MATHEMATICS OF COMPUTATION
Volume 74, Number 251, Pages 1545–1557
S 0025-5718(04)01709-0
Article electronically published on September 10, 2004
COMPUTING WEIGHT 2 MODULAR FORMS OF LEVEL p2
ARIEL PACETTI AND FERNANDO RODRIGUEZ VILLEGAS,
WITH AN APPENDIX BY B. GROSS
Abstract. For a prime p we describe an algorithm for computing the Brandt
matrices giving the action of the Hecke operators on the space V of modular
forms of weight 2 and level p2 . For p ≡ 3 mod 4 we define a special Hecke stable
subspace V0 of V which contains the space of modular forms with CM by the
√
ring of integers of Q( −p) and we describe the calculation of the corresponding
Brandt matrices.
1. Introduction
The main goal of this paper is to describe an effectively computable Hecke stable
subspace V0 of the space V of modular forms of weight 2 and level p2 , with p ≡
3 mod√4 prime, containing the space VCM of forms with CM by the ring of integers
of Q( −p). The space V0 is constructed in terms of the Brandt matrices associated
to ideal classes of an order (of index p in a maximal order) in the quaternion algebra
over Q ramified at p and ∞.
Computationally this approach to study VCM has several positive features. First,
the total space V has dimension that grows proportionally to p2 whereas V0 has
dimension that grows proportionally to p. This means that in practice calculations
with V0 can be carried out for much larger primes p than with V itself. Second,
the space V0 is indeed effectively computable; more concretely, V0 can be cut out
from V in a straightforward manner.
Ultimately, the reason for studying the questions discussed here is to effectively
compute a Shimura lift of the CM forms of level p2 . In the present paper we describe
how to compute the corresponding eigenvector of all Brandt matrices, In a later
publication we will describe how this can be used, in a generalization of methods
of Gross for level p, to obtain a Shimura lift.
In conclusion the main computational principle in this paper is that by using
Brandt matrices it is possible (say, for nonsquarefree level) to effectively work with
smaller dimensional Hecke stable subspaces of modular forms. This appears to be
a useful principle that could be exploited further.
Received by the editor February 18, 2003 and, in revised form, December 16, 2003.
2000 Mathematics Subject Classification. Primary 11F11; Secondary 11E20, 11Y99.
The first and second authors were supported in part by grants from TARP and NSF (DMS99-70109); they would like to thank the Department of Mathematics at Harvard University, where
part of this work was done, for its hospitality.
c
2004
American Mathematical Society
1545
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1546
ARIEL PACETTI AND FERNANDO RODRIGUEZ VILLEGAS
2. Preliminaries on quaternion algebras
Notation. Fix a prime p > 2 and let B be the quaternion algebra over Q ramified
at p and at ∞ (such an algebra is unique up to isomorphism). We write N (x) for
the reduced norm of an element x ∈ B, and we write Tr(x) for its reduced trace.
Definitions.
(1) A lattice I ⊂ B is a Z-module of rank 4.
(2) An order O ⊂ B is a ring which is a lattice.
(3) Given a lattice I, its left order is Ol (I) := {x ∈ B / xI ⊂ I}; similarly, its
right order is Or (I) := {x ∈ B / Ix ⊂ I}.
(4) For a lattice I and a prime q we let Iq := I ⊗ Zq .
(5) Given an order O, a left O-ideal is a lattice I such that I is locally principal;
i.e., for all primes q we have Iq = Oq aq for some aq ∈ (B ⊗ Qq )× .
(6) For a left O-ideal I of B, its norm N (I) is the positive generator of the
ideal of Z generated by N (x) with x ∈ I.
(7) Given a left O-ideal I of B, we define NI : I −→ Z as x → N (x)/N (I).
(8) Given a lattice I, its dual is I # := {b ∈ B / Tr(bI) ⊂ Z}.
(9) A lattice is integral if it is contained in its left and right orders.
We fix a maximal order O once and for all.
Proposition 1. If I is a lattice such that Ol (I) is maximal, then I is a left Ol (I)ideal.
Proof. See [Vi, p. 86].
#
#
Theorem 1. Let I be a left O-ideal and I its dual. Then I is a right O-ideal
#
Z/pZ ⊕ Z/pZ as
and I ι := N (I)pI is a left O-ideal contained in I with I/I ι
abelian groups and N (I ι ) = N (I)p. If I = O, then O/Oι Fp2 as rings.
