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Fake Congruence Modular Curves and Subgroups of the Modular Group
Journal of Algebra 214, 276᎐300 Ž1999. Article ID jabr.1998.7678, available online at http:rrwww.idealibrary.com on
Fake Congruence Modular Curves and Subgroups of the Modular GroupU Gabriel Berger † Department of Mathematics, Uni¨ ersity of California at Ir¨ ine, Ir¨ ine, California 92697-3875 E-mail: [email protected] Communicated by Walter Feit Received April 13, 1998
We construct a special class of noncongruence modular subgroups and curves, analogous in some ways to the usual congruence ones. The subgroups are obtained via the Burau representation, and the associated quotient curves have a natural moduli space interpretation. In fact, they are reduced Hurwitz spaces corresponding to covers with 4 branch points and monodromy group equal to semi-direct products of a cyclic and an abelian group. Furthermore, they form a modular tower in the sense of Fried. We study representations on the cohomology of these fake congruence modular curves and also calculate the genera of certain quotient curves. 䊚 1999 Academic Press Key Words: Burau representation; fake congruence subgroup; Hurwitz space; modular tower.
1. INTRODUCTION The present work has two starting points. The first is the work of Oda and Terasoma on the Burau representation of the Artin braid group Bn . This representation is, in general, a homomorphism
n : Bn ª GLny1 Ž ⺪ w t , ty1 x . . U
Ž 1.
Supported in part by JSPS Grant P94015 and NSA Grant 032596. The author expresses his thanks and acknowledges his debts to Takayuki Oda for introducing the topics considered in this paper and to Mike Fried for valuable discussions regarding Hurwitz spaces and connections with the modular group. We note that similar topics have been considered earlier by Helmut Voelklein, whose primary interest was the realization of certain finite groups of Lie type as Galois groups over ⺡. †
276 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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However, in wO-Tx and this paper, it is shown that under certain conditions and for certain prime powers q, the map 3 induces a surjective homomorphism
: PSL2 Ž ⺪ . ª PSL2 Ž ⺖q . ,
Ž 2.
where ⺖q is the finite field with q elements in it. ŽIn fact, we can define such a in more generality. But then all we would know about its image is N ., where is a root that it is a certain unitary subgroup of PGL2 Ž⺪w xrN w x . of unity and N is an ideal in ⺪ . Recall that PSL2 Ž⺪. is generated by the images mod ² " 1: of Ss
ž
1 0
1 1
/
Us
and
ž
1 y1
0 , 1
/
with the relations Ž SU . 3 s Ž SUS . 2 s 1. ŽHere and subsequently, all matrices will be taken mod ² "1:.. Thus B3 maps surjectively onto PSL2 Ž⺪. via
1 ¬ S,
2 ¬ U.
Ž 3.
The kernel is the center ZŽ B3 ., which is generated by Ž 1 2 . 3. The second starting point is the following result of Fried wFx, which gives a new, combinatorial group-theoretic way to approach congruence subgroups. Suppose N is an odd integer and DN is the dihedral group ²␥ , t : ␥ 2 s t N s 1, ␥ t␥y1 s ty1 : .
Ž 4.
Observe that DN is the semi-direct product of ⺪rN and ⺪r2. Embed DN in the symmetric group SN via the permutation representation on the N cosets of ²␥ :, and take C to be the conjugacy class of involutions. Let X be the set
Ž a, b, c, d . g C 4 : ² a, b, c, d : s DN ,
and
a ⭈ b ⭈ c ⭈ d s 1 4 rNS N Ž C . ,
Ž 5. where NS NŽ C . is the normalizer of C in S N . We can define a right action of PSL2 Ž⺪. on X via
Ž a, b, c, d . S s Ž abay1 , a, c, d .
Ž 6.
Ž a, b, c, d . U s Ž a, bcby1 , b, d . .
Ž 7.
Fried’s theorem then states that the stabilizer of the element
Ž ␥ , ␥ , t␥ , t␥ . is ⌫0 Ž N ..
mod NS N Ž C .
Ž 8.
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GABRIEL BERGER
In fact, in wF, B-Fx it is shown that given any transitive subgroup G of Sn and conjugacy classes Ž C1 , . . . , C4 . of G, we can obtain a finite-index subgroup ⌬ of SL 2 Ž⺪.. Moreover, the associated quotient curve ᒅr⌬ has a moduli space interpretation. We therefore obtain a way of generating noncongruence subgroups of SL 2 Ž⺪. with additional structure associated to them. In wB2x, the following is shown. Suppose G is the semi-direct product N i Ž⺪w xrN N .U , for d an integer and N an ideal of ⺪w x relatively ⺪w xrN prime to d. Let Ž C1 , . . . , C4 . s Ž C , C , C , Cy3 ., with C the conjugacy class N .. of . Then our ⌬ is in fact equal to the stabilizer of ⬁ g ⺠ 1 Ž⺪w xrN That is, ⌬ s y1 Ž B ., where B is the standard Borel subgroup
ž
) 0
) )
/
N . and is the map obtained from the Burau representaof PGL2 Ž⺪w xrN tion as above. Equivalently, ᒅr⌬ is a course moduli space for Kummer N. covers X of ⺠ 1 together with a subgroup of J Ž X . isomorphic to ⺪w xrN If G s DN , then ⌬ is just ⌫0 Ž N .. We thus obtain a noncongruence modular curve with a moduli space interpretation which generalizes X 0 Ž N .. Furthermore, if we fix d and let N vary through powers of a prime p, then for certain p we obtain a modular tower Žsee wF2x.. It is our hope that these so-called ‘‘fake congruence subgroups and modular curves’’ will provide a fertile testing ground for many of the objects and theories associated with the usual congruence ones ŽHecke operators, action of Frobenius, . . . .. Accordingly, we have begun their study in this paper. The contents are as follows: in Section 2, we outline the basic framework in which we intend this work to be considered, including a review of Hurwitz spaces and their relation to noncongruence modular curves. In Section 3, we use the Oda᎐Terasoma theorem to construct fake congruence subgroups and fake congruence modular curves. In Section 4, we determine the character multiplicities of the representation of PSL2 Ž⺖q . on the cohomology of our fake congruence modular curves. Finally, in Section 5, we determine the genera of various quotient curves.
2. HURWITZ SPACES OF FOUR-BRANCH POINT COVERS AND FAKE CONGRUENCE SUBGROUPS 2.1. Re¨ iew of Hurwitz Spaces. The references for this section are wB-F, F-Vx. Let G be a finite group embedded as a transitive subgroup of Sn for some positive integer n. Let C s Ž C1 , . . . , C4 . e a quadruple of conjugacy classes of G. We are interested in parametrizing covers of ⺠ 1 Žover ⺓.
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ramified over four points with monodromy group G and ramification data C. To do so, we introduce the following definitions. DEFINITION 1. We set U 4 s Ž z1 , . . . , z 4 . g Ž⺠ 1 . 4 : z i / z j for i / j4 , and let U4 s U 4rS 4 denote the natural quotient. The Nielsen class associated to the data Ž G, C. is NiŽ G, C. and is
Ž g 1 , . . . , g 4 . g G 4 : ² g 1 , . . . , g 4 : s G, g 1 and
⭈⭈⭈ g 4 s 1
g i g Ci
for some g S4 4 .
We also set NiŽ G, C. ab s NiŽ G, C.rNS nŽC., where NS nŽC. is the normalizer of C in S n . Finally, the Hurwitz monodromy group H4 is the fundamental group of U4 . It has the presentation ² Q1 , Q2 , Q3 : Q i Q iq1 Q i s Q iq1 Q i Q iq1 , Q1 Q3 s Q3 Q1 , Q1 Q2 Q3 Q3 Q2 Q1 s 1: . Note that H4 is very closely related to B3 , since B3 is the fundamental group of unordered, distinct triples of ⺓ 1 s ⺠ 1 y ⬁4 . In fact, Hurwitz space theory can be formulated using B3 instead of H4 . We have an action of H4 on NiŽ G, C. ab as follows: if Ž g 1 , . . . , g 4 . g NiŽ G, C. ab , then
Ž g 1 , . . . , g 4 . Qi s Ž g 1 , . . . , g iy1 , g i g iq1 gy1 i , g i , g iq2 , . . . , g 4 . . ŽThus Q i sends g i to g i g iq1 gy1 i , sends g iq1 to g i , and fixes the other two elements of the quadruple.. Since H4 is the fundamental group of U4 , each of its orbits on NiŽ G, C. ab corresponds to a connected cover of U4 . Let H Ž G, C. denote the disjoint union of these covers. THEOREM 1 wF-Vx. H Ž G, C. is a coarse moduli space for co¨ ers of ⺠ 1 with monodromy group G and ramification data C. In fact, this theorem holds for covers with an arbitrary number of branch points. 2.2. PSL 2 Ž⺓. Action. We have canonical PSL2 Ž⺓. actions on U 4 , U4 , and H s H Ž G, C. as above: if ␥ g PSL2 Ž⺓., then ␥ Ž a, b, c, d . s Ž␥ a, ␥ b, ␥ c, ␥ d .. Also, given a cover : X ª ⺠ 1, ␥ acts on this cover by taking it to the composite ␥ . We denote the quotients of these actions by U 4 r e d, U4r e d, and H r e d, respectively. The following result is due to Thompson Žsee wFx..
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PROPOSITION 1. There is a surjecti¨ e homomorphism
: H4 ª PSL2 Ž ⺪ . gi¨ en by
Ž Q1 . s S s
ž
1 0
1 , 1
/
Ž Q2 . s U s
Ž Q3 . s S s
ž
1 0
ž
1 y1
0 , 1
/
1 . 1
/
The kernel is the quaternion group Q8 of order 8. THEOREM 2 wB-Fx. Suppose HO is a connected component of H . Then there exists a subgroup ⌬ of finite index in PSL2 Ž⺪. such that we ha¨ e the following commutati¨ e diagram: H rOe d ª ᒅr⌬ x x U4r e d ªᒅrPSL2 Ž⺪. The horizontal maps are isomorphisms, and the bottom isomorphism is obtained by taking any a, b, c, d4 to the unique Ž up to isomorphism. elliptic cur¨ e ramified o¨ er a, b, c, d4 . Furthermore, ⌬ arises as follows: by construction, H r0e d corresponds to a subgroup of finite index ⌺ in H4 . Then ⌬ s Ž ⌺ ., where is the homomorphism of Proposition 1. 2.3. The Semidirect Product of Two Abelian Groups and the Burau Representation. Our general set up is as follows. Suppose d G 1 is an integer, and is a primitive dth root of unity. Let N be an ideal of ⺪w x coprime to d, and by abuse of notation, continue to denote the image of in N by . Subsequently, by ⺪w xrN N we will mean the additive group of ⺪w xrN N unless otherwise noted; Ž⺪w xrN N .U will denote the multiplicative ⺪w xrN N. We can embed Ž⺪w xrN N .U into the automorgroup of Žthe ring. ⺪w xrN N by setting ␥ Ž x . s ␥ x for ␥ g Ž⺪w xrN N .U and phism group of ⺪w xrN N. Thus we may form the semidirect product x g ⺪w xrN U
N i Ž ⺪ w x rN N. . ⺪ w x rN N and symbols w ␥ x for ␥ g Ž⺪w xrN N .U , This group is generated by ⺪w xrN with the relations
w ␥ xw ␣ x s w ␥␣ x for ␣ g Ž ⺪ w x rNN . and
U
w ␥ x y1 t w ␥ x s ␥ t for t g ⺪ w x rNN .
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281
ŽIn the second equality, the left-hand multiplication is in ⺪w xrN Ni Ž⺪w xrN N .U , and the right-hand multiplication is in the ring ⺪w xrN N.. Let N i ² : of ⺪w xrN N i Ž⺪w xrN N .U . Notice that G be the subgroup ⺪w xrN U N i Ž⺪w xrN N . acts on ⺪w xrN N as a set: the additive group ⺪w xrN N ⺪w xrN N .U acts acts upon itself by addition, and the multiplicative group Ž⺪w xrN N by multiplication. Thus if 噛⺪w xrN N s N, we obtain an embedon ⺪w xrN N i Ž⺪w xrN N .U and its subgroup G into SN , given as ding of ⺪w xrN N and x w ␥ x g ⺪w xrN N i Ž⺪w xrN N .U , then follows: if t g ⺪w xrN t Ž xw␥ x. s Ž t q x . w␥ x . N i Ž⺪w xrN N .U and G with their images in SN . Let C be Identify ⺪w xrN the conjugacy class of w x. Set C s Ž C1 , . . . , C4 . s Ž C , C , C , Cy3 .. Now let ab
X s Ni Ž G, C . rQ8 ,
Ž 9.
where Q8 is the kernel of as in Proposition 1. Then we have a well-defined action of PSL2 Ž⺪. on X. THEOREM 3 wB2x. Ž1. The action of PSL2 Ž⺪. on X is transiti¨ e. Ž2. We can identify X with ⺠ 1 Ž⺪w xrN N .. Furthermore, we ha¨ e the following commutati¨ e diagram, where the right hand map is as defined in Subsection 3.2 and the left hand map is as described in the Introduction. id
6
PSL 2 Ž⺪.
