wheretrue thisgroupwould
0 0 0
be isomorphic to Z/2 Π where Π is the permutation group of n elements. It
4 n
×
is clear that the center of this group is Z/2 1 , generated by the Atkin-Lehner
involution w , but this involution is not the×hy{pe}relliptic one. (cid:3)
3
The problem of N = 48 is that S does not commute with the Atkin-Lehner
4
involution w ; thus the direct product decomposition of Claim 2 is not possible.
3
This problem appears also for powers of 3 one can prove,
Lemma10. LetN =3v3(N) ipnii wherepi aredifferentprimes ofQ. Imposethat
TS3he∈reNfoorremi(fΓs0o(mNe))p.nTi hen1S(3mQcoodmumlou3te)stwheithCwlapimnii i2fiasnndootntlryuief.pnii ≡1(modulo3).
i ≡−
NORMALIZER OF Γ0(N) 5
Proof. LetusshowthatS3doesnotcommutewithwpnii ifandonlyifpnii ≡−1(mod
pnik 1
3). Observe the equality wpnii = √1pnii (cid:18) Ni t pnii (cid:19):
wpniS3wpniS32 =
i i
1 (pnik)2+Nt(1+ pniik) pnik(2pniik +1)+(pniik +1)(2Nt +pni)
i 3 i 3 3 3 i .
pnii Nt(pniik)+Nt(N3t +pnii) Nt(2pni3ik +1)+pnii(N3t +pnii)(2N3t +pnii) !
2k2pni pnik
For this element to belong to Γ (N) one needs to impose i + i Z. Since
pni 1 o 1(mod 3) it is need0ed that k 1(mod 3). Now3from de3t(w∈) =1 we
obita≡in that−pnik 1(mod 3); therefore pni≡ 1(mod 3). pi (cid:3)
i ≡ i ≡
5. The group structure of Norm(Γ (N))/Γ (N) revisited.
0 0
In this section we correct Claim 2. We prove here that the quotient
Norm(Γ (N))/Γ (N)
0 0
is the product of some groups associated every one of them to the primes which
divide N. See for the explicit result theorem 16.
Theorem 11. Any element w Norm(Γ (N)) has an expression of the form
0
∈
w =w Ω,
m
where w is an Atkin-Lehner involution of Γ (N) with (m,6) = 1 and Ω be-
m 0
longs to the subgroup generated by S and the Atkin Lehner involutions w ,
v(N) 2v2(N)
w . Moreover for gcd(v(N),23) 2 the group structure for the subgroup
3v3(N) ≤
< S ,w > and < S ,w > of < S ,w ,w >
v2(v(N)) 2v2(N) v3(v(N)) 3v3(N) v(N) 2v2(N) 3v3(N)
is the predicted by Atkin-Lehner at Claim 2, but these two subgroups do not neces-
sary commute withe each other element-wise.
Proof. Let us take any element w of the Norm(Γ (N)). By Theorem 1 we can
0
express w as follows,
r∆ u 1 rδ∆2 u
w =√δ vδ∆ = v
sN l∆ ∆√δ sN lδ∆2
(cid:18) vδ∆ (cid:19) (cid:18) v (cid:19)
LetusdenotebyU =2v2(N)3v3(N). Write∆′ =gcd(∆,N/U)andδ′ =gcd(δ,N/U);
then we obtain
wδ′∆′2w = ∆∆′ 1δ/δ′ r′vδNδ(′Nt∆∆′)′22 v′vδδ(u′N′∆∆)′22 !
Observe that if v(N)=1 we alreapdy finish and we reobtain proposition 3. This is
clear if gcd(N,6)=1; if not, the matrix wwδ′∆′2 is the Atkin-Lehner involution at
(∆)2 δ N.