Proof. If O is an order, then, by definition, Oι is its different. Since B has only one
ramified prime, P = Oι is the unique maximal 2-sided prime over p. Since all ideals
are locally principal, we have that if Iq = Oq aq , then Iq = aq Oq = aq Oq for all
primes q; also, it is not hard to check that Iqι = Oqι aq . By [Vi, Lemma 4.7, p. 24],
the different is a bilateral O-ideal of norm p. It follows that O/Oι Op /Opι Fp2
and it is now easy to finish the proof.
Remark. It is not hard to verify that I ι = PI, where P is the different, which could
have been used as its definition.
Proposition 2. If I is a lattice, then (I # )# = I.
Proof. This is standard.
Corollary 1. If I is a lattice, then Ol (I # ) = Or (I) and Or (I # ) = Ol (I).
Proof. It is clear that if α is in I # and x is in Ol (I), then αx ∈ I # , which implies
¯ we get that Or (I) ⊂
that Ol (I) ⊂ Or (I # ); using that I # = I¯# and replacing I by I,
#
#
Oi (I ). Applying the same argument to I and using the previous proposition,
we get the other inclusion.
Lemma 1. Let J ⊂ I be two left O-ideals. Then (N (J)/N (I))2 = |I/J| = [I : J].
Proof. It is enough to check locally the case I = O. If Jq = Oq αq , then N (Jq ) =
N (αq ). Since J is integral, Oq αq ⊂ Oq ; its index is up to a unit in Z×
q the determi
nant of multiplication by αq , which equals N (αq )2 .
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COMPUTING WEIGHT 2 MODULAR FORMS OF LEVEL p2
1547
Lemma 2. Let I be a left O-ideal and J ⊂ I a sublattice of index p, such that
I ι ⊂ J ⊂ I. Then Ol (J) = Z + Oι ⊂ O with index p. Furthermore OJ = I and
N (J) = N (I).
Proof. Clearly Ol (J) contains Z + Oι . Since I/I ι is a 1-dimensional vector space
over O/Oι Fp2 (by Theorem 1) and J/I ι is a submodule of index p, necessarily
Ol (J) must equal the proper submodule Z + Oι (of index p in O).
Definition. An order has level p2 if it has index p in some maximal order.
We denote by Õ = Z + P the unique suborder of level p2 in O (see [Pi, Lemma
1.4, p. 181]) and by h, h̃ the class numbers of O, Õ, respectively.
Proposition 3. Any lattice I with Ol (I) = Õ is an Õ-ideal.
Proof. Let I be such a lattice. By Proposition 1, for all primes q = p, Iq is principal,
since Õq = Oq . For the ramified prime, since Zp is a PID, there exists ap ∈ Ip with
(N (ap )) = N (Ip ). Therefore, Õp ap ⊂ Ip ⊂ Op Ip . Since Op Ip is an ideal for Op of
the same norm as Ip , we have by Lemma 1 that Op Ip = Op ap . On the other hand,
the index of Õp in Op is p; hence, Ip = O˜p ap .
Proposition 4. Let I be a left Õ-ideal. Then the following hold.
(1) If x ∈ I is such that p NI (x), then NIp(x) is independent of x, where
(÷) denotes
the Kronecker symbol.
NI (x)
(2)
only depends on the equivalence class of I.
p
(3) If Iis principal, then NIp(x) = 1.
Proof. The proofs are quite elementary; see [Pi, Proposition 5.1, p. 198].
Elements
x ∈I as in the proposition always exist; we let χ(I) denote the common
value of NIp(x) . It is easy to check that χ(I) = χ(I) where the bar denotes
conjugation and χ(I −1 ) = χ(I).
Corollary 2. Given two orders Oj of level p2 for j = 1, 2 and left Oj -ideals Ij for
j = 1, 2 such that Or (I1 ) = O2 , then χ(I1 I2 ) = χ(I1 )χ(I2 ).
Proof. Pick xj ∈ Ij for j = 1, 2 with p NIj (xj ) and take x1 x2 ∈ I1 I2 ; note that
N (I1 I2 ) = N (I1 )N (I2 ).
3. Computing left Õ-ideal representatives
Proposition 5. Let p be a prime and let B = (a, b) be the quaternion algebra
ramified at p and infinity with i2 = a and j 2 = b. Then a Õ order is given by the
basis:
• 12 (1 + j), 12 (pi + k), j, k with a = −1, b = −p if p ≡ 3 mod 4,
• 13 (1 + j + k), 14 (pi + 2j + k), j, k with a = −2, b = −p if p ≡ 5 mod 8,
pj
i
k
sk
• 12 + pj
2 , 2 + 2 , k, q + q with a = −p, b = −q if p ≡ 1 mod 8 where
q is a prime such that pq = −1, q ≡ 3 mod 4 and s is an integer with
s2 ≡ −p mod q and s ≡ −q mod p.