6
PSL2 Ž⺪. 6
N. PGL2 Ž⺪w d xrN
6
AutŽ X .
In addition, the stabilizer of the image of Ž␥ , ␥y1 , t␥ , ␥y1 ty1 . in X is equal to the subgroup ⌫0 Ž N . s ␣ g PSL2 Ž ⺪ . : Ž ␣ . ⬁ s ⬁ 4 . Ž3. The cur¨ e Y0 Ž N . s ᒅr⌫0 Ž N . is a reduced Hurwitz space for co¨ ers of ⺠ 1 with monodromy group G and ramification data C. Thus each point on Y0 Ž N . corresponds to an equi¨ alence class of diagrams of the following form: X ␥
6

⺠1
6
6
W
␣
6
Y
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GABRIEL BERGER
We require all maps to be ramified o¨ er 4 points, and the ¨ ertical maps to be Galois, with monodromy group ⺪rd⺪ s ² :. In addition, the map ␥ should ha¨ e ramification data of the form Ž , , , y3 . Ž here, is its own conjugacy N. class in ⺪rd⺪ s ² :.. Furthermore, ␣ must be Galois with group ⺪w xrN Finally, we require  to ha¨ e monodromy group G and ramification data Ž C , C , C , Cy3 .. We note that once we specify the map  , Y and X will be uniquely determined. Two diagrams are considered to be equi¨ alent if and only if the lower horizontal maps are equi¨ alent mod PSL2 Ž⺓.. 3. CONSTRUCTION OF THE ⌫ Ž N . 3.1. Fake Congruence Subgroups. To begin with, we fix natural numbers n G 3, d G 7, a primitive dth root of unity , K s ⺡Ž ., F s ⺡Ž q y1 ., N q; OF an ideal, and N s N q OK . Let Bn denote the Artin braid group ² 1 , . . . ny1 : i j s j i
for and
i iq1 i s iq1 i iq1 : .
The reduced Burau representation Žsee wBix. is a homomorphism n : Bn ª GLny1Ž⺪w t, ty1 x.. It can be obtained, among other methods, via the Fox free calculus, or from the action of Bn on the homology of an infinite cyclic extension of ⺠ 1 ramified over n q 1 points. Its action on the generators of Bn is given as yt 0 1 ¬ 0 0 0
⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈ 0
1 1 0 ⭈⭈⭈ 0
0 0 1 ⭈⭈⭈ 0
0 1 t 0 0
⭈⭈⭈ 0 yt 0 ⭈⭈⭈
0 ⭈⭈⭈ 1 1 0
1 0 ny 1 ¬ 0 0 0
0 1 ⭈⭈⭈ 0 0
1 ry2 0 r ¬ ⭈⭈⭈ ⭈⭈⭈ 0
0 0 0 , 0 1 0 0 0 0
Ž 1 - r - n y 1.
1 ny ry2
and ⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈
0 0 ⭈⭈⭈ 1 t
0 0 0 . 0 yt
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283
N , and Setting t s and reducing by N gives us a map ⺪w t, ty1 x ª OKrN N .. We then obtain a new map hence a map GLnŽ⺪w t, ty1 x. ª GLnŽ OKrN N. n , N : Bn ª GLny1 Ž OKrN by composing with the Burau representation. Now suppose n s 3. In this case we can describe our map as follows. First, notice that B3 has presentation ² 1 , 2 : 1 2 1 s 2 1 2 :. Then 3, N is given by
1 ª 2 ª
ž ž
y 0 1
1 1
/ /
Ž 10 .
0 . y
Ž 11 .
N. where we understand the matrix entries to be elements of OKrN Recall that ZŽ B3 . is generated by Ž 1 2 . 3 and observe that
3, N Ž Ž 1 2 .
3
.s
ž
3
0
0
3
/
.
Thus 3, N induces a map N. . N : PSL2 Ž ⺪ . s B3rZ Ž B3 . ª PGL2 Ž OKrN
Ž 12 .
DEFINITION 2. We have the following analogs of congruence groups and modular curves: Ž1. ⌫ Ž N . s kerŽN . Ž2. ⌫0 Ž N . s ␥ g PSL2 Ž⺪.: N Ž␥ .⬁ s ⬁ Žwhere we take ⬁ g N .. ⺠ 1 Ž OKrN Ž3. X Ž N . s ᒅU r⌫ Ž N . Ž4. X 0 Ž N . s ᒅU r⌫0 Ž N . Ž5. Y Ž N . s ᒅr⌫ Ž N . Ž6. Y0 Ž N . s ᒅr⌫0 Ž N .. Any subgroup of PSL2 Ž⺪. that contains ⌫ Ž N . for some N will be called a fake congruence subgroup. If d s 2, we obtain the usual congruence objects with N replaced by N. Recall that PSL2 Ž⺪. is the free product of the group of order 2 generated by USU s Ž 01 y10 . and the group of order 3 generated by SU s Ž y10 11 ..
284
GABRIEL BERGER
DEFINITION 3. Let k be a positive integer. The Hecke Triangle Group T Ž2, 3, k . is the group ²  2 ,  3 ,  k :  22 s  33 s  kk s  2  3  k s 1: . We have a canonical surjection : PSL2 Ž⺪. ª T Ž2, 3, k . given by 0 y1 0 1 SU s ¬ ␥3. ¬ ␥2 , 1 0 y1 1 y1 Ž . Since S s ŽUSUSU .y1 and ␥ k s ␥y1 3 ␥ 2 , we also see that S s ␥ k . N ., we have Now note that in our map N : PSL2 Ž⺪. ª PGL2 Ž⺪w xrN USU s
ž
/
ž
N Ž S r . s 3, N Ž 1r . s
ž
Ž y .
r
Ž Ž y .
0
/
r
y 1 . r Ž y y 1 . . Ž 13 . 1
/
Thus if we set k s 噛² y :, then N Ž S k . s 1, and so N factors through T Ž2, 3, k .. We set N. .. ⌬ Ž N . s ker Ž T Ž 2, 3, k . ª PGL2 Ž ⺪ w x rN
Ž 14 .
3.2. The Oda᎐Terasoma Theorem. By the results of wO-Tx, the image of Bn under n is unitary with respect to a certain hermitian form H g GLny 1Ž⺪w t, ty1 x.. ŽHere, conjugation is the map t ¬ ty1 .. H is given by y
1 y
1 y1
t
⭈⭈⭈
y
0
tq1 1
q1
0
1
y
1 y1
t
q1 ⭈⭈⭈
1 tq1 1 ⭈⭈⭈
y
0
⭈⭈⭈
0
⭈⭈⭈
1 tq1 ⭈⭈⭈
.
Ž 15 .
0 ⭈⭈⭈
N . Žalso denoted H .. This form then descends to a form in GLny 1Ž OKrN In fact, one can show that the image of n, N is contained in N . s g g GLny1 Ž OKrN N . : g t Hg s H , det g g ² y : 4 . Uny 1 Ž , H , OKrN The main theorem of wO-Tx is that under certain conditions, Bn actually N .. surjects onto Uny 1Ž , H, OKrN THEOREM 4 wO-Tx. Assume n G 3, d G 7 if d is odd, d G 14 if d is e¨ en, N is coprime with d and ŽŽ1 y i .rŽ1 y .. Ž2 F i F n., the prime factors ᒎ of N di¨ iding 6 satisfy NK r ⺡ Ž ᒎ . G 10, N is coprime to n y 4, and d is odd when n s 4. Then N. n , N : Bn ª Uny1 Ž , H , OKrN is surjecti¨ e. A special case of this is the following.
FAKE CONGRUENCE MODULAR CURVES
285
THEOREM 5. Hypotheses as abo¨ e, suppose n s 3 and Nq is a prime ideal ᒍq. Set ᒍ s ᒍqOK . Then the map
ᒍ : PSL2 Ž ⺪ . ª PU2 Ž , H , OKrᒍ . is surjecti¨ e. 3.3. Analysis of 3 . We must now analyze 3 in more detail. LEMMA 1. Let f s ord d p s min a g ⺞: p a ' 1 mod d4 . Let ᒍq; OF be a prime o¨ er p. Then ᒍq Ž1. remains prime in OK if f is e¨ en, and 噛Ž OKrᒍq OK . s p f , 噛O OF r ᒍ s p f r2 , Ž2. splits into two primes in OK if f is odd, and 噛Ž OF rᒍq. s p f. q
Proof. Let Tq Ž X . s
Ł Ž X y Ž i q yi . .
Ž i , d .s1 0-i-dr2
be the irreducible polynomial of q y1 over ⺪. We must factor Tq in ⺖p w X x; the degree of each factor will be the residue field index of ᒍq Žsee wCx... Choose a primitive dth root of unity ␥ g ⺖p . Then Tq Ž X . s
Ł Ž X y Ž ␥ i q ␥yi . .
Ž i , d .s1 0-i-dr2
is a factorization of Tq over ⺖p . Consider
Ł
0Fj-f
ž X y Ž␥
pj
q ␥yp
j
. / g ⺖p w X x .
This is irreducible if and only if f is odd Žotherwise, it will be a square.. Thus f Ž ᒍqrp . s fr2 if f is even and f Ž ᒍqrp . s f if f is odd. The same argument shows, upon factoring TŽ X. s
Ł Ž X y ␥ i.
Ž i , d .s1 0-i-d
in ⺖p w X x, that any prime ᒍ ; OK over p satisfies f Ž ᒍrp . s f. Since ᒍq cannot ramify in OK by assumption, the result follows.
286
GABRIEL BERGER
LEMMA 2.
PU2 Ž OKrᒍ . : PSU2 Ž OKrᒍ . s
½
2, 1,
if q is odd if q is e¨ en.
Proof. We prove the lemma for OKrᒍ s ⺖q 2 ; the proof for OKrᒍ s ⺖q = ⺖q is similar. Note that if x g U2 Ž⺖q 2 . and detŽ x . s a, then aa s a qq 1 s 1. Thus the image x of x in PU2 Ž⺖q 2 . is actually in PSU2 Ž⺖q 2 . if and only if there exists some g ⺖q 2 with 2 s a, qq 1 s 1. Thus if q is odd, then x g PSU2 Ž⺖q 2 . m aŽ qq1.r2 s 1. So in this case, PU2 Ž ⺖q 2 . s PSU2 Ž ⺖q 2 . j PSU2 Ž ⺖q 2 . x, where x is any element of U2 Ž⺖q 2 . satisfying detŽ x .Ž qq1.r2 s y1. If q s 2 r and a qq 1 s 1, set s a q r2q1. Then 2 s a and qq 1 s 1. So PU2 Ž⺖q 2 . s PSU2 Ž⺖q 2 .. We thus observe that when we factor through to PSL2 Ž⺪., the worst we can do is surject onto a group in which PSL2 Ž⺖q . has index 2. We now list the cases in which we land precisely on PSL2 Ž⺖q .. PROPOSITION 2. With the notation as abo¨ e, if Ž p . s ᒍ l ⺪, then PU2 Ž , OKrᒍ . s PSU2 Ž OKrᒍ . if and only if the following conditions hold: ᒍ is prime and p is odd. 2 ¦ d: q ' y1 mod 4 2 5 d: always 4 N d: 2 N Ž q q 1.rd. Case II. ᒍ s ᑜ ᑜ and p is odd. Ž1. 2 ¦ d: q ' 1 mod 4 Ž2. 2 5 d: always Ž3. 4 N d: 2 N Ž q y 1.rd. Case III. p s 2: always.
Case I. Ž1. Ž2. Ž3.
Proof. Recall that all elements in PU2 Ž , OKrᒍ . have determinant contained in ² y :. Case I. We must have Žy .Ž qq1.r2 q ᒍ s 1 q ᒍ, or equivalently, Žy .Ž qq1.r2 s 1. Since ² y : has order 2 d, dr2, or d depending on whether 2 ¦ d, 2 5 d, or 4 N d, the result follows.
FAKE CONGRUENCE MODULAR CURVES
287
Case II. Proceed as in Case I. Assumption. From now on, assume that our map ᒍ surjects onto PU2 Ž , OKrᒍ . ( PSL2 Ž⺖q ., that q is odd, and that Ž6, k . s 1, where k s 噛² y :. PROPOSITION 3.
⌫ Ž ᒍ . is noncongruence.
Proof. We first quote the following lemma. LEMMA 3 wWx. Suppose ⌬ is a finite index subgroup of PSL2 Ž⺪.. Let N denote the least common multiple of the cusp widths of ⌬. Then ⌬ is congruence if and only if ⌬ = ⌫ Ž N .. Completion of Proof. Since ⌫ Ž ᒍ . is normal, all cusp widths will equal k s 噛² y :. Now if ⌫ Ž ᒍ . = ⌫ Ž k ., then we will have a homomorphism PSL2 Ž⺪rk⺪. ª PSL2 Ž⺖q .. But this is impossible: Since q ¦ k by assumption, no composition factor of PSL2 Ž⺪rk⺪. could be equal to PSL2 Ž⺖q ..