∆′ δ′ ∈
Now we need only to check that any matrix of the form
(1) Ω= ∆∆′ 1δ/δ′ r′δδv′N((Nt∆∆′)′)2 v′δδv′(u(N′∆∆)′)2 !
p
6 FRANCESCBARS
is generated by S and the Atkin-Lehner involutions at 2 and 3 which are the
v(N)
factors of δ(∆)2. To check this observe that Ω=Ω Ω with
δ′ ∆′ 2 3
(2) Ω2= 2v2(∆∆′1√δ/δ′) 0B@ r′′22vv22(N(δvδt′(′N′(∆∆))′)2) v′′22vv22((uδvδ′′(′N(∆∆))′)2) 1CA
Ω3= 3v3(∆∆′1√δ/δ′) 0B@ r′′′33vv33N((vδtδ′(′′N′(∆)∆)′)2) v′′′33vv33(u(vδ′δ′(′′N(∆)∆)′)2) 1CA.
We only consider the case for Ω , the case for the Ω is similar. We can assume
2 3
that 2v2(∆∆′√δ/δ′) = 1 substituting Ω2 by w2v2(N)Ω2 if necessary. Thus, we are
r u′
reducedto a matrix of the form Ω˜2 = 2v2N(vt′(′N)) 2v2(vv′(N)) !. Now for some i we
r u
can obtain S2iv2(v(N))Ω˜2 = (cid:18) 2v2N(vt′(′N)) v′′ (cid:19); name this matrix by Ω2. Then, it is
easy to check that w Si w Ω Γ (N) for some i.
2v2(N) 2v2(v(N)) 2v2(N) 2 ∈ 0
Similar argument as above are obtained if we multiply w by w on the right,
m
i.e. ww is also some Ω as above obtaining similar conclusion.
m
LetusseenowthatthegroupgeneratedbyS andtheAtkin-Lehnerinvo-
v2(v(N))
lutionsat2,andthe groupgeneratedbyS andthe Atkin-Lehnerinvolution
v3(v(N))
at 3 have the structure predicted in Claim 2 when gcd(v(N),23) 2. We only
≤
need to check when v(N) is a power of 2 or 3 by (2). For v(N) = 1 the matrix
(1) is w (we denote w := id) (we have in this case a much deeper result,
δδ′(∆∆′)2 1
see proposition 3). Take now v(N) = 2. If l = gcd(3,δ/δ ) let Ω = w Ω; the
′ l ′
matrix Ω is as (1) but with gcd(3,δ/δ ) = 1, and δ ∆2 is only a power of 2.
′ ′ δ′∆′2
Then Ω < S ,w >, let us to precise the group structure. For v(N) = 2
′ ∈ 2 2v2(N)
we have v (N)=2 or 3, and we have already proved the group structure of Claim
2
[1] in lemmas 7,8 (we have moreover that Claim 2 is true because S commutes
2
with the Atkin-Lehner involutions wpni if (pi,2) = 1, see proposition 5). Assume
i
now v(N) = 3. If l = gcd(2,δ/δ ) and Ω = w Ω then Ω is as (1) but with
′ l ′ ′
gcd(2,δ/δ )= 1, and δ ∆2 is only a power of 3. Then Ω < S ,w >, let us
′ δ′∆′2 ′ ∈ 3 3v3(N)
toprecisethegroupstructure. Forv(N)=3wehavev (N) 2. Letusbeginwith
3
≥
v (N)=2, then Ω is of the form
3 ′
r u′
Ω′ = N3t′′ v3′ !=:a(r′,u′,t′,v′)
(from the formulation of Theorem 1 we can consider ∆ = 1 = δ because the
∆′ δ′