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1548
ARIEL PACETTI AND FERNANDO RODRIGUEZ VILLEGAS
Proof. This is just an easy but tedious computation. Note that in the case p ≡
.
1 mod 8 the maximal order we are considering is M = 12 + 2j , 2i + k2 , k, qj + (s+q)k
pq
The conditions on s make this an order, and it is easy to check that it is maximal,
but it differs from the one defined in [Pi, Proposition 5.2].
Given a left O-ideal I, by Lemma 2 and Proposition 3 there are p + 1 Õ-left
ideals J with
(1)
I ι ⊂ J ⊂ I,
[I : J] = [J : I ι ] = p.
We call any such J a p-subideal of I.
Proposition 6. Any p-subideal J is of the form I ι [v] for some v ∈ I and for any
such v we have p NI (v).
Proof. Since J has index p in I, it is clear that J = I ι [v] for some v ∈ I, v ∈ I ι ,
and locally all these ideals are equal for all primes q = p. Let Ip = Op ap . Then we
saw that Ipι = Opι ap ; since v ∈ I, v = uap with u ∈ Op . If p|NI (v), then p|N (u);
hence u ∈ Opι and we would have that J ⊂ I ι .
We now show how to obtain a set of representatives of left Õ-ideals by considering
these index p sublattices for a set of representatives of left O-ideals. We then use
these ideals to construct the Brandt matrices for Õ.
Proposition 7. Let Ii for i = 1, 2 be left O-ideals and let Ji ⊂ Ii for i = 1, 2
corresponding p-subideals. If I1 and I2 are nonequivalent, then so are J1 and J2 .
Proof. If J1 = J2 α for some α ∈ Õ, then I1 = OJ1 = OJ2 α = I2 α (by Lemma 2)
which is a contradiction.
We fix a set of representatives I 1 , . . . , I h of left O-ideals.
Proposition 8. Every Õ-ideal is equivalent to some p-subideal J ⊂ I j for some j.
Proof. The left O-ideal OJ is equivalent to some I j ; i.e., OJ = I j α for some α and
hence OJα−1 = I j . Therefore Jα−1 ⊂ I j and OJα−1 = I j . A simple calculation
shows that Jα−1 has index p in I j . For a prime q = p, we have that Oq = Õq .
Then Oq Jq = Jq = Iqj , so no primes other than p appear in the index. As for the
ramified prime, let us say that Jp α−1 = Õp ap , and Ipj = Op cp . Since
Op Jp = Ip , we
j
j
−1
have that Op ap = Op cp so Ip = Op ap ; therefore Ip /(Jp α ) = Op ap /Õp ap = p.
Since Jα−1 ⊂ I j with index p, to see that I ι ⊂ Jα−1 , it is enough to check
locally at p. Let Jp = Õp bp , Ip = Op ap . Without loss of generality we may assume
that bp α−1 = ap . By the proof of Theorem 1, we see that Ipι = Opι ap . Also Opι ⊂ Õp ;
therefore Opι ap ⊂ Õp ap = Õp bp α−1 = Jp α−1 .
The following lemma is easy to check.
Lemma 3. Two p-subideals J, J ⊂ I are equivalent if and only if Ju = J for
u ∈ Or (I)× .
Corollary 3. Given a left O-ideal I, the number of nonequivalent p-subideals J ⊂ I
is (p + 1)|Or (J)× |/|Or (I)× |.
Proposition 9. If p > 3, then the number of units in Õ is 2, and if p = 3 the
number of units is 2 or 6.
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COMPUTING WEIGHT 2 MODULAR FORMS OF LEVEL p2
1549
Proof. See Proposition 5.12 of [Pi2].
For j = 1, . . . , h we let Oj = Or (I j ) and let Õj be its suborder of index p.
Corollary 4. We have
(2)
h̃ = (p + 1)
h
|Õj× |
j=1
|Oj× |
.
If p > 3, then h̃ = (p2 − 1)/12.
Proof. This is clear from Proposition 9 and Eichler’s mass formula for maximal
ideals.
There are the same number of Õ-ideals with character χ equal to 1 as with
character −1. The proof given in [Pi, Proposition 5.6, p. 199] uses the action of a
certain element α of the idele group of B on ideals. We now describe an algorithmic
version of this action.
The components αq of α are as follows: for q = p we set αq = 1 and for q = p
we want αp with zero trace such that
a
= −1,
p
where a = N (αp )/pn and n = vp (N (αp )) with vp the valuation at p. We then have
that χ(αp J) = −χ(J). We denote by δ the involution
(3)
δ:
J → αJ.