4. CHARACTER MULTIPLICITIES IN THE REPRESENTATION OF PSL2 Ž⺖q . ON COHOMOLOGY GROUPS 4.1. The Calculation on H 1. We continue our assumption that PSL2 Ž⺪r⌫ Ž ᒍ . ( PSL2 Ž⺖q .. Recall that if k s 噛² y :, then our map ᒍ : PSL2 Ž⺪. ª PSL2 Ž⺖q . factors through T Ž2, 3, k . with kernel ⌬Ž ᒍ .. Thus T Ž2, 3, k .r⌬Ž ᒍ . ( PSL2 Ž⺖q . as well. Note that for k ) 6, T Ž2, 3, k . ¨ PSUŽ1, 1.. ŽWe can obtain this injection from the Burau representation as follows: for an appropriate choice of a dth root of unity,
1 Hs
y1
y1 q 1
y1
q1 1
will have negative determinant. Now one may easily verify that the image of B3 under the Burau representation, evaluation t ¬ , and reduction mod scalars is isomorphic to T Ž2, 3, k .. Thus we have T Ž2, 3, k . embedded as a subgroup of a unitary group which is conjugate to PSUŽ1, 1... Thus
288
GABRIEL BERGER
T Ž2, 3, k . acts on the Poincare disk ⺔ 1. Set RŽ ᒍ . s ⺔ 1r⌬Ž ᒍ . and recall that ⺔ 1rT Ž2, 3, k . ( ᒅU rPSL2 Ž⺪. ( ⺠ 1. We thus obtain two isomorphic PSL2 Ž⺖q . covers of ⺠ 1, X Ž ᒍ . s ᒅUr⌫ Ž ᒍ . ª ᒅUrPSL 2 Ž ⺪ . s ⺠ 1 and R Ž ᒍ . s ⺔ 1r⌬ Ž ᒍ . ª ⺔ 1rT Ž 2, 3, k . s ⺠ 1 . We wish to determine the structures of H 1 Ž X Ž ᒍ ., ⺓. and H 1, 0 Ž X Ž ᒍ ., ⺓. as ⺓w PSL2 Ž⺖q .x-modules. Observe that the same results hold with X Ž ᒍ . replaced by RŽ ᒍ ., since the covers are isomorphic. Now the cover X Ž ᒍ . ª ⺠ 1 is ramified over the points p 2 s PSL2 Ž ⺪ . e 2 i r4 ,
p 3 s PSL2 Ž ⺪ . e 2 i r6 ,
pk s PSL2 Ž ⺪ . i⬁.
and
Ž 16 .
Lying over them are the points P2 s ⌫ Ž ᒍ . e 2 i r4 ,
P3 s ⌫ Ž ᒍ . e 2 i r6 ,
and
Pk s ⌫ Ž ᒍ . i⬁. Ž 17 .
Observe that the inertia group of P2 over p 2 is generated by
ž
0 1
ž
0 y1
y1 0
/
mod ᒍ ,
1 1
/
mod ᒍ ,
that of P3 over p 3 by
and that of Pk over pk by
ž
1 0
1 1
mod ᒍ
/
Žsee subsection 3.1.. Call these generators ␥ 2 , ␥ 3 , and ␥ k , respectively. ŽNote that ␥ i s i mod ⌬Ž ᒍ ... We can approach H 1 Ž X Ž ᒍ ., ⺓. by means of the Lefschetz fixed point theorem: for g g PSL2 Ž⺖q ., 2
噛Fix Ž g . s 噛 Ž ⌬ X Ž ᒍ . ⭈ ⌫g . s
Ý Ž y1. i tr Ž g N H i Ž X Ž ᒍ . , ⺓. . , is0
where ⌬ X Ž ᒍ . is the diagonal of X Ž ᒍ . and ⌫g is the graph of g acting on X Ž ᒍ .. We know that trŽ g N H i Ž X Ž ᒍ ., ⺓. s 1 for i s 0, 2, and we calculate 噛FixŽ g . as follows.
289
FAKE CONGRUENCE MODULAR CURVES
Note that our above comments tell us that if g is not conjugate to ␥ i , i s 2, 3, k, then 噛FixŽ g . s 0. Furthermore, the only points in X Ž ᒍ . with nontrivial stabilizers are the points lying over p 2 , p 3 , pk . Thus if ␥ i x s x, then since ␥ 2 , ␥ 3 , ␥ k all have relatively prime orders, we must have x s pi for some g PSL2 Ž⺖q .. So
␥ i pi s pi m y1␥ i pi s pi m y1␥ i g ²␥ i : m g N Ž ²␥ i : . . ŽFor a subgroup H of PSL2 Ž⺖q ., N Ž H . denotes the normalizer of H in PSL2 Ž⺖q ... Also,
pi s pi m y1 g ²␥ i : . This implies that 噛Fix Ž ␥ i . s
噛N Ž ²␥ i : . 噛²␥ i :
.
We now recall the following Žsee wF-Hx.: Any element of PSL2 Ž⺖q . is conjugate to either
LEMMA 4. I,
ž
1 0
1 1 , 1 0
/ž
a ⑀ , 1 0
/ž
0 y1
a
/
,
or
ž
x y
y⑀ , x
/
where ⺖qU s ² ⑀ :. An element of the fourth type is said to be split Cartan and an element of the fifth type is said to be nonsplit Cartan. The nonsplit Cartan elements form a subgroup of order Ž q q 1.r2. From examining the orders of each conjugacy class, we see that if 2 i N Ž q y 1., then ␥ i is conjugate to a split Cartan element, and if 2 i N Ž q q 1., then ␥ i is conjugate to a nonsplit Cartan element. Now by wF-Hx, 噛N²␥ : s
½
q y 1, q q 1,
␥ split ␥ nonsplit.
Also, half the conjugates will equal ␥ and half will equal ␥y1 . So
噛Fix Ž ␥ . s
噛N²␥ : 噛²␥ :
s
¡q y 1 ,
␥ ; ␥ i , ␥ split
¢
␥ ; ␥ i , ␥ nonsplit.
~
i qq1 i
,
Putting this information together, we obtain the following.
290
GABRIEL BERGER
PROPOSITION 4. If ␥ ¤ ␥ i , i s 2, 3, k, then trŽ␥ N H 1 Ž X Ž ᒍ ., ⺓.. s 2. If ␥ ; ␥ i for some i as abo¨ e, then
tr Ž ␥ N H 1 Ž X Ž ᒍ . , ⺓ . . s
¡2 y q y 1 ,
2 i N Ž q y 1.
¢2 y
2 i N Ž q q 1. .
~
i qq1 i
,
Remark. Our fake congruence groups are generalizations of regular congruence groups, since
ž
y 0
1 1 , 1
/ž
0 y
/
mod p
are the usual generators of SL 2 Ž⺖p . when d s 2. But in this case, the image Ž 10 11 . of 1 g B3 under p is unipotent, not Cartan. Thus our subsequent calculations differ in this case. The list of nontrivial irreducible finite-dimensional representations of SL 2 Ž⺖q . is quite short Žsee wF-Hx.: Ž1. The complement V of ⌺ g in the permutation representation is irreducible. ⺖qU
b .4 Ž2. Let B s Ž 0a ay1 be the Borel subgroup of PSL2 Ž⺖q .. Suppose 2 i rŽ qy1. ² : s ⑀ and s e . The map
ž
⑀n 0
) ª n ⑀yn
/
gives a 1-dimensional representation Wn of B; Ind PBS L 2 Ž⺖ q . Wn \ Yn is irreducible if n / Ž q y 1.r2; and for n s Ž q y 1.r2, Yn splits into two irreducible factors Yq and Yy. Ž3. If C denotes the nonsplit Cartan subgroup Ž xy
y⑀ x
.4 and
m : C ª ⺓ is a character, then V m Ym ' Ind CP S L 2 Ž⺖ q . [ Ym [ Zm for some representation Zm which is irreducible if m / Ž q s 1.r2 and splits into two irreducible factors Zq and Zy otherwise. We would like to know the multiplicity of each irreducible representation in our representation ᒍ : PSL2 Ž⺖q . ª AutŽ H 1 Ž X Ž ᒍ ., ⺓... We may use the standard character formula m s Ý g P S L 2 Ž⺖ q . ᒍ Ž␥ . Ž ␥ . , where ᒍ is the character of ᒍ and is irreducible.
291
FAKE CONGRUENCE MODULAR CURVES
TABLE I Character Multiplicities qy
V
Ý ai y T i
qq1yTy
Y2 r
Ý Ž ai q 1. i iNr
qy1yTy
Z2 m
qq1
Yq, Y y
2
qy1
Zq, Zy
2
Ý
Ž1 y a i .
i i mod m
y
y
1 2
1 2
Ty
Ý
1
Ý
1
a is1 qq1 even 2i
Ty
a isy1 qq1 even 2i
THEOREM 6. Assumptions as abo¨ e, define a i g ² " 1: by q ' a i mod i for i s 2, 3, k. Also, let q y ai Ts Ý . i is2, 3, 7 Then the character multiplicities are as listed in Table I, where i runs through 2, 3, k, Yq and Yy appear only for q ' 1 mod 4, and Zq and Zy appear only for q ' y1 mod 4. Proof. We make use of the character table of wF-Hx. We note that representations of PSL2 Ž⺖q . are the same as representations of SL 2 Ž⺖q . that map y1 2 to the identity. In the representation V, the character of "1 2 is q, the character of a split Žrespectively, nonsplit. Cartan element is 1 Žrespectively, y1., and there are Ž q y 3.r2 Žrespectively, Ž q y 1.r2. conjugacy classes with q 2 q q Žrespectively, q 2 y q . elements in each class. Our multiplicity then becomes 1 噛PSL2 Ž ⺖q .
⭈ 4 qg q Ž q 2 q q .
ž
⭈ 2
ž
qy3 2
y
Ý Xi
yŽ q 2 y q . ⭈ 2
ž
/
q
qy1 2
2y
Ý
ž
y
Ý Yi
qy1
/ / Ýž i
q
Xi
2y
qq1 i
/ / Yi
.
292
GABRIEL BERGER
Here, X i Žrespectively, Yi . is the number of conjugacy classes of elements in the cyclic subgroup ²␥ i : that are split Žnonsplit. Cartan. Thus,
Xi s
½
i y 1, 0,
ai s 1 a i s y1
and Yi s d y 1 y X i . Summing up gives us the desired result. Further use of the character tables gives us the following. The multiplicity for Y2 r is 1 噛PSL2 Ž ⺖q .
⭈ 4 Ž q q 1. g q 4 Ž q 2 y 1. q Ž q 2 q q .
ž
⭈ 2 Ž y2 y
Ý Ai . q Ý
ž
qy1
2y
i
/ / Ai
,
where
¡0, ¢y2 q 2 i ,
A s~y2, i
a i s y1 a i s 1, i ¦ r a i s 1, i N r .
The multiplicity for Z2 m is 1 噛PSL2 Ž ⺖q .
⭈ 4 Ž q y 1. g y 4 Ž q 2 y 1.
ž
qŽ q 2 y q . ⭈ 2 Ž2 y
Ý Bi . q Ý
ž
where
¡0, ¢2 y 2 i ,
B q~2, i
ai s 1 a i s y1, i ¦ m a i s y1, i N m.
2y
qy1 i
/ / Bi
,
293
FAKE CONGRUENCE MODULAR CURVES
The multiplications for Yq and Yy are 1 噛PSL2 Ž ⺖q .
⭈ 2 Ž q q 1. g q 2 Ž q 2 y 1.
ž
q Ž q 2 q q . ⭈ 2 Ž y1 y
Ý Ci . q Ý
ž
2y
qy1 i
/ / Ci
,
where
¡0, Ci s
~y1 q i ,
¢y1,
a i s y1 qy1 a i s 1, even 2i qy1 a i s 1, odd. 2i
The multiplicities for Zq and Zy are 1 噛PSL2 Ž ⺖q .
⭈ 2 Ž q y 1. g y 2 Ž q 2 y 1.
ž
qŽ q 2 y q . ⭈ 2 Ž1 y
Ý Di . q Ý
ž
2y
qq1 i
/ / Di
,
where
¡0, 1 y i, D s~
ai s 1 a i s y1,
i
¢1,
a i s 1,
qq1
2i qq1 2i
even
odd.
Again, summing up gives us the desired results. 4.2. The Calculation on H 1, 0 . Note that if g g PSL2 Ž⺖q ., p g X Ž ᒍ ., and g Ž p . s p, then g induces an automorphism g# , p : TX Ž ᒍ ., p ª TX Ž ᒍ ., p of the tangent space of X Ž ᒍ . at p. Since TX Ž ᒍ ., p is 1-dimensional, g#, p s e i for some g ⺢. We can calculate e i as follows: choose ˜ g g PSL2 Ž⺪.
294
GABRIEL BERGER
such that ⌫ Ž ᒍ . ˜ g s g and z p g ᒅU such that ⌫ Ž ᒍ . z p s p. Then in local coordinates, g# , p
⭸ ⭸z
s˜ gX Ž z p .
⭸
,
⭸z
and so if ˜ g s Ž ac db ., then 1
g# , p s
Ž cz p q d .
2
.