factors outside 3 does not appear if we multiply for a convenient Atkin-Lehner
involution, and for 3 observe that under our condition ∆=1) and we have
a(r ,u,t,v ) Γ (N) t u 0(mod 3)
′ ′ ′ ′ 0 ′ ′
∈ ⇔ ≡ ≡
a(r′,u′,t′,v′)w9 Γ0(N) r′ v′ 0(mod 3)
∈ ⇔ ≡ ≡
a(r ,u,t,v )S Γ (N) r +u t 0(mod 3)
′ ′ ′ ′ 3 0 ′ ′ ′
∈ ⇔ ≡ ≡
a(r ,u,t,v )S2 Γ (N) 2r +u t 0(mod 3)
′ ′ ′ ′ 3 ∈ 0 ⇔ ′ ′ ≡ ′ ≡
a(r ,u,t,v )S w Γ (N) r qt +v 0(mod 3)
′ ′ ′ ′ 3 9 0 ′ ′ ′
∈ ⇔ ≡ ≡
a(r ,u,t,v )S2w Γ (N) r 2qt +v 0(mod 3)
′ ′ ′ ′ 3 9 ∈ 0 ⇔ ′ ≡ ′ ′ ≡
a(r′,u′,t′,v′)w9S32 ∈Γ0(N)⇔r′+u′ ≡v′ ≡0(mod 3)
NORMALIZER OF Γ0(N) 7
a(r ,u,t,v )w S Γ (N) r +2u v 0(mod 3)
′ ′ ′ ′ 9 3 0 ′ ′ ′
∈ ⇔ ≡ ≡
a(r ,u,t,v )w S2w Γ (N) u qt +v 0(mod 3)
′ ′ ′ ′ 9 3 9 ∈ 0 ⇔ ′ ≡ ′ ′ ≡
a(r ,u,t,v )S2w S2 Γ (N) u 2qt +v 0(mod 3)
′ ′ ′ ′ 3 9 3 ∈ 0 ⇔ ′ ≡ ′ ′ ≡
a(r ,u,t,v )S2w S Γ (N) r +u 2tq+v 0(mod 3)
′ ′ ′ ′ 3 9 3 ∈ 0 ⇔ ′ ′ ≡ ′ ′ ≡
a(r ,u,t,v )S w S2 Γ (N) 2r +u qt +v 0(mod 3)
′ ′ ′ ′ 3 9 3 ∈ 0 ⇔ ′ ′ ≡ ′ ′ ≡
and these are all the possibilities, proving that the group is S ,w S3 = w2 =
{ 3 9| 3 9
(w S )3 = 1 of order 12. Observe that S does not commute with w (see for
9 3 3 2
}
example lemma 7).
Suppose now that v (N) 3. We distinguish the cases v (N) odd and v (N)
3 3 3
≥
even. Suppose v (N) is even, then δ =1 and Ω has the following form
3 δ′ ′
1 r (∆)2 u′
′ ∆′ 3
∆∆′ N3t′ v′(∆∆′)2 !
with α := ∆/∆′ dividing 3[v3(N)/2]−1. Since this last matrix has determinant 1
we see that α satisfies gcd(α,N/(32α2)) = 1; thus α = 1 or α = 3[v3(N)/2]−1.
r u′
Write a(r′,u′,t′,v′) = N3t′′ v3′ ! when we take α = 1 and b(r′,u′,t′,v′) =
r′(33[v[v33N((NNt′))//22]]−1) v′(33[v[v33(u(NN′))//22]]−1) ! when α = 3[v3(N)/2]−1. It is easy to check
that b(r ,u,t,v ) = w a(r ,u,t,v ) and that the group structure is the pre-
′ ′ ′ ′ 3v3(N) ′ ′ ′ ′
dicted in a similar way as the one done above for v(N) = 2. Suppose now that
v3(N) is odd, then δδ′ is 1 or 3 and ∆∆′ divides 3[v3(N)/2]−1. Now from det() = 1
we obtain that the only possibilities are δ = 1 = ∆ name the matrices for this
δ′ ∆′
case following equation 1 by a(r ,u,t,v ), and the other possibility is δ = 3
′ ′ ′ ′ δ′
and ∆∆′ = 3[v3(N)/2]−1, write the matrices for this case following equation 1 by
c(r ,u,t,v ). It is also easy to check that c(r ,u,t,v ) = w a(r ,u ,t ,v ),
′ ′ ′ ′ ′ ′ ′ ′ 3v3(N) ′′ ′′ ′′ ′′
and that the group structure is the predicted.