Note that if J and J are equivalent, then so are δJ and δJ .
3.1. Construction of αp . From now on we fix the specific basis i, j for the algebra
B and the maximal order O as in [Pi2, Proposition 5.2, p.369].
There are two cases.
(1) If p ≡ 1 mod 4, then by our very choice of basis for the quaternion algebra
we may take αp to be one of i or j.
(2) If p ≡ 3 mod 4, then −1 is a nonsquare and we look for αp with norm −p.
If α = x1 i + x2 j + x3 k, with i2 = −1, j 2 = −p = k 2 , then N (αp ) = x21 +
p(x22 + x33 ). We can take x1 = 0 and look for a solution to the equation
x22 + x23 = −1 in Zp , which is achieved by finding a solution to x22 + x23 ≡
−1 mod p and then lifting the solution using Hensel’s lemma.
3.2. Action of αp on I. We will follow [Ei, Theorem 7, p. 34]. First we need to
compute an r such that αp J ⊃ Jpr .
Lemma 4. Let n = vp (N (αp )) be the p-valuation of the norm of αp . Then αp J ⊃
Jpn/2+1 .
s
Proof. In order that αp J ⊃ Jps , we must have α−1
p p ∈ Õ. Note that if β ∈ O,
−1 s−1
∈ O or, equivalently, when
then pβ ∈ Õ; hence it is enough to check when αp p
s−1
vp (N (α−1
p
))
≥
0.
It
is
now
straightforward
to
verify
that it is enough to take
p
vp (N (αp ))
+ 1.
s≥
2
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1550
ARIEL PACETTI AND FERNANDO RODRIGUEZ VILLEGAS
Set r = n/2 + 1. Starting with a global basis for Jpr , we start adjoining
elements until we find a generating set for αJ. Say J = u1 , u2 , u3 , u4 so that
αp J = αp u1 , αp u2 , αp u3 , αp u4 . It is not hard to see that we have
αp uj ≡ vj mod pr Jp ,
s
j = 1, . . . , 4,
r
with p vj ∈ J for some s. We set J = Jp , v1 , v2 , v3 , v4 . Clearly αp Jp = Jp and
for a prime q = p we have vi ∈ Jq for i = 1, . . . , 4 and hence Jq = Jq .
Having computed representatives for some maximal order (respectively, an order
of level p2 ), we can get representatives for any other order, if needed, by simply
multiplying on the right by an appropriate ideal (see [Pi2, Proposition 1.21, p. 348]
for a proof of this elementary fact).
To perform the above computations accurately, we need to know a priori how
many terms of the p-expansion of αp to use.
Lemma 5. Given a left Õ-ideal J, let αp be as constructed above. In order to
compute αp J, it is enough to know αp to order O(pr+1 ), where r = vp (N (αp ))/2+
1.
Proof. For our choice of O, i, j we have {1, i, j, k} ⊂ O and hence {p, pi, pj, pk} ⊂ Õ.
Then, with the notation as in the proof of Lemma 4, {piut, pjut , pkut } ⊂ I for
1 ≤ t ≤ 4; hence, pr+1 αp ut ∈ pr I and the denominator of the xj is at most
r + 1.
Note that with our choice of αp we have r = 1 for p ≡ 1 mod 4 and r = 2 for
p ≡ 3 mod 4. By Lemma 5, therefore, it is enough to compute the first two terms
in the p-adic expansion of αp .
3.3. Further structure. There is more structure on the ideals J that we are going
to use to prove some properties of the Brandt matrices.
It is clear that Op /Opι is isomorphic to Fp2 and Õp /Opι to Fp . Let S :=
(Op /Opι )× , a cyclic group of order p2 − 1. Given a Õ-ideal J and u ∈ S, we
define uJ, with some notation abuse, by regarding Op as a subring of the adeles.
It is easy to check that this gives rise to a (left) action of S on left Õ-ideals with
stabilizer (Õp )× /(Opι )× . It is also easy to check that S acts on the set of p-subideals
making it a principal homogeneous space for G := (Op /Opι )× /(Õp /Opι )× , a cyclic
group of order p + 1.
Let u be a generator of G and let J be some p-subideal of I. Then {ui J}pi=0 are
all the p-subideals of I. By Proposition 5.6 of [Pi] we know that if Jp = Õp αp , then
χ(J) is the quadratic symbol of N (αp )/N (J) modulo p. Since the norm map from
Fp2 to Fp is surjective, we must have that N (u) is a nonsquare modulo p and hence
χ(ui J) = (−1)i χ(J).