So choose
␥ ˜2 s
ž
0 1
y1 , 0
␥˜3 s
/
ž
0 y1
1 , 1
/
␥˜k s
and
ž
1 0
1 . 1
/
Let z 2 s e 2 i r4 , z 3 s e 2 i r6 , and z k s i⬁. Then one may easily check that ␥ jU , p j s e 2 i r j for j s 2, 3. For j s k, use the transformation z ¬ e 2 i z r k . In these coordinates, z ¬ ␥ ˜k z s z q 1 becomes the map y ¬ e 2 i r k y. 2 i r k Thus ␥ kU , p k s e as well. Noting that 6
g
XŽᒍ.
y1
6
g
y1
XŽᒍ.
6
6
XŽᒍ.
XŽᒍ.
induces an equality g#, p s Ž y1 g .#, p , we see that
␥j
U
, pj
s
½
e 2 i r j ,
y1␥ j s ␥ j
ey2 i r j ,
y1␥ j s ␥y1 j .
Now we make use of the following: THEOREM 7 ŽHolomorphic Lefschetz Fixed Point Theorem.. Lef Ž g , OX Ž ᒍ . . s
1
Ý pgFix Ž g . g# , pse i
1 y eyi
.
Then Lef Ž g , OX Ž ᒍ . . s
⬁
Ý Ž y1. i tr Ž g U N H i , 0 Ž X Ž ᒍ . , ⺓. . . is0
Let
295
FAKE CONGRUENCE MODULAR CURVES
THEOREM 8. Let ᒍ1, 0 denote the representation g ª AutŽ H 1, 0 Ž X Ž ᒍ ., ⺓... Then ᒍ1, 0 s 12 ᒍ . Note. This differs sharply from the congruence case. Proof. From our above calculations, we see that for any ␣ g PSL2 Ž⺖q ., Lef Ž ␥ j␣ , OX Ž ᒍ . . s s
q"1 2j 1 2
ž
1 y i r j
1ye
q
1 y i r j
1qe
/
s
q"1 2j
噛Fix Ž ␥ j␣ . ,
Ž 18 . Ž 19 .
since y1␥ j s ␥ j or ␥y1 for q " 1r2 j elements modulo ²␥ j :. j Then by the holomorphic Lefschetz fixed point theorem, tr Ž ␥ U N H 1, 0 Ž X Ž ᒍ . , ⺓ . . s 1 y Lef Ž ␥ , X X Ž ᒍ . . s1y
1 2
噛Fix Ž ␥ . s
Ž 20 . 1 2
ᒍ Ž ␥ .
᭙␥ . Ž 21 .
5. GENERA OF QUOTIENT CURVES 5.1. The Calculation. Let J Ž ᒍ . denote the Jacobian of X Ž ᒍ .. We would like to understand the decomposition of J Ž ᒍ . into simple factors. As a first step in this direction, we calculate the genera of certain quotient curves of X Ž ᒍ .. Suppose L is a subgroup of ⺖qU with w⺖qU r² "1:: "Lr² "1:x s ␣ L . Define a subgroup BL of PSL2 Ž⺖q . by BL s
½ž
a 0
b : agL . ay1
/
5
Let C be the nonsplit Cartan subgroup of PSL2 Ž⺖q . and S be the split 0 .4 . Denote the genus of X Ž ᒍ .rH by g Ž H . for any Cartan subgroup Ž ay1 0 a Ž . subgroup of PSL2 ⺖q . PROPOSITION 5. We ha¨ e the following: Ž1. Ž2. Ž3. Ž4.
2 g Ž 14. y 2 s 噛PSL2 Ž⺖q .Ž1 y Ý i Ž1ri .. 2 g Ž BL . y 2 s ␣ LŽ q q 1 y Ý i ⑀ i . 2 g Ž C . y 2 s q Ž q y 1. y Ý i ␦ i 2 g Ž S . y 2 s q Ž q q 1. y Ý i ␥ i ,
296
GABRIEL BERGER
where all summations are o¨ er i s 2, 3, k and
⑀ i Ž resp., ␦ i , ␥ i .
¡q y 1 q 2
s~
¢
i
qq1 i
ž ž
resp., resp.,
q Ž q y 1. i
,
Ž q y 1. Ž q q 2. i
Ž q q 1. Ž q y 2. i
q 2,
q 2 , q ' 1Ž i .
qŽ q q 1 i
/ . /
, q ' y1 Ž i . .
Proof. Let H be any subgroup of PSL2 Ž⺖q ., and let pi , ␥ i , be as in Section 4. Using the Hurwitz formula, 2 g Ž H . y 2 s PSL2 Ž ⺖q . : H ⭈ Ž y2 . q Ý Ž PSL2 Ž ⺖q . : H y 噛 points over pi .
Ž 22 .
i
s PSL2 Ž ⺖q . : H y
Ý
噛points over pi .
Ž 23 .
i
Now if PSL2 Ž⺖q . s jH j , the ramification index of  j pi over pi is ²␥ i j : : ²␥ i j : l H . We calculate these indices in the case H s BL ; the remaining cases are similar. First, note that a set of coset representatives for BL in PSL2 Ž⺖q . is given by
½ž
a y1
a
0 t
y1
a
/
s A a, t ,
ž
0 y1
a
ya 0
/5
,
where a runs over a set of coset representatives for L in ⺖qU and t is an arbitrary element of ⺖q . We may assume by Lemma 5 that ␥ i is split or nonsplit Cartan; we will examine these cases separately. So assume that ␥ i 0 . is split, and hence of the form Ž 0x xy1 . Then A a, t ⭈ ␥ i ⭈ Ay1 a, t s
ž
y2
a
x t Ž x y xy1 .
0 y1
x
/
.
297
FAKE CONGRUENCE MODULAR CURVES
We thus see that ²␥ i : A a, t l BL is trivial unless t s 0, in which case ²␥ i : A a, t : BL . Also,
ž
ya 0
0 y1
a
/ž
x 0
0 y1
x
/ž
0 yay1
y1 a s x 0 0
0 . x
/ ž
/
Hence in this case, the index in question will always be 1. So in 2 ␣ L cases our index is 1, and in w PSL2 Ž⺖q .: BL x y 2 ␣ L s Ž q y 1. ␣ L cases, our index is i. Thus we have 2 ␣ L unramified points and Ž q y 1. ␣ Lri points with ramification index i. Our proposition then follows in this case. Now assume that ␥ i is nonsplit, and hence of the form Ž yx⑀ xy .. Then A a, t ⭈ ␥ i ⭈ Ay1 a, t s
ž
x y yt
a2 y
ay2 y Ž ⑀ y t 2 .
x q yt
/
.
We thus see that ²␥ i : A a, t l BL is always trivial, since y is never 0 Žas the order of ␥ i is i . and ⑀ cannot be a square. Also,
ž
0 ay1
ya 0
/ž
x y⑀
y x
/ž
0 yay1
x a s 0 yay2 y
/
ž
ya2 y⑀ x
/
.
Thus in all Ž q q 1. ␣ L cases the ramification index is i, and we thus obtain Ž q q 1. ␣ Lri points in this case. This completes the proof of the proposition for BL . 5.2. AN EXAMPLE. Suppose d s 14, and choose s ye 2 i r7. As usual, set F s ⺡Ž q y1 . and K s ⺡Ž .. Let ᒍqs Ž13, q y1 q 7. ; OF . Then one may verify that ᒍq remains prime in OK , that OF rᒍqs ⺪rŽ13., and that the map X ¬ y induces an isomorphism ⺪ w X x r Ž 13, X 2 y 7X q 1 . ( OKrᒍq OK . Let ⑀ s X q 3 modŽ13, X 2 y 7X q 1.. Then ⑀ 2 s x 2 q 6 X q 9, which equals 8 mod Ž13, X 2 y 7X q 1.. Now conjugation in ⺪w X xrŽ13, X 2 y 7X q 1. is given by X modŽ 13, X 2 y 7X q 1 . ¬ Xy1 modŽ 13, X 2 y 7X q 1 . s 7 y X modŽ 13, X 2 y 7X q 1 . .
298
GABRIEL BERGER
So ⑀ is equal to Ž7 y X . q 3 modŽ13, X 2 y 7X q 1., which is just y⑀ . Recall that our map : PSL2 Ž⺪. ª PSUŽ , H, ⺖ 13 2 . is given by Ss Us
ž ž
1 0
1 ⑀y3 ¬ 1 0
/ ž / ž
1 y1
1 1
/
Ž 24 .
0 1 ¬ 1 y⑀ q 3
0 . ⑀y3
/
Ž 25 .
since ⑀ y 3 s X modŽ13, X 2 y 7X q 1. corresponds to y under our isomorphism. Now conjugate each matrix by
ž
6⑀ q 1 0
7⑀ q 2 4
/
and multiply by 9⑀ q 5. ŽThe latter is permitted since we’re working mod scalars.. Then
ž⑀ ž
y3 0
1 5 q 4⑀ ¬ 1 0
0 5 y 4⑀
/ ž / ž
0 5 y 3⑀ ¬ ⑀y3 6 y 2⑀
1 y⑀ q 3
6q2 5q3
/ ⑀ ⑀/
Ž 26 . ,
Ž 27 .
and we thus obtain generators for PSU2 Ž⺖ 13 2 .. Under the canonical isomorphism PSU2 Ž⺖ 13 2 . ª PSL2 Ž⺖ 13 . given by
ž
a q b⑀ c y d⑀
aqc c q d⑀ ¬ a y b⑀ bqd
/ ž
⑀ 2 Ž b y d. , ayc
/
we observe that
ž ž
5 q 4⑀ 0
0 5 ¬ 5 y 4⑀ 4
5 y 3⑀ 6 y 2⑀
6q2 5q3
/ ž ⑀ ž ⑀/ ¬
y2 y1
6 5
/
Ž 28 .
y1 . y1
/
Ž 29 .
299
FAKE CONGRUENCE MODULAR CURVES
Our composite map is then ⍀ : PSL2 Ž ⺪ . ª PSL2 Ž ⺖13 . : S¬ U¬
ž ž
5 4
6 5
y2 y1
Ž 30 .
/
Ž 31 .
y1 . y1
/
Ž 32 .
Setting ␥ 2 s ⍀ ŽUSU ., ␥ 3 s ⍀ Ž SU ., and ␥ 7 s ⍀ Ž S ., we finally obtain the following images of the generators of T Ž2, 3, 7.:
␥2 s
ž
7 10
8 , 6
/
␥3 s
ž
9 0
2 , 3
/
␥7 s
ž
5 4
6 . 5
/
Using our genus formulas, we see that our associated curve has genus 14, and that the representation into AutŽ H 1 Ž R N , ⺓.. is just Y2 [ Y2 . REFERENCES wA-Swx O. Atkin and H. P. F. Swinnerton-Dyer, Modular forms on noncongruence subgroups, in Proc. Sympos. Pure Math., Vol. 19, pp. 1᎐25, Amer. Math. Soc., Providence, 1971. wBex G. V. Belyi, On extensions of the maximal cyclotomic field having a given Galois group, J. Reine. Angew. Math. 341 Ž1983., 1᎐25. wBx G. Berger, Hecke operators on non-congruence subgroups, C. R. Acad. Sci. Paris Ser. ´ I. Math. 319 Ž1994., 915᎐919. wB2x G. Berger, Fake congruence subgroups and the Hurwitz monodromy group, draft, 1997. wB-Fx G. Berger and M. Fried, The Hurwitz monodromy group H4 , in preparation, 1997. wCx H. Cohen, ‘‘A Course in Computational Algebraic Number Theory,’’ Grad. Texts in Math., Vol. 138, Springer-Verlag, New YorkrBerlin, 1993. wBix J. Birman, Braids, links, and mapping class groups, Princeton, 1973. wFx M. Fried, Arithmetic of 3 and 4 branch point covers: A bridge provided by noncongruence subgroups of SLŽ2, Z ., in ‘‘Sem. de Theorie des Nombres, Paris, 1987-88,’’ Prog. Math., Vol. 81, pp. 77᎐117, Birkhauser, Basel, 1990. ¨ wF-Vx M. Fried and H. Volklein, The inverse Galois problem and rational points on moduli spaces, Math. Ann. 290, No. 4 Ž1991., 771᎐800. wF2x M. Fried, Introduction to modular towers: Generalizing dihedral group-modular curve connections, in ‘‘Recent Developments in the Inverse Galois Problem,’’ Contemporary Mathematics, Vol. 186, pp. 111᎐173, Amer. Math. Soc., Providence, 1995. wF-Hx W. Fulton and J. Harris, ‘‘Representation Theory: A First Course,’’ Springer-Verlag, New York, 1991. wMx A. M. Macbeath, Generators of the linear fractional groups, in Proc. Sympos. Pure Math., Vol. 12, pp. 14᎐32, Amer. Math. Soc., Providence, 1968. wO-Tx T. Oda and T. Terasoma, Surjectivity of reductions of the Burau representation of Artin braid groups, draft, 1995.
300 wSchx wVx wWx
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A. J. Scholl, Modular forms and de Rham cohomology; Atkin᎐Swinnerton᎐Dyer congruences, In¨ ent. Math. 79 Ž1985., 49᎐77. H. Volklein, Braid group action via GLŽ n, q . and UŽ n, q ., and Galois realizations, Israel J. Math. 82 Ž1993., 405᎐427. K. Wolfahrt, An extension of F. Klein’s level concept, Internat. J. Math. 8 Ž1967., 529᎐535.