(cid:3)
Corollary12. LetN =3v3(N) ipnii,withpidifferentprimessuchthatgcd(pi,6)=
1. Suppose that v(N)=3 and pni 1(mod 3) for all i. Then Claim 2 is true.
Qi ≡
Proof. Fromtheproofoftheabovetheorem11forv(N)=3withv (N) 2,lemma
3
≥
10, and that the general observation that the Atkin-Lehner involutions commute
one with each other we obtain that the direct product decomposition of Claim 2 is
true obtaining the result. (cid:3)
Now we showsthe correctionsto Claim2 forv(N)=4andv(N)=8,aboutthe
groupstructureofthe subgroupof Norm(Γ (N))/Γ (N)generatedfor S andthe
0 0 2k
Atkin-Lehner involution at prime 2.
Proposition13. Supposev(N)=4,observethatinthissituationv (N)=4,or5.
2
Then the group structure of the subgroup < w ,S > of Norm(Γ (N))/Γ (N)
2v2(N) 4 0 0
is given by the relations:
(1) For v (N)=4 we have S4 =w2 =(w S )3 =1.
2 4 16 16 4
(2) For v (N)=5 we have S4 =w2 =(w S )4 =1.
2 4 32 32 4
8 FRANCESCBARS
Proof. It is a straightforward computation. Observe that for v (N)=4 the state-
2
ment coincides with Claim2 but not for v (N)=5, where one checks that S does
2 4
not commute with w S w . (cid:3)
32 4 32
Proposition 14. Suppose v(N) = 8 and v (N) even (this is the case (3)(c) in
2
Claim 2). Then the group < w ,S > Norm(Γ (N))/Γ (N) satisfies the
2v2(N) 8 ⊆ 0 0
following relations: S8 =w2 =1, and
8 2v2(N)
(1) for v (N)=6 we have (w S )3 =1,
2 64 8
(2) for v (N) 8 we do not have the relation (w S )3 =1,
2 ≥ 2v2(N) 8
(3) for v (N) 10 we have the relation: S commutes with w S w ,
2 ≥ 8 2v2(N) 8 2v2(N)
(4) for v (N) = 6 or 8 we do not have the relation: S commutes with the
2 8
element w S w .
2v2(N) 8 2v2(N)
(5) For v (N)=8 we have the relation: w S w S w S3w S3 =1.
2 256 8 256 8 256 8 256 8
Proof. Straightforward. (cid:3)
Proposition15. Supposev(N)=8andv (N)odd(thisisthecase(3)(d)inClaim
2
2). Then the group < w ,S > Norm(Γ (N))/Γ (N) satisfies the following
2v2(N) 8 ⊆ 0 0
relations: S8 =w2 =1, and
8 2v2(N)
(1) for v (N)=7 (w S )4 =1,
2 128 8
(2) for v (N) 9 we do not have the relation (w S )4 =1,
2 ≥ 2v2(N) 8
(3) for v (N) 9 we have the Atkin-Lehner relation: S commutes with
2 8
≥
w S w ,
2v2(N) 8 2v2(N)
(4) for v (N)=7 we do not have that S commutes with w S w .
2 8 128 8 128
Proof. Straightforward. (cid:3)
Let us finally write the revisited results concerning Claim 2 that we prove;
Theorem 16. The quotient Norm(Γ (N))/Γ (N) is a product of the following
0 0
groups:
(1) w for every prime q, q 5 q N.