We form a set of inequivalent p-subideals J = {J, uJ, . . . , ur−1 J} where r is the
smallest positive integer such that ur J is equivalent to J. Note that r is necessarily
even since J decomposes into two subsets, according to the value of χ, which are in
bijection by δ. Also, if u ∈ G, then u is its inverse since uu = N (u) and N (u) ∈ Õ1 .
4. Constructing the Brandt matrices
Now we can describe the calculation of the Brandt matrices themselves. We
should point out that the software package Magma [Ma] includes routines for calculations of Brandt matrices due to D. Kohel and these are described in [Ko] (note,
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COMPUTING WEIGHT 2 MODULAR FORMS OF LEVEL p2
1551
however, that the paper does not treat the case of level p2 though the routines in
Magma do).
We pick a maximal order O and we calculate representatives {I 1 , . . . , I h(O) }
of left O-ideal classes (using Pizer’s algorithm for the level p case) and we fix
a generator u of G. To compute inequivalent p-subideals of each I k , we follow
Section 2 and we order them as follows. Dropping the k from the notation, we pick
a p-subideal J0 with χ(J0 ) = 1 and consider
(4)
J0 , J2 , . . . , Jr−2 , J1 , J3 , . . . , Jr−1
where J1 = δJ0 and Ji+2j = u2j Ji for i = 0, 1, r is the number of nonequivalent
p-subideals of I k and δ is the involution defined in (3). Note that by construction
χ(Ji ) = (−1)i .
We will consider the Brandt matrices B(q) defined using the following ordering
of classes of p-subideals. First we put the classes with χ = +1 as
(5)
J01 , J21 , . . . , Jr11 −2 , J02 , J22 , . . . , Jr22 −2 , . . . J0h , J2h , . . . , Jrhh −2 ,
followed by those with χ = −1,
(6)
J11 , J31 , . . . , Jr11 −1 , J12 , J32 , . . . , Jr22 −1 , . . . J1h , J3h , . . . , Jrhh −1 ,
where h = h(O) and Jji are the representatives for the p-subideals of I i as described
in (4).
For every prime q we consider the Brandt matrix B(q) with respect to the above
chosen basis. One of the important things of ordering the basis in this form is the
following.
Proposition 10. For q = p write the Brandt matrix B(q) in block form
A B
B(q) =
,
C D
where each A, B, C, D has size h(Õ)/2 × h(Õ)/2. Then the following hold.
(1) If pq = 1, then B = C = 0, and A = D.
(2) If pq = −1, then A = D = 0, and B = C.
Proof. This is just a special case of [Pi, Theorem 5.15, Theorem 5.18 p. 203].
The above proposition shows that to find the eigenvectors and eigenvalues of
B(q) we just need to work with A or B, depending on the case, which have half the
size of B(q).
We now restrict to the case ( pq ) = 1 (the other is completely analogous). It is
not hard to see that the group G and the involution δ, acting on Õ-ideals generate
a dihedral group D of order 2(p + 1). Concretely, δuδ = u−1 . In particular, this
relation allows us to restrict our attention to the matrix A. We let Ai,j be the
i
with
ri /2 × rj /2 submatrix of A corresponding to the columns Jlj and the rows Jm
l = 0, 2, . . . , rj and m = 0, 2, . . . , ri .
We index the rows and columns of Ai,j by indices l, m modulo ri /2 and rj /2,
respectively.
Proposition 11. The matrix Ai,j has the following properties:
Let r = gcd(ri /2, rj /2). Then there exist coefficients c(k) indexed by k mod r
such that the l, m entry of Ai,j equals c(m − l).
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1552
ARIEL PACETTI AND FERNANDO RODRIGUEZ VILLEGAS
In practice this fact means, in particular, that the successive rows of Ai,j are
obtained from the first by a shift of one step to the right.
Lemma 6. For v ∈ G we have v Õp = Õp v.
Proof. The order v Õp v −1 is a suborder of Op of index p; hence v Õp v −1 = Õp .
Proof of Proposition 11. The entry [l, m] of the matrix Ai,j corresponds to the ideal
i
. The p-subideal (J1j )p = Õp αp for some element αp ∈ Op and since
(Jlj )−1 Jm
we assume that p does not divide the norm of the ideal class representatives, αp
determines an element ua ∈ G. Hence (J1j )p = ua Õp and similarly (J1i )p = ub Õp
i
for some 0 ≤ a, b < p + 1. Therefore, (Jlj )p = ua+2l Õp and (Jm
)p = ub+2m Õp . It
j −1 i
b−a+2m−2l
Õp , by Lemma 6. We
follows that the p-subideal ((Jl ) Jm )p equals u
i
have then that (Jlj )−1 Jm
= u2(m−l) ((J1j )−1 J1i ). Since, by definition, uri sends J1i
to an equivalent p-subideal and analogously for urj and J1j , the [l, m] entry of Ai,j
depends only on the residue of m − l modulo r.