Fake Congruence Modular Curves and Subgroups of the Modular GroupU Gabriel Berger † Department of Mathematics, Uni¨ ersity of California at Ir¨ ine, Ir¨ ine, California 92697-3875 E-mail: [email protected] Communicated by Walter Feit Received April 13, 1998
We construct a special class of noncongruence modular subgroups and curves, analogous in some ways to the usual congruence ones. The subgroups are obtained via the Burau representation, and the associated quotient curves have a natural moduli space interpretation. In fact, they are reduced Hurwitz spaces corresponding to covers with 4 branch points and monodromy group equal to semi-direct products of a cyclic and an abelian group. Furthermore, they form a modular tower in the sense of Fried. We study representations on the cohomology of these fake congruence modular curves and also calculate the genera of certain quotient curves. 䊚 1999 Academic Press Key Words: Burau representation; fake congruence subgroup; Hurwitz space; modular tower.
1. INTRODUCTION The present work has two starting points. The first is the work of Oda and Terasoma on the Burau representation of the Artin braid group Bn . This representation is, in general, a homomorphism
n : Bn ª GLny1 Ž ⺪ w t , ty1 x . . U
Ž 1.
Supported in part by JSPS Grant P94015 and NSA Grant 032596. The author expresses his thanks and acknowledges his debts to Takayuki Oda for introducing the topics considered in this paper and to Mike Fried for valuable discussions regarding Hurwitz spaces and connections with the modular group. We note that similar topics have been considered earlier by Helmut Voelklein, whose primary interest was the realization of certain finite groups of Lie type as Galois groups over ⺡. †
276 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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FAKE CONGRUENCE MODULAR CURVES
However, in wO-Tx and this paper, it is shown that under certain conditions and for certain prime powers q, the map 3 induces a surjective homomorphism
: PSL2 Ž ⺪ . ª PSL2 Ž ⺖q . ,
Ž 2.
where ⺖q is the finite field with q elements in it. ŽIn fact, we can define such a in more generality. But then all we would know about its image is N ., where is a root that it is a certain unitary subgroup of PGL2 Ž⺪w xrN w x . of unity and N is an ideal in ⺪ . Recall that PSL2 Ž⺪. is generated by the images mod ² " 1: of Ss
ž
1 0
1 1
/
Us
and
ž
1 y1
0 , 1
/
with the relations Ž SU . 3 s Ž SUS . 2 s 1. ŽHere and subsequently, all matrices will be taken mod ² "1:.. Thus B3 maps surjectively onto PSL2 Ž⺪. via
1 ¬ S,
2 ¬ U.
Ž 3.
The kernel is the center ZŽ B3 ., which is generated by Ž 1 2 . 3. The second starting point is the following result of Fried wFx, which gives a new, combinatorial group-theoretic way to approach congruence subgroups. Suppose N is an odd integer and DN is the dihedral group ²␥ , t : ␥ 2 s t N s 1, ␥ t␥y1 s ty1 : .
Ž 4.
Observe that DN is the semi-direct product of ⺪rN and ⺪r2. Embed DN in the symmetric group SN via the permutation representation on the N cosets of ²␥ :, and take C to be the conjugacy class of involutions. Let X be the set
Ž a, b, c, d . g C 4 : ² a, b, c, d : s DN ,
and
a ⭈ b ⭈ c ⭈ d s 1 4 rNS N Ž C . ,
Ž 5. where NS NŽ C . is the normalizer of C in S N . We can define a right action of PSL2 Ž⺪. on X via
Ž a, b, c, d . S s Ž abay1 , a, c, d .
Ž 6.
Ž a, b, c, d . U s Ž a, bcby1 , b, d . .
Ž 7.
Fried’s theorem then states that the stabilizer of the element
Ž ␥ , ␥ , t␥ , t␥ . is ⌫0 Ž N ..
mod NS N Ž C .
Ž 8.
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GABRIEL BERGER
In fact, in wF, B-Fx it is shown that given any transitive subgroup G of Sn and conjugacy classes Ž C1 , . . . , C4 . of G, we can obtain a finite-index subgroup ⌬ of SL 2 Ž⺪.. Moreover, the associated quotient curve ᒅr⌬ has a moduli space interpretation. We therefore obtain a way of generating noncongruence subgroups of SL 2 Ž⺪. with additional structure associated to them. In wB2x, the following is shown. Suppose G is the semi-direct product N i Ž⺪w xrN N .U , for d an integer and N an ideal of ⺪w x relatively ⺪w xrN prime to d. Let Ž C1 , . . . , C4 . s Ž C , C , C , Cy3 ., with C the conjugacy class N .. of . Then our ⌬ is in fact equal to the stabilizer of ⬁ g ⺠ 1 Ž⺪w xrN That is, ⌬ s y1 Ž B ., where B is the standard Borel subgroup
ž
) 0
) )
/
N . and is the map obtained from the Burau representaof PGL2 Ž⺪w xrN tion as above. Equivalently, ᒅr⌬ is a course moduli space for Kummer N. covers X of ⺠ 1 together with a subgroup of J Ž X . isomorphic to ⺪w xrN If G s DN , then ⌬ is just ⌫0 Ž N .. We thus obtain a noncongruence modular curve with a moduli space interpretation which generalizes X 0 Ž N .. Furthermore, if we fix d and let N vary through powers of a prime p, then for certain p we obtain a modular tower Žsee wF2x.. It is our hope that these so-called ‘‘fake congruence subgroups and modular curves’’ will provide a fertile testing ground for many of the objects and theories associated with the usual congruence ones ŽHecke operators, action of Frobenius, . . . .. Accordingly, we have begun their study in this paper. The contents are as follows: in Section 2, we outline the basic framework in which we intend this work to be considered, including a review of Hurwitz spaces and their relation to noncongruence modular curves. In Section 3, we use the Oda᎐Terasoma theorem to construct fake congruence subgroups and fake congruence modular curves. In Section 4, we determine the character multiplicities of the representation of PSL2 Ž⺖q . on the cohomology of our fake congruence modular curves. Finally, in Section 5, we determine the genera of various quotient curves.
2. HURWITZ SPACES OF FOUR-BRANCH POINT COVERS AND FAKE CONGRUENCE SUBGROUPS 2.1. Re¨ iew of Hurwitz Spaces. The references for this section are wB-F, F-Vx. Let G be a finite group embedded as a transitive subgroup of Sn for some positive integer n. Let C s Ž C1 , . . . , C4 . e a quadruple of conjugacy classes of G. We are interested in parametrizing covers of ⺠ 1 Žover ⺓.
FAKE CONGRUENCE MODULAR CURVES
279
ramified over four points with monodromy group G and ramification data C. To do so, we introduce the following definitions. DEFINITION 1. We set U 4 s Ž z1 , . . . , z 4 . g Ž⺠ 1 . 4 : z i / z j for i / j4 , and let U4 s U 4rS 4 denote the natural quotient. The Nielsen class associated to the data Ž G, C. is NiŽ G, C. and is
Ž g 1 , . . . , g 4 . g G 4 : ² g 1 , . . . , g 4 : s G, g 1 and
⭈⭈⭈ g 4 s 1
g i g Ci
for some g S4 4 .
We also set NiŽ G, C. ab s NiŽ G, C.rNS nŽC., where NS nŽC. is the normalizer of C in S n . Finally, the Hurwitz monodromy group H4 is the fundamental group of U4 . It has the presentation ² Q1 , Q2 , Q3 : Q i Q iq1 Q i s Q iq1 Q i Q iq1 , Q1 Q3 s Q3 Q1 , Q1 Q2 Q3 Q3 Q2 Q1 s 1: . Note that H4 is very closely related to B3 , since B3 is the fundamental group of unordered, distinct triples of ⺓ 1 s ⺠ 1 y ⬁4 . In fact, Hurwitz space theory can be formulated using B3 instead of H4 . We have an action of H4 on NiŽ G, C. ab as follows: if Ž g 1 , . . . , g 4 . g NiŽ G, C. ab , then
Ž g 1 , . . . , g 4 . Qi s Ž g 1 , . . . , g iy1 , g i g iq1 gy1 i , g i , g iq2 , . . . , g 4 . . ŽThus Q i sends g i to g i g iq1 gy1 i , sends g iq1 to g i , and fixes the other two elements of the quadruple.. Since H4 is the fundamental group of U4 , each of its orbits on NiŽ G, C. ab corresponds to a connected cover of U4 . Let H Ž G, C. denote the disjoint union of these covers. THEOREM 1 wF-Vx. H Ž G, C. is a coarse moduli space for co¨ ers of ⺠ 1 with monodromy group G and ramification data C. In fact, this theorem holds for covers with an arbitrary number of branch points. 2.2. PSL 2 Ž⺓. Action. We have canonical PSL2 Ž⺓. actions on U 4 , U4 , and H s H Ž G, C. as above: if ␥ g PSL2 Ž⺓., then ␥ Ž a, b, c, d . s Ž␥ a, ␥ b, ␥ c, ␥ d .. Also, given a cover : X ª ⺠ 1, ␥ acts on this cover by taking it to the composite ␥ . We denote the quotients of these actions by U 4 r e d, U4r e d, and H r e d, respectively. The following result is due to Thompson Žsee wFx..
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GABRIEL BERGER
PROPOSITION 1. There is a surjecti¨ e homomorphism
: H4 ª PSL2 Ž ⺪ . gi¨ en by
Ž Q1 . s S s
ž
1 0
1 , 1
/
Ž Q2 . s U s
Ž Q3 . s S s
ž
1 0
ž
1 y1
0 , 1
/
1 . 1
/
The kernel is the quaternion group Q8 of order 8. THEOREM 2 wB-Fx. Suppose HO is a connected component of H . Then there exists a subgroup ⌬ of finite index in PSL2 Ž⺪. such that we ha¨ e the following commutati¨ e diagram: H rOe d ª ᒅr⌬ x x U4r e d ªᒅrPSL2 Ž⺪. The horizontal maps are isomorphisms, and the bottom isomorphism is obtained by taking any a, b, c, d4 to the unique Ž up to isomorphism. elliptic cur¨ e ramified o¨ er a, b, c, d4 . Furthermore, ⌬ arises as follows: by construction, H r0e d corresponds to a subgroup of finite index ⌺ in H4 . Then ⌬ s Ž ⌺ ., where is the homomorphism of Proposition 1. 2.3. The Semidirect Product of Two Abelian Groups and the Burau Representation. Our general set up is as follows. Suppose d G 1 is an integer, and is a primitive dth root of unity. Let N be an ideal of ⺪w x coprime to d, and by abuse of notation, continue to denote the image of in N by . Subsequently, by ⺪w xrN N we will mean the additive group of ⺪w xrN N unless otherwise noted; Ž⺪w xrN N .U will denote the multiplicative ⺪w xrN N. We can embed Ž⺪w xrN N .U into the automorgroup of Žthe ring. ⺪w xrN N by setting ␥ Ž x . s ␥ x for ␥ g Ž⺪w xrN N .U and phism group of ⺪w xrN N. Thus we may form the semidirect product x g ⺪w xrN U
N i Ž ⺪ w x rN N. . ⺪ w x rN N and symbols w ␥ x for ␥ g Ž⺪w xrN N .U , This group is generated by ⺪w xrN with the relations
w ␥ xw ␣ x s w ␥␣ x for ␣ g Ž ⺪ w x rNN . and
U
w ␥ x y1 t w ␥ x s ␥ t for t g ⺪ w x rNN .
FAKE CONGRUENCE MODULAR CURVES
281
ŽIn the second equality, the left-hand multiplication is in ⺪w xrN Ni Ž⺪w xrN N .U , and the right-hand multiplication is in the ring ⺪w xrN N.. Let N i ² : of ⺪w xrN N i Ž⺪w xrN N .U . Notice that G be the subgroup ⺪w xrN U N i Ž⺪w xrN N . acts on ⺪w xrN N as a set: the additive group ⺪w xrN N ⺪w xrN N .U acts acts upon itself by addition, and the multiplicative group Ž⺪w xrN N by multiplication. Thus if 噛⺪w xrN N s N, we obtain an embedon ⺪w xrN N i Ž⺪w xrN N .U and its subgroup G into SN , given as ding of ⺪w xrN N and x w ␥ x g ⺪w xrN N i Ž⺪w xrN N .U , then follows: if t g ⺪w xrN t Ž xw␥ x. s Ž t q x . w␥ x . N i Ž⺪w xrN N .U and G with their images in SN . Let C be Identify ⺪w xrN the conjugacy class of w x. Set C s Ž C1 , . . . , C4 . s Ž C , C , C , Cy3 .. Now let ab
X s Ni Ž G, C . rQ8 ,
Ž 9.
where Q8 is the kernel of as in Proposition 1. Then we have a well-defined action of PSL2 Ž⺪. on X. THEOREM 3 wB2x. Ž1. The action of PSL2 Ž⺪. on X is transiti¨ e. Ž2. We can identify X with ⺠ 1 Ž⺪w xrN N .. Furthermore, we ha¨ e the following commutati¨ e diagram, where the right hand map is as defined in Subsection 3.2 and the left hand map is as described in the Introduction. id
6
PSL 2 Ž⺪.
6
PSL2 Ž⺪. 6
N. PGL2 Ž⺪w d xrN
6
AutŽ X .