{ qυq(N)} ≥ |
(2) (a) If υ (N)=0, 1
3
{ }
(b) If υ (N)=1, w
3 3
{ }
(c) If υ (N) = 2, w ,S ; satisfying w2 = S3 = (w S )3 = 1 (factor of
3 { 9 3} 9 3 9 3
order 12)
(d) Ifυ (N) 3; w ,S ;wherew2 =S3 =1andw S w
3 ≥ { 3υ3(N) 3} 3υ3(N) 3 3υ3(N) 3 3υ3(N)
commute with S (factor group with 18 elements)
3
(3) Let be λ=υ (N) and µ=min(3,[λ]) and denote by υ =2µ the we have:
2 2 ′′
(a) If λ=0 ; 1
{ }
(b) If λ=1; w
2
{ }
(c) If λ = 2µ and 2 ≤ λ ≤ 6; {w2υ2(N),Sυ′′} with the relations w22υ2(N) =
Sυυ′′′′ = (w2υ2(N)Sυ′′)3 = 1, where they have orders 6,24, and 96 for
υ =2,4,8 respectively.
(d) If λ > 2µ and 2 ≤ λ ≤ 7; { w2υ2(N),Sυ′′}; w22υ2(N) = Sυυ′′′′ = 1.
Moreover, (w2υ2(N)Sυ′′)4 =1.
(c˜),(d˜) If λ 9; w ,S with the relations w2 = S8 = 1 and S
≥ { 2υ2(N) 8} 2υ2(N) 8 8
commutes with w S w .
2v2(N) 8 2v2(N)
(cˆ) If λ = 8; w ,S with the relations w2 = S8 = 1 and
{ 2υ2(N) 8} 2υ2(N) 8
w S w S w S3w S3 =1.
256 8 256 8 256 8 256 8
NORMALIZER OF Γ0(N) 9
Observation 17. One needs to warn that for the situation v(N) = 8 possible the
relations does not define totally the factor group, but it is a computation more.
Observation 18. The product between the different groups appearing in theorem
16 is easily computable. Effectively, we know that the Atkin-Lehner involutions
commute, and S commutes with S . Moreover S commutes with
2v2(v(N)) 3v3(v(N)) 2
any element from lemma 4. Consider w an Atkin-Lehner involution for X (N)
pn 0
with p a prime. One obtains the following results by using the same arguments
appearing in the proof of lemma 10;
(1) let p be coprime with 3 and 3v(N). S commutes with w if and only if
3 pn
|
pn 1(modulo 3). If pn 1(modulo 3) then w S =S2w .
≡ ≡− pn 3 3 pn
(2) Let p be coprime with 2 and 4v(N). S commutes with w if and only if
4 pn
|
pn 1(modulo 4). If pn 1(modulo 4) then w S =S3w .
≡ ≡− pn 4 4 pn
(3) Let p be coprime with 2 and 8v(N). Then, w S = Skw if pn
| pn 8 8 pn ≡
k(modulo 8), in particular S commutes with w if and only if pn
8 pn
≡
1(modulo 8).
References
[1] Atkin,A.O.L.,Lehner,J.: Heckeoperator onΓ0(N). Math.Ann.185,134-160(1970).
[2] Bars,F.: Determinacio´ de les corbes X0(N) biel.l´ıptiques.Tesina UniversitatAuto`noma de
Barcelona.September 1997.Findacopyathttp://mat.uab.es/∼francesc/publicac.html
[3] Kenku,M.A.,Momose,F.: AutomorphismsgroupsofthemodularcurvesX0(N).Compositio
Math.65,51-80,(1988).
[4] Lehner,J.,Newman,M.: WeierstrasspointsofΓ0(N).Ann.ofMath.79,360-368(1964).
[5] Newman,M.: Structuretheorem formodularsubgroups.DukeMath.J.22,25-32(1955).
[6] Newman,M.: Conjugacy, genus andclassnumbers.Math.Ann.196m198-217(1975).
[7] Ogg,A.P.: Hyperellipticmodularcurves.Bull.Soc.math.France102,449-462(1974).
FrancescBarsCortina,Depart. Matema`tiques,UniversitatAut`onomadeBarcelona,
08193 Bellaterra. Catalonia. Spain. E-mail: [email protected]