5. The subspace V0
Let V be the vector space of complex valued functions on the classes of left Õideals. The dihedral group D generated by δ and G defined earlier has a left action
on V by means of
γ ∈ D.
γf (J) := f (γ −1 J),
We consider the subspace V0 of V of functions f0 satisfying
f0 (u2 J) = −f0 (J),
where u is any generator of G.
Note that if p ≡ 1 mod 4, this space is identically zero as G has order p + 1.
For p ≡ 3 mod 4 we may describe V0 in a more conceptual way as the ρ-isotypical
component of V with ρ the 2-dimensional irreducible representation of D induced
from any of the two characters of G of order 4.
We may further split the space V0 into two subspaces V0± where δ acts as ±1.
It is easy to verify that any generator u of G takes V0+ isomorphically into V0− and
vice versa.
Theorem 2. The subspaces V0± are stable under the action of all Brandt matrices
B(q).
Proof. We first prove that V0 is stable under the Brandt matrices. Let vi =
(1, −1, . . . , −1) of length ri /2 and similarly let vj = (1, −1, . . . , −1) of length rj /2.
We consider the case where ( pq ) = 1; the other case is completely analogous. Using
the choice of basis above, it is enough to prove that Ai,j vj = λvi for some λ ∈ Z
and this is clear from the form of the matrix Ai,j given by Proposition 11. It is also
easy to see that λ = 0 if ri /2 is odd.
Since δ commutes with B(q) (see Proposition 10), the subspaces V0± are also
stable under the action of the Brandt matrices.
We let B0 (q) be the matrix B(q) restricted to V0+ . One of the main motivations
for considering this subspace is that it contains, for p > 3, a copy of the space
√ of
modular forms of weight 2 and level p2 with CM by the ring of integers of Q( −p).
The proof of this fact is given by Benedict Gross in the appendix and uses the
local and global Jacquet-Langlands correspondence. Concretely, it is the subspace
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COMPUTING WEIGHT 2 MODULAR FORMS OF LEVEL p2
1553
+
VCM
⊂ V0+ characterized by the vanishing of B0 (q) for all primes q with ( pq ) = −1;
+
+
clearly, VCM
is stable under the Hecke
algebra. For p > 3, VCM
has dimension
√
h(−p), the class number of K = Q( −p), and it can be identified with the tangent
space of an abelian variety B(p)/Q obtained as the restriction of scalars of a certain
elliptic curve A(p) with CM by the ring of integers of K (see [Gr]). For p = 3 both
+
VCM
and V0+ are zero.
We now obtain a formula for the dimension of V0+ .
Proposition 12. For a prime p > 3 and congruent to 3 modulo 4 the dimension
of V0+ is given by
1
−3
2
1
1
+
dim(V0 ) =
(p + 5) +
1−
−
1−
.
12
3
p
2
p
Proof. Note that the first part of the formula is the number of ideals for the maximal
order (for p ≡ 3 mod 4). By Corollary 3 to compute the number of nonequivalent
p-subideals of a given ideal I = Ij , we need to compute w = |O× |/|Õ× | where O
is the right order of I. We claim that w = 1, 2 or 3. Let u ∈ O be a unit. Since all
elements in B satisfy a quadratic polynomial, the field F = Q[u] is an imaginary
quadratic field. If u = ±1, then u is a primitive root of order 3 or 4. In both cases,
if there is an embedding of Z[u] into O , it is unique up to conjugation because
the class number of Z[u] is one. The existence of such an embedding into some
−4
maximal order is determined by the quadratic symbols ( −3
p ) and ( p ), respectively.
√
It is known that Z[i] and Z[(1 + −3)/2] embed into the same maximal order only
for p = 2 or 3. Hence, in the first case w = 3 and in the second w = 2 since (by
Proposition 9) Õ× is of order 2. By Corollary 3, rj = (p + 1)/w ; hence if w = 3,
then rj is always even and if w = 2, then rj is even if and only if p ≡ 7 mod 8.
The formula now follows.
6. Tables
The calculations in Table 1 were made with PARI-GP [GP] (check the website
[PRV] for the corresponding routines).
Table 1.
p
7
11
19
23
31
43
47
59
67
71
79
83
103
dim V0+
1
1
1
3
3
3
5
5
5
7
7
7
9
+
dim VCM
1
1
1
3
3
1
5
3
1
7
5
3
5
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1554
ARIEL PACETTI AND FERNANDO RODRIGUEZ VILLEGAS
Example 1. Let p = 11. In this case the class number for maximal orders is 2;
hence the matrix Ai,j will have four blocks. The first Brandt matrices are below.