In addition, the stabilizer of the image of Ž␥ , ␥y1 , t␥ , ␥y1 ty1 . in X is equal to the subgroup ⌫0 Ž N . s ␣ g PSL2 Ž ⺪ . : Ž ␣ . ⬁ s ⬁ 4 . Ž3. The cur¨ e Y0 Ž N . s ᒅr⌫0 Ž N . is a reduced Hurwitz space for co¨ ers of ⺠ 1 with monodromy group G and ramification data C. Thus each point on Y0 Ž N . corresponds to an equi¨ alence class of diagrams of the following form: X ␥
6

⺠1
6
6
W
␣
6
Y
282
GABRIEL BERGER
We require all maps to be ramified o¨ er 4 points, and the ¨ ertical maps to be Galois, with monodromy group ⺪rd⺪ s ² :. In addition, the map ␥ should ha¨ e ramification data of the form Ž , , , y3 . Ž here, is its own conjugacy N. class in ⺪rd⺪ s ² :.. Furthermore, ␣ must be Galois with group ⺪w xrN Finally, we require  to ha¨ e monodromy group G and ramification data Ž C , C , C , Cy3 .. We note that once we specify the map  , Y and X will be uniquely determined. Two diagrams are considered to be equi¨ alent if and only if the lower horizontal maps are equi¨ alent mod PSL2 Ž⺓.. 3. CONSTRUCTION OF THE ⌫ Ž N . 3.1. Fake Congruence Subgroups. To begin with, we fix natural numbers n G 3, d G 7, a primitive dth root of unity , K s ⺡Ž ., F s ⺡Ž q y1 ., N q; OF an ideal, and N s N q OK . Let Bn denote the Artin braid group ² 1 , . . . ny1 : i j s j i
for and
i iq1 i s iq1 i iq1 : .
The reduced Burau representation Žsee wBix. is a homomorphism n : Bn ª GLny1Ž⺪w t, ty1 x.. It can be obtained, among other methods, via the Fox free calculus, or from the action of Bn on the homology of an infinite cyclic extension of ⺠ 1 ramified over n q 1 points. Its action on the generators of Bn is given as yt 0 1 ¬ 0 0 0
⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈ 0
1 1 0 ⭈⭈⭈ 0
0 0 1 ⭈⭈⭈ 0
0 1 t 0 0
⭈⭈⭈ 0 yt 0 ⭈⭈⭈
0 ⭈⭈⭈ 1 1 0
1 0 ny 1 ¬ 0 0 0
0 1 ⭈⭈⭈ 0 0
1 ry2 0 r ¬ ⭈⭈⭈ ⭈⭈⭈ 0
0 0 0 , 0 1 0 0 0 0
Ž 1 - r - n y 1.
1 ny ry2
and ⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈
0 0 ⭈⭈⭈ 1 t
0 0 0 . 0 yt
FAKE CONGRUENCE MODULAR CURVES
283
N , and Setting t s and reducing by N gives us a map ⺪w t, ty1 x ª OKrN N .. We then obtain a new map hence a map GLnŽ⺪w t, ty1 x. ª GLnŽ OKrN N. n , N : Bn ª GLny1 Ž OKrN by composing with the Burau representation. Now suppose n s 3. In this case we can describe our map as follows. First, notice that B3 has presentation ² 1 , 2 : 1 2 1 s 2 1 2 :. Then 3, N is given by
1 ª 2 ª
ž ž
y 0 1
1 1
/ /
Ž 10 .
0 . y
Ž 11 .
N. where we understand the matrix entries to be elements of OKrN Recall that ZŽ B3 . is generated by Ž 1 2 . 3 and observe that
3, N Ž Ž 1 2 .
3
.s
ž
3
0
0
3
/
.
Thus 3, N induces a map N. . N : PSL2 Ž ⺪ . s B3rZ Ž B3 . ª PGL2 Ž OKrN
Ž 12 .
DEFINITION 2. We have the following analogs of congruence groups and modular curves: Ž1. ⌫ Ž N . s kerŽN . Ž2. ⌫0 Ž N . s ␥ g PSL2 Ž⺪.: N Ž␥ .⬁ s ⬁ Žwhere we take ⬁ g N .. ⺠ 1 Ž OKrN Ž3. X Ž N . s ᒅU r⌫ Ž N . Ž4. X 0 Ž N . s ᒅU r⌫0 Ž N . Ž5. Y Ž N . s ᒅr⌫ Ž N . Ž6. Y0 Ž N . s ᒅr⌫0 Ž N .. Any subgroup of PSL2 Ž⺪. that contains ⌫ Ž N . for some N will be called a fake congruence subgroup. If d s 2, we obtain the usual congruence objects with N replaced by N. Recall that PSL2 Ž⺪. is the free product of the group of order 2 generated by USU s Ž 01 y10 . and the group of order 3 generated by SU s Ž y10 11 ..
284
GABRIEL BERGER
DEFINITION 3. Let k be a positive integer. The Hecke Triangle Group T Ž2, 3, k . is the group ²  2 ,  3 ,  k :  22 s  33 s  kk s  2  3  k s 1: . We have a canonical surjection : PSL2 Ž⺪. ª T Ž2, 3, k . given by 0 y1 0 1 SU s ¬ ␥3. ¬ ␥2 , 1 0 y1 1 y1 Ž . Since S s ŽUSUSU .y1 and ␥ k s ␥y1 3 ␥ 2 , we also see that S s ␥ k . N ., we have Now note that in our map N : PSL2 Ž⺪. ª PGL2 Ž⺪w xrN USU s
ž
/
ž
N Ž S r . s 3, N Ž 1r . s
ž
Ž y .
r
Ž Ž y .
0
/
r
y 1 . r Ž y y 1 . . Ž 13 . 1
/
Thus if we set k s 噛² y :, then N Ž S k . s 1, and so N factors through T Ž2, 3, k .. We set N. .. ⌬ Ž N . s ker Ž T Ž 2, 3, k . ª PGL2 Ž ⺪ w x rN
Ž 14 .
3.2. The Oda᎐Terasoma Theorem. By the results of wO-Tx, the image of Bn under n is unitary with respect to a certain hermitian form H g GLny 1Ž⺪w t, ty1 x.. ŽHere, conjugation is the map t ¬ ty1 .. H is given by y
1 y
1 y1
t
⭈⭈⭈
y
0
tq1 1
q1
0
1
y
1 y1
t
q1 ⭈⭈⭈
1 tq1 1 ⭈⭈⭈
y
0
⭈⭈⭈
0
⭈⭈⭈
1 tq1 ⭈⭈⭈
.
Ž 15 .
0 ⭈⭈⭈
N . Žalso denoted H .. This form then descends to a form in GLny 1Ž OKrN In fact, one can show that the image of n, N is contained in N . s g g GLny1 Ž OKrN N . : g t Hg s H , det g g ² y : 4 . Uny 1 Ž , H , OKrN The main theorem of wO-Tx is that under certain conditions, Bn actually N .. surjects onto Uny 1Ž , H, OKrN THEOREM 4 wO-Tx. Assume n G 3, d G 7 if d is odd, d G 14 if d is e¨ en, N is coprime with d and ŽŽ1 y i .rŽ1 y .. Ž2 F i F n., the prime factors ᒎ of N di¨ iding 6 satisfy NK r ⺡ Ž ᒎ . G 10, N is coprime to n y 4, and d is odd when n s 4. Then N. n , N : Bn ª Uny1 Ž , H , OKrN is surjecti¨ e. A special case of this is the following.
FAKE CONGRUENCE MODULAR CURVES
285
THEOREM 5. Hypotheses as abo¨ e, suppose n s 3 and Nq is a prime ideal ᒍq. Set ᒍ s ᒍqOK . Then the map
ᒍ : PSL2 Ž ⺪ . ª PU2 Ž , H , OKrᒍ . is surjecti¨ e. 3.3. Analysis of 3 . We must now analyze 3 in more detail. LEMMA 1. Let f s ord d p s min a g ⺞: p a ' 1 mod d4 . Let ᒍq; OF be a prime o¨ er p. Then ᒍq Ž1. remains prime in OK if f is e¨ en, and 噛Ž OKrᒍq OK . s p f , 噛O OF r ᒍ s p f r2 , Ž2. splits into two primes in OK if f is odd, and 噛Ž OF rᒍq. s p f. q
Proof. Let Tq Ž X . s
Ł Ž X y Ž i q yi . .
Ž i , d .s1 0-i-dr2
be the irreducible polynomial of q y1 over ⺪. We must factor Tq in ⺖p w X x; the degree of each factor will be the residue field index of ᒍq Žsee wCx... Choose a primitive dth root of unity ␥ g ⺖p . Then Tq Ž X . s
Ł Ž X y Ž ␥ i q ␥yi . .
Ž i , d .s1 0-i-dr2
is a factorization of Tq over ⺖p . Consider
Ł
0Fj-f
ž X y Ž␥
pj
q ␥yp
j
. / g ⺖p w X x .
This is irreducible if and only if f is odd Žotherwise, it will be a square.. Thus f Ž ᒍqrp . s fr2 if f is even and f Ž ᒍqrp . s f if f is odd. The same argument shows, upon factoring TŽ X. s
Ł Ž X y ␥ i.
Ž i , d .s1 0-i-d
in ⺖p w X x, that any prime ᒍ ; OK over p satisfies f Ž ᒍrp . s f. Since ᒍq cannot ramify in OK by assumption, the result follows.
286
GABRIEL BERGER
LEMMA 2.
PU2 Ž OKrᒍ . : PSU2 Ž OKrᒍ . s
½
2, 1,
if q is odd if q is e¨ en.
Proof. We prove the lemma for OKrᒍ s ⺖q 2 ; the proof for OKrᒍ s ⺖q = ⺖q is similar. Note that if x g U2 Ž⺖q 2 . and detŽ x . s a, then aa s a qq 1 s 1. Thus the image x of x in PU2 Ž⺖q 2 . is actually in PSU2 Ž⺖q 2 . if and only if there exists some g ⺖q 2 with 2 s a, qq 1 s 1. Thus if q is odd, then x g PSU2 Ž⺖q 2 . m aŽ qq1.r2 s 1. So in this case, PU2 Ž ⺖q 2 . s PSU2 Ž ⺖q 2 . j PSU2 Ž ⺖q 2 . x, where x is any element of U2 Ž⺖q 2 . satisfying detŽ x .Ž qq1.r2 s y1. If q s 2 r and a qq 1 s 1, set s a q r2q1. Then 2 s a and qq 1 s 1. So PU2 Ž⺖q 2 . s PSU2 Ž⺖q 2 .. We thus observe that when we factor through to PSL2 Ž⺪., the worst we can do is surject onto a group in which PSL2 Ž⺖q . has index 2. We now list the cases in which we land precisely on PSL2 Ž⺖q .. PROPOSITION 2. With the notation as abo¨ e, if Ž p . s ᒍ l ⺪, then PU2 Ž , OKrᒍ . s PSU2 Ž OKrᒍ . if and only if the following conditions hold: ᒍ is prime and p is odd. 2 ¦ d: q ' y1 mod 4 2 5 d: always 4 N d: 2 N Ž q q 1.rd. Case II. ᒍ s ᑜ ᑜ and p is odd. Ž1. 2 ¦ d: q ' 1 mod 4 Ž2. 2 5 d: always Ž3. 4 N d: 2 N Ž q y 1.rd. Case III. p s 2: always.
Case I. Ž1. Ž2. Ž3.
Proof. Recall that all elements in PU2 Ž , OKrᒍ . have determinant contained in ² y :. Case I. We must have Žy .Ž qq1.r2 q ᒍ s 1 q ᒍ, or equivalently, Žy .Ž qq1.r2 s 1. Since ² y : has order 2 d, dr2, or d depending on whether 2 ¦ d, 2 5 d, or 4 N d, the result follows.
FAKE CONGRUENCE MODULAR CURVES
287
Case II. Proceed as in Case I. Assumption. From now on, assume that our map ᒍ surjects onto PU2 Ž , OKrᒍ . ( PSL2 Ž⺖q ., that q is odd, and that Ž6, k . s 1, where k s 噛² y :. PROPOSITION 3.
⌫ Ž ᒍ . is noncongruence.
Proof. We first quote the following lemma. LEMMA 3 wWx. Suppose ⌬ is a finite index subgroup of PSL2 Ž⺪.. Let N denote the least common multiple of the cusp widths of ⌬. Then ⌬ is congruence if and only if ⌬ = ⌫ Ž N .. Completion of Proof. Since ⌫ Ž ᒍ . is normal, all cusp widths will equal k s 噛² y :. Now if ⌫ Ž ᒍ . = ⌫ Ž k ., then we will have a homomorphism PSL2 Ž⺪rk⺪. ª PSL2 Ž⺖q .. But this is impossible: Since q ¦ k by assumption, no composition factor of PSL2 Ž⺪rk⺪. could be equal to PSL2 Ž⺖q ..
4. CHARACTER MULTIPLICITIES IN THE REPRESENTATION OF PSL2 Ž⺖q . ON COHOMOLOGY GROUPS 4.1. The Calculation on H 1. We continue our assumption that PSL2 Ž⺪r⌫ Ž ᒍ . ( PSL2 Ž⺖q .. Recall that if k s 噛² y :, then our map ᒍ : PSL2 Ž⺪. ª PSL2 Ž⺖q . factors through T Ž2, 3, k . with kernel ⌬Ž ᒍ .. Thus T Ž2, 3, k .r⌬Ž ᒍ . ( PSL2 Ž⺖q . as well. Note that for k ) 6, T Ž2, 3, k . ¨ PSUŽ1, 1.. ŽWe can obtain this injection from the Burau representation as follows: for an appropriate choice of a dth root of unity,
1 Hs
y1
y1 q 1
y1
q1 1
will have negative determinant. Now one may easily verify that the image of B3 under the Burau representation, evaluation t ¬ , and reduction mod scalars is isomorphic to T Ž2, 3, k .. Thus we have T Ž2, 3, k . embedded as a subgroup of a unitary group which is conjugate to PSUŽ1, 1... Thus
288
GABRIEL BERGER
T Ž2, 3, k . acts on the Poincare disk ⺔ 1. Set RŽ ᒍ . s ⺔ 1r⌬Ž ᒍ . and recall that ⺔ 1rT Ž2, 3, k . ( ᒅU rPSL2 Ž⺪. ( ⺠ 1. We thus obtain two isomorphic PSL2 Ž⺖q . covers of ⺠ 1, X Ž ᒍ . s ᒅUr⌫ Ž ᒍ . ª ᒅUrPSL 2 Ž ⺪ . s ⺠ 1 and R Ž ᒍ . s ⺔ 1r⌬ Ž ᒍ . ª ⺔ 1rT Ž 2, 3, k . s ⺠ 1 . We wish to determine the structures of H 1 Ž X Ž ᒍ ., ⺓. and H 1, 0 Ž X Ž ᒍ ., ⺓. as ⺓w PSL2 Ž⺖q .x-modules. Observe that the same results hold with X Ž ᒍ . replaced by RŽ ᒍ ., since the covers are isomorphic. Now the cover X Ž ᒍ . ª ⺠ 1 is ramified over the points p 2 s PSL2 Ž ⺪ . e 2 i r4 ,
p 3 s PSL2 Ž ⺪ . e 2 i r6 ,
pk s PSL2 Ž ⺪ . i⬁.
and
Ž 16 .
Lying over them are the points P2 s ⌫ Ž ᒍ . e 2 i r4 ,
P3 s ⌫ Ž ᒍ . e 2 i r6 ,
and
Pk s ⌫ Ž ᒍ . i⬁. Ž 17 .
Observe that the inertia group of P2 over p 2 is generated by
ž
0 1
ž
0 y1
y1 0
/
mod ᒍ ,
1 1
/
mod ᒍ ,
that of P3 over p 3 by
and that of Pk over pk by
ž
1 0
1 1
mod ᒍ
/
Žsee subsection 3.1.. Call these generators ␥ 2 , ␥ 3 , and ␥ k , respectively. ŽNote that ␥ i s i mod ⌬Ž ᒍ ... We can approach H 1 Ž X Ž ᒍ ., ⺓. by means of the Lefschetz fixed point theorem: for g g PSL2 Ž⺖q ., 2
噛Fix Ž g . s 噛 Ž ⌬ X Ž ᒍ . ⭈ ⌫g . s
Ý Ž y1. i tr Ž g N H i Ž X Ž ᒍ . , ⺓. . , is0
where ⌬ X Ž ᒍ . is the diagonal of X Ž ᒍ . and ⌫g is the graph of g acting on X Ž ᒍ .. We know that trŽ g N H i Ž X Ž ᒍ ., ⺓. s 1 for i s 0, 2, and we calculate 噛FixŽ g . as follows.
289
FAKE CONGRUENCE MODULAR CURVES
Note that our above comments tell us that if g is not conjugate to ␥ i , i s 2, 3, k, then 噛FixŽ g . s 0. Furthermore, the only points in X Ž ᒍ . with nontrivial stabilizers are the points lying over p 2 , p 3 , pk . Thus if ␥ i x s x, then since ␥ 2 , ␥ 3 , ␥ k all have relatively prime orders, we must have x s pi for some g PSL2 Ž⺖q .. So
␥ i pi s pi m y1␥ i pi s pi m y1␥ i g ²␥ i : m g N Ž ²␥ i : . . ŽFor a subgroup H of PSL2 Ž⺖q ., N Ž H . denotes the normalizer of H in PSL2 Ž⺖q ... Also,
pi s pi m y1 g ²␥ i : . This implies that 噛Fix Ž ␥ i . s
噛N Ž ²␥ i : . 噛²␥ i :
.
We now recall the following Žsee wF-Hx.: Any element of PSL2 Ž⺖q . is conjugate to either
LEMMA 4. I,
ž
1 0
1 1 , 1 0
/ž
a ⑀ , 1 0
/ž
0 y1
a
/
,
or
ž
x y
y⑀ , x
/
where ⺖qU s ² ⑀ :. An element of the fourth type is said to be split Cartan and an element of the fifth type is said to be nonsplit Cartan. The nonsplit Cartan elements form a subgroup of order Ž q q 1.r2. From examining the orders of each conjugacy class, we see that if 2 i N Ž q y 1., then ␥ i is conjugate to a split Cartan element, and if 2 i N Ž q q 1., then ␥ i is conjugate to a nonsplit Cartan element. Now by wF-Hx, 噛N²␥ : s
½
q y 1, q q 1,
␥ split ␥ nonsplit.
Also, half the conjugates will equal ␥ and half will equal ␥y1 . So
噛Fix Ž ␥ . s
噛N²␥ : 噛²␥ :
s
¡q y 1 ,
␥ ; ␥ i , ␥ split
¢
␥ ; ␥ i , ␥ nonsplit.
~
i qq1 i
,
Putting this information together, we obtain the following.
290
GABRIEL BERGER
PROPOSITION 4. If ␥ ¤ ␥ i , i s 2, 3, k, then trŽ␥ N H 1 Ž X Ž ᒍ ., ⺓.. s 2. If ␥ ; ␥ i for some i as abo¨ e, then
tr Ž ␥ N H 1 Ž X Ž ᒍ . , ⺓ . . s
¡2 y q y 1 ,
2 i N Ž q y 1.
¢2 y
2 i N Ž q q 1. .
~
i qq1 i
,
Remark. Our fake congruence groups are generalizations of regular congruence groups, since
ž
y 0
1 1 , 1
/ž
0 y
/
mod p
are the usual generators of SL 2 Ž⺖p . when d s 2. But in this case, the image Ž 10 11 . of 1 g B3 under p is unipotent, not Cartan. Thus our subsequent calculations differ in this case. The list of nontrivial irreducible finite-dimensional representations of SL 2 Ž⺖q . is quite short Žsee wF-Hx.: Ž1. The complement V of ⌺ g in the permutation representation is irreducible. ⺖qU
b .4 Ž2. Let B s Ž 0a ay1 be the Borel subgroup of PSL2 Ž⺖q .. Suppose 2 i rŽ qy1. ² : s ⑀ and s e . The map
ž
⑀n 0
) ª n ⑀yn
/
gives a 1-dimensional representation Wn of B; Ind PBS L 2 Ž⺖ q . Wn \ Yn is irreducible if n / Ž q y 1.r2; and for n s Ž q y 1.r2, Yn splits into two irreducible factors Yq and Yy. Ž3. If C denotes the nonsplit Cartan subgroup Ž xy
y⑀ x
.4 and
m : C ª ⺓ is a character, then V m Ym ' Ind CP S L 2 Ž⺖ q . [ Ym [ Zm for some representation Zm which is irreducible if m / Ž q s 1.r2 and splits into two irreducible factors Zq and Zy otherwise. We would like to know the multiplicity of each irreducible representation in our representation ᒍ : PSL2 Ž⺖q . ª AutŽ H 1 Ž X Ž ᒍ ., ⺓... We may use the standard character formula m s Ý g P S L 2 Ž⺖ q . ᒍ Ž␥ . Ž ␥ . , where ᒍ is the character of ᒍ and is irreducible.
291
FAKE CONGRUENCE MODULAR CURVES
TABLE I Character Multiplicities qy
V
Ý ai y T i
qq1yTy
Y2 r
Ý Ž ai q 1. i iNr
qy1yTy
Z2 m
qq1
Yq, Y y
2
qy1
Zq, Zy
2
Ý
Ž1 y a i .
i i mod m
y
y
1 2
1 2
Ty
Ý
1
Ý
1
a is1 qq1 even 2i
Ty
a isy1 qq1 even 2i
THEOREM 6. Assumptions as abo¨ e, define a i g ² " 1: by q ' a i mod i for i s 2, 3, k. Also, let q y ai Ts Ý . i is2, 3, 7 Then the character multiplicities are as listed in Table I, where i runs through 2, 3, k, Yq and Yy appear only for q ' 1 mod 4, and Zq and Zy appear only for q ' y1 mod 4. Proof. We make use of the character table of wF-Hx. We note that representations of PSL2 Ž⺖q . are the same as representations of SL 2 Ž⺖q . that map y1 2 to the identity. In the representation V, the character of "1 2 is q, the character of a split Žrespectively, nonsplit. Cartan element is 1 Žrespectively, y1., and there are Ž q y 3.r2 Žrespectively, Ž q y 1.r2. conjugacy classes with q 2 q q Žrespectively, q 2 y q . elements in each class. Our multiplicity then becomes 1 噛PSL2 Ž ⺖q .
⭈ 4 qg q Ž q 2 q q .
ž
⭈ 2
ž
qy3 2
y
Ý Xi
yŽ q 2 y q . ⭈ 2
ž
/
q
qy1 2
2y
Ý
ž
y
Ý Yi
qy1
/ / Ýž i
q
Xi
2y
qq1 i
/ / Yi
.
292
GABRIEL BERGER
Here, X i Žrespectively, Yi . is the number of conjugacy classes of elements in the cyclic subgroup ²␥ i : that are split Žnonsplit. Cartan. Thus,
Xi s
½
i y 1, 0,
ai s 1 a i s y1
and Yi s d y 1 y X i . Summing up gives us the desired result. Further use of the character tables gives us the following. The multiplicity for Y2 r is 1 噛PSL2 Ž ⺖q .
⭈ 4 Ž q q 1. g q 4 Ž q 2 y 1. q Ž q 2 q q .
ž
⭈ 2 Ž y2 y
Ý Ai . q Ý
ž
qy1
2y
i
/ / Ai
,
where
¡0, ¢y2 q 2 i ,
A s~y2, i
a i s y1 a i s 1, i ¦ r a i s 1, i N r .
The multiplicity for Z2 m is 1 噛PSL2 Ž ⺖q .
⭈ 4 Ž q y 1. g y 4 Ž q 2 y 1.
ž
qŽ q 2 y q . ⭈ 2 Ž2 y
Ý Bi . q Ý
ž
where
¡0, ¢2 y 2 i ,
B q~2, i
ai s 1 a i s y1, i ¦ m a i s y1, i N m.
2y
qy1 i
/ / Bi
,
293
FAKE CONGRUENCE MODULAR CURVES
The multiplications for Yq and Yy are 1 噛PSL2 Ž ⺖q .
⭈ 2 Ž q q 1. g q 2 Ž q 2 y 1.
ž
q Ž q 2 q q . ⭈ 2 Ž y1 y
Ý Ci . q Ý
ž
2y
qy1 i
/ / Ci
,
where
¡0, Ci s
~y1 q i ,
¢y1,
a i s y1 qy1 a i s 1, even 2i qy1 a i s 1, odd. 2i
The multiplicities for Zq and Zy are 1 噛PSL2 Ž ⺖q .
⭈ 2 Ž q y 1. g y 2 Ž q 2 y 1.
ž
qŽ q 2 y q . ⭈ 2 Ž1 y
Ý Di . q Ý
ž
2y
qq1 i
/ / Di
,
where
¡0, 1 y i, D s~
ai s 1 a i s y1,
i
¢1,
a i s 1,
qq1
2i qq1 2i
even
odd.
Again, summing up gives us the desired results. 4.2. The Calculation on H 1, 0 . Note that if g g PSL2 Ž⺖q ., p g X Ž ᒍ ., and g Ž p . s p, then g induces an automorphism g# , p : TX Ž ᒍ ., p ª TX Ž ᒍ ., p of the tangent space of X Ž ᒍ . at p. Since TX Ž ᒍ ., p is 1-dimensional, g#, p s e i for some g ⺢. We can calculate e i as follows: choose ˜ g g PSL2 Ž⺪.
294
GABRIEL BERGER
such that ⌫ Ž ᒍ . ˜ g s g and z p g ᒅU such that ⌫ Ž ᒍ . z p s p. Then in local coordinates, g# , p
⭸ ⭸z
s˜ gX Ž z p .
⭸
,
⭸z
and so if ˜ g s Ž ac db ., then 1
g# , p s
Ž cz p q d .
2
.