B(2) =
B(3) =
0
0
0
0
0
1
0
0
1
1
0
0
0
0
0
0
1
0
1
1
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
1
1
1
0
0
1
0
0
1
1
0
0
0
0
0
0
1
0
1
1
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
2
0
0
1
1
0
0
0
0
0
0
2
0
1
1
0
0
0
0
0
0
0
2
1
1
0
0
0
0
0
1
1
1
0
1
0
0
0
0
0
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
2
0
0
1
1
0
0
0
0
0
0
2
0
1
1
0
0
0
0
0
0
0
2
1
1
0
0
0
0
0
1
1
1
0
1
0
0
0
0
0
1
1
1
1
0
,
.
+
Example 2. Let p = 47. In this case V0+ = VCM
is of dimension 5. We give some
q
+
= V0+ , we know that
examples of the matrices B0 (q) for q with ( p ) = 1; since VCM
q
B0 (q) vanishes for ( p ) = −1.
1 2 0
0 0
1 0 1
0 1
1 0
B0 (2) = 0 1 1
0 0 1 −2 0
0 3 0
0 0
and
B0 (3) =
0
0
1
1
0
0
2
2 0
1
1 −2 0
1 −2
0 0
.
−2
0
0 1
0
0
3 1
+
Table 2 shows the abelian varieties B(p) corresponding to VCM
for small p labeled
as in William Stein’s list. Table 3 is the corresponding table for subspaces of V0+
+
stable by the Hecke algebra in the complement of VCM
.
The case of p = 79 is interesting. It is the only case with p ≤ 400 where the
+
in V0+ contains 1-dimensional Hecke stable subspaces. By
complement of VCM
calculations of Cremona the two subspaces correspond to the elliptic curve of the
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COMPUTING WEIGHT 2 MODULAR FORMS OF LEVEL p2
1555
Table 2.
p
7
11
19
23
31
43
47
59
67
71
79
83
Label
49A
121A
361A
529F
961G
1849A
2209F
3481C
4489A
5041F
dim
1
1
1
3
3
1
5
3
1
7
5
3
Table 3.
p
43
59
67
79
79
83
Label
1849E
3481A
4489E
6241A
6241B
dim
2
2
4
1
1
4
equation
y 2 + xy = x3 − x2 − 64x − 179
√
and its quadratic twist by Q( −79).
Appendix
We will prove that the space of CM modular forms of weight 2 and level p2 injects
into the space V0 . Let Õp× (respectively Op× ) be the group of invertible elements
of Õp (respectively of Op ). Then the quotient Op× /Õp× is isomorphic to the group
G; hence Op× contains a unique subgroup Kp such that Op× /Kp is cyclic of order 4.
Note that the group Õp× is equal to Z×
p (1 + P) where P is the unique integral order
of norm p in Bp . Define
Ol× \B̂ × /B × → C}.
(7)
M := {f : Kp ×
l=p
Translating back to the language of ideals of B as in the body of the paper, we can
identify M with the subspace of V where u4 acts trivially with u a generator of G.
Recall that we have defined
l
(8)
VCM := f ∈ M : f |Tl = 0 for all
= −1
p
where Tl is the l-th Hecke operator. Recall the involution δ defined in (3); it acts
on M commuting with the Tl and hence also gives an involution of MCM . We may
±
therefore decompose M and MCM into their eigenspaces M ± , MCM
with respect to
δ.
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1556
ARIEL PACETTI AND FERNANDO RODRIGUEZ VILLEGAS
Let [u] be a generator of Op× /Kp . Then we can identify the space V0 with the
functions f ∈ M + such that f |[u2 ] = −f .
±
Theorem 3. MCM
⊂ V0± and has dimension h(−p) if p ≥ 7.
Proof. We know that the space of cusp forms F of weight 2 for Γ0 (p2 ) with complex
√
multiplication by Q( −p) has dimension h(−p) (see [Gr]). By a theorem of Serre
(see Theorem 17, [Se])
space is characterized by the condition that F |Tl = 0
this
for all primes l with
l
p
= −1. This gives h(−p) automorphic representations
π = π∞ ⊗ πp ⊗
πl
l=p
of P Gl2 (A) with
• π∞ a discrete series of weight 2 for P Gl2 (R),
• πp an irreducible representation of P Gl2 (Qp ) of conductor p2 ,
representation of P Gl2 (Ql ) with Hecke eigenval• πl an irreducible unramified
ues al = 0 if
l
p
= −1.