So choose
␥ ˜2 s
ž
0 1
y1 , 0
␥˜3 s
/
ž
0 y1
1 , 1
/
␥˜k s
and
ž
1 0
1 . 1
/
Let z 2 s e 2 i r4 , z 3 s e 2 i r6 , and z k s i⬁. Then one may easily check that ␥ jU , p j s e 2 i r j for j s 2, 3. For j s k, use the transformation z ¬ e 2 i z r k . In these coordinates, z ¬ ␥ ˜k z s z q 1 becomes the map y ¬ e 2 i r k y. 2 i r k Thus ␥ kU , p k s e as well. Noting that 6
g
XŽᒍ.
y1
6
g
y1
XŽᒍ.
6
6
XŽᒍ.
XŽᒍ.
induces an equality g#, p s Ž y1 g .#, p , we see that
␥j
U
, pj
s
½
e 2 i r j ,
y1␥ j s ␥ j
ey2 i r j ,
y1␥ j s ␥y1 j .
Now we make use of the following: THEOREM 7 ŽHolomorphic Lefschetz Fixed Point Theorem.. Lef Ž g , OX Ž ᒍ . . s
1
Ý pgFix Ž g . g# , pse i
1 y eyi
.
Then Lef Ž g , OX Ž ᒍ . . s
⬁
Ý Ž y1. i tr Ž g U N H i , 0 Ž X Ž ᒍ . , ⺓. . . is0
Let
295
FAKE CONGRUENCE MODULAR CURVES
THEOREM 8. Let ᒍ1, 0 denote the representation g ª AutŽ H 1, 0 Ž X Ž ᒍ ., ⺓... Then ᒍ1, 0 s 12 ᒍ . Note. This differs sharply from the congruence case. Proof. From our above calculations, we see that for any ␣ g PSL2 Ž⺖q ., Lef Ž ␥ j␣ , OX Ž ᒍ . . s s
q"1 2j 1 2
ž
1 y i r j
1ye
q
1 y i r j
1qe
/
s
q"1 2j
噛Fix Ž ␥ j␣ . ,
Ž 18 . Ž 19 .
since y1␥ j s ␥ j or ␥y1 for q " 1r2 j elements modulo ²␥ j :. j Then by the holomorphic Lefschetz fixed point theorem, tr Ž ␥ U N H 1, 0 Ž X Ž ᒍ . , ⺓ . . s 1 y Lef Ž ␥ , X X Ž ᒍ . . s1y
1 2
噛Fix Ž ␥ . s
Ž 20 . 1 2
ᒍ Ž ␥ .
᭙␥ . Ž 21 .
5. GENERA OF QUOTIENT CURVES 5.1. The Calculation. Let J Ž ᒍ . denote the Jacobian of X Ž ᒍ .. We would like to understand the decomposition of J Ž ᒍ . into simple factors. As a first step in this direction, we calculate the genera of certain quotient curves of X Ž ᒍ .. Suppose L is a subgroup of ⺖qU with w⺖qU r² "1:: "Lr² "1:x s ␣ L . Define a subgroup BL of PSL2 Ž⺖q . by BL s
½ž
a 0
b : agL . ay1
/
5
Let C be the nonsplit Cartan subgroup of PSL2 Ž⺖q . and S be the split 0 .4 . Denote the genus of X Ž ᒍ .rH by g Ž H . for any Cartan subgroup Ž ay1 0 a Ž . subgroup of PSL2 ⺖q . PROPOSITION 5. We ha¨ e the following: Ž1. Ž2. Ž3. Ž4.
2 g Ž 14. y 2 s 噛PSL2 Ž⺖q .Ž1 y Ý i Ž1ri .. 2 g Ž BL . y 2 s ␣ LŽ q q 1 y Ý i ⑀ i . 2 g Ž C . y 2 s q Ž q y 1. y Ý i ␦ i 2 g Ž S . y 2 s q Ž q q 1. y Ý i ␥ i ,
296
GABRIEL BERGER
where all summations are o¨ er i s 2, 3, k and
⑀ i Ž resp., ␦ i , ␥ i .
¡q y 1 q 2
s~
¢
i
qq1 i
ž ž
resp., resp.,
q Ž q y 1. i
,
Ž q y 1. Ž q q 2. i
Ž q q 1. Ž q y 2. i
q 2,
q 2 , q ' 1Ž i .
qŽ q q 1 i
/ . /
, q ' y1 Ž i . .
Proof. Let H be any subgroup of PSL2 Ž⺖q ., and let pi , ␥ i , be as in Section 4. Using the Hurwitz formula, 2 g Ž H . y 2 s PSL2 Ž ⺖q . : H ⭈ Ž y2 . q Ý Ž PSL2 Ž ⺖q . : H y 噛 points over pi .
Ž 22 .
i
s PSL2 Ž ⺖q . : H y
Ý
噛points over pi .
Ž 23 .
i
Now if PSL2 Ž⺖q . s jH j , the ramification index of  j pi over pi is ²␥ i j : : ²␥ i j : l H . We calculate these indices in the case H s BL ; the remaining cases are similar. First, note that a set of coset representatives for BL in PSL2 Ž⺖q . is given by
½ž
a y1
a
0 t
y1
a
/
s A a, t ,
ž
0 y1
a
ya 0
/5
,
where a runs over a set of coset representatives for L in ⺖qU and t is an arbitrary element of ⺖q . We may assume by Lemma 5 that ␥ i is split or nonsplit Cartan; we will examine these cases separately. So assume that ␥ i 0 . is split, and hence of the form Ž 0x xy1 . Then A a, t ⭈ ␥ i ⭈ Ay1 a, t s
ž
y2
a
x t Ž x y xy1 .
0 y1
x
/
.
297
FAKE CONGRUENCE MODULAR CURVES
We thus see that ²␥ i : A a, t l BL is trivial unless t s 0, in which case ²␥ i : A a, t : BL . Also,
ž
ya 0
0 y1
a
/ž
x 0
0 y1
x
/ž
0 yay1
y1 a s x 0 0
0 . x
/ ž
/
Hence in this case, the index in question will always be 1. So in 2 ␣ L cases our index is 1, and in w PSL2 Ž⺖q .: BL x y 2 ␣ L s Ž q y 1. ␣ L cases, our index is i. Thus we have 2 ␣ L unramified points and Ž q y 1. ␣ Lri points with ramification index i. Our proposition then follows in this case. Now assume that ␥ i is nonsplit, and hence of the form Ž yx⑀ xy .. Then A a, t ⭈ ␥ i ⭈ Ay1 a, t s
ž
x y yt
a2 y
ay2 y Ž ⑀ y t 2 .
x q yt
/
.
We thus see that ²␥ i : A a, t l BL is always trivial, since y is never 0 Žas the order of ␥ i is i . and ⑀ cannot be a square. Also,
ž
0 ay1
ya 0
/ž
x y⑀
y x
/ž
0 yay1
x a s 0 yay2 y
/
ž
ya2 y⑀ x
/
.
Thus in all Ž q q 1. ␣ L cases the ramification index is i, and we thus obtain Ž q q 1. ␣ Lri points in this case. This completes the proof of the proposition for BL . 5.2. AN EXAMPLE. Suppose d s 14, and choose s ye 2 i r7. As usual, set F s ⺡Ž q y1 . and K s ⺡Ž .. Let ᒍqs Ž13, q y1 q 7. ; OF . Then one may verify that ᒍq remains prime in OK , that OF rᒍqs ⺪rŽ13., and that the map X ¬ y induces an isomorphism ⺪ w X x r Ž 13, X 2 y 7X q 1 . ( OKrᒍq OK . Let ⑀ s X q 3 modŽ13, X 2 y 7X q 1.. Then ⑀ 2 s x 2 q 6 X q 9, which equals 8 mod Ž13, X 2 y 7X q 1.. Now conjugation in ⺪w X xrŽ13, X 2 y 7X q 1. is given by X modŽ 13, X 2 y 7X q 1 . ¬ Xy1 modŽ 13, X 2 y 7X q 1 . s 7 y X modŽ 13, X 2 y 7X q 1 . .
298
GABRIEL BERGER
So ⑀ is equal to Ž7 y X . q 3 modŽ13, X 2 y 7X q 1., which is just y⑀ . Recall that our map : PSL2 Ž⺪. ª PSUŽ , H, ⺖ 13 2 . is given by Ss Us
ž ž
1 0
1 ⑀y3 ¬ 1 0
/ ž / ž
1 y1
1 1
/
Ž 24 .
0 1 ¬ 1 y⑀ q 3
0 . ⑀y3
/
Ž 25 .
since ⑀ y 3 s X modŽ13, X 2 y 7X q 1. corresponds to y under our isomorphism. Now conjugate each matrix by
ž
6⑀ q 1 0
7⑀ q 2 4
/
and multiply by 9⑀ q 5. ŽThe latter is permitted since we’re working mod scalars.. Then
ž⑀ ž
y3 0
1 5 q 4⑀ ¬ 1 0
0 5 y 4⑀
/ ž / ž
0 5 y 3⑀ ¬ ⑀y3 6 y 2⑀
1 y⑀ q 3
6q2 5q3
/ ⑀ ⑀/
Ž 26 . ,
Ž 27 .
and we thus obtain generators for PSU2 Ž⺖ 13 2 .. Under the canonical isomorphism PSU2 Ž⺖ 13 2 . ª PSL2 Ž⺖ 13 . given by
ž
a q b⑀ c y d⑀
aqc c q d⑀ ¬ a y b⑀ bqd
/ ž
⑀ 2 Ž b y d. , ayc
/
we observe that
ž ž
5 q 4⑀ 0
0 5 ¬ 5 y 4⑀ 4
5 y 3⑀ 6 y 2⑀
6q2 5q3
/ ž ⑀ ž ⑀/ ¬
y2 y1
6 5
/
Ž 28 .
y1 . y1
/
Ž 29 .
299
FAKE CONGRUENCE MODULAR CURVES
Our composite map is then ⍀ : PSL2 Ž ⺪ . ª PSL2 Ž ⺖13 . : S¬ U¬
ž ž
5 4
6 5
y2 y1
Ž 30 .
/
Ž 31 .
y1 . y1
/
Ž 32 .
Setting ␥ 2 s ⍀ ŽUSU ., ␥ 3 s ⍀ Ž SU ., and ␥ 7 s ⍀ Ž S ., we finally obtain the following images of the generators of T Ž2, 3, 7.:
␥2 s
ž
7 10
8 , 6
/
␥3 s
ž
9 0
2 , 3
/
␥7 s
ž
5 4
6 . 5
/
Using our genus formulas, we see that our associated curve has genus 14, and that the representation into AutŽ H 1 Ž R N , ⺓.. is just Y2 [ Y2 . REFERENCES wA-Swx O. Atkin and H. P. F. Swinnerton-Dyer, Modular forms on noncongruence subgroups, in Proc. Sympos. Pure Math., Vol. 19, pp. 1᎐25, Amer. Math. Soc., Providence, 1971. wBex G. V. Belyi, On extensions of the maximal cyclotomic field having a given Galois group, J. Reine. Angew. Math. 341 Ž1983., 1᎐25. wBx G. Berger, Hecke operators on non-congruence subgroups, C. R. Acad. Sci. Paris Ser. ´ I. Math. 319 Ž1994., 915᎐919. wB2x G. Berger, Fake congruence subgroups and the Hurwitz monodromy group, draft, 1997. wB-Fx G. Berger and M. Fried, The Hurwitz monodromy group H4 , in preparation, 1997. wCx H. Cohen, ‘‘A Course in Computational Algebraic Number Theory,’’ Grad. Texts in Math., Vol. 138, Springer-Verlag, New YorkrBerlin, 1993. wBix J. Birman, Braids, links, and mapping class groups, Princeton, 1973. wFx M. Fried, Arithmetic of 3 and 4 branch point covers: A bridge provided by noncongruence subgroups of SLŽ2, Z ., in ‘‘Sem. de Theorie des Nombres, Paris, 1987-88,’’ Prog. Math., Vol. 81, pp. 77᎐117, Birkhauser, Basel, 1990. ¨ wF-Vx M. Fried and H. Volklein, The inverse Galois problem and rational points on moduli spaces, Math. Ann. 290, No. 4 Ž1991., 771᎐800. wF2x M. Fried, Introduction to modular towers: Generalizing dihedral group-modular curve connections, in ‘‘Recent Developments in the Inverse Galois Problem,’’ Contemporary Mathematics, Vol. 186, pp. 111᎐173, Amer. Math. Soc., Providence, 1995. wF-Hx W. Fulton and J. Harris, ‘‘Representation Theory: A First Course,’’ Springer-Verlag, New York, 1991. wMx A. M. Macbeath, Generators of the linear fractional groups, in Proc. Sympos. Pure Math., Vol. 12, pp. 14᎐32, Amer. Math. Soc., Providence, 1968. wO-Tx T. Oda and T. Terasoma, Surjectivity of reductions of the Burau representation of Artin braid groups, draft, 1995.
300 wSchx wVx wWx
GABRIEL BERGER
A. J. Scholl, Modular forms and de Rham cohomology; Atkin᎐Swinnerton᎐Dyer congruences, In¨ ent. Math. 79 Ž1985., 49᎐77. H. Volklein, Braid group action via GLŽ n, q . and UŽ n, q ., and Galois realizations, Israel J. Math. 82 Ž1993., 405᎐427. K. Wolfahrt, An extension of F. Klein’s level concept, Internat. J. Math. 8 Ž1967., 529᎐535.