The local Jaquet-Langlands correspondence gives a bijection between irreducible,
square-integrable, representations πv of P Gl2 (Qv ) and finite dimensional, irreducible representations πv of Bv× /Q×
v , where Bv is the quaternion division algebra over Qv . The local correspondence is characterized by the identity Tr(t|πv ) +
Tr(t|πv ) = 0 for all regular elliptic conjugacy classes t.
is the trivial represenIf π∞ is the weight 2 discrete series of P Gl2 (R), then π∞
×
×
tation of H /R = SO3 .
If πp has conductor pn+1 , then πp is trivial on the subgroup 1 + πpn Op of Bp× .
We want to apply this to the local component πp of our CM forms. First,
we must check that πp is square-integrable. In fact we will show it is a cuspidal
representation by checking that its Langlands parameter σ(πp ) : W (Qp ) → Gl2 (C)
gives an irreducible 2-dimension representation of the local Weil group.
By construction of the CM forms, we have
W (Q )
σ(πp ) = IndW (kpp) χp
√
where kp = Qp (ηp ), with ηp = −p,
χp is the local component of our Hecke
and
= −1, we have χp (−1) = −1. Hence if τ
characters of conductor (ηp ). Since −1
p
is the nontrivial automorphism of kp over Qp ,
(9)
(10)
χτp (ηp ) = χp (ηpτ ) = χp (−ηp ) = −χp (ηp )
and χτp = χp . This shows that σ(πp ) is irreducible by Mackey’s criterion for induced
representations.
We will now determine the corresponding irreducible representations πp of D =
×
Bp /(1 + ηp Op )Q×
p . D is a dihedral group of order 2(p + 1), with normal subgroup
×
G = Op× /(1 + ηp Op )Z×
F×
p
p2 /Fp . Hence any irreducible representation of D has
dimension 1 or 2.
Since πp satisfies πp ⊗ p (det) √
πp , where p is the quadratic character of Qp
associated to the extension kp = Qp ( −p), the same holds for πp : πp ⊗ p (G) πp .
This is false if πp is 1-dimensional, so we must have
(11)
D −1
πp = IndD
)
G (γ) = IndG (γ
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COMPUTING WEIGHT 2 MODULAR FORMS OF LEVEL p2
1557
for some character γ of G with γ = γ −1 (so γ 2 = 1). (This is the representation
of D denoted by ρ at the beginning of §4.) Since (G) on F×
p2 is just the quadratic
character β of G, we have that γβ = γ −1 . Equivalently γ 2 = γ −2 = β and (γ, γ −1 )
are the two characters of order 4 of G. Hence the subgroup Kp of index 4 in Op×
acts trivially on πp . Let [u] be a generator of G. Since the action of G on IndD
G (γ)
γ
0
2
is given by 0 γ −1 in an appropriate basis, [u ] acts as −1. Therefore, the CM
modular forms are actually in the space V0 .
Any D-subrepresentation W ⊂ V0 splits as a sum W = W + ⊕ W − of spaces W ±
of half the dimension where δ acts by ±1.
To recapitulate, the local representations π∞ and πp occur in the local Jaquet
Langlands correspondence, and
we have identified π∞ and πp . By the global correspondence if π = π∞ ⊗ πp ⊗ l=p πl is an automorphic cuspidal representation of
P Gl2 (A), then π = π∞
⊗ πp ⊗ l=p πl is an automorphic cuspidal representation
of BA× /A× which appears with multiplicity one in the space of automorphic forms.
Since we have h(−p) such irreducible π ’s and each contributes a 2-dimensional
space to MCM , we get a space of dimension 2h(−p). Taking ±-eigenspaces under
±
±
δ, we conclude that VCM
⊂ V0± with VCM
of dimension h(−p) as claimed.
References
M. Eichler, Lectures on modular correspondences, Bombay, Tata Institute of Fundamental
Research, 1955-56.
[Gr] B. Gross, Arithmetic on elliptic curves with complex multiplication, with an appendix by
B. Mazur, Lecture Notes in Mathematics, 776, Springer, Berlin, 1980. MR81f:10041
[Ma] Magma computational algebra system http://magma.maths.usyd.edu.au/magma/.
[GP] PARI-GP http://www.parigp-home.de/.
[Ko] D. Kohel, Hecke module structure of quaternions, Class field theory—its centenary and
prospect (Tokyo, 1998), 177–195, Adv. Stud. Pure Math., 30, Math. Soc. Japan, Tokyo,
2001. MR2002i:11059
[Pi]
A. Pizer, Theta Series and Modular Forms of Level p2 M , Compositio Mathematica, Vol.
40, Fasc. 2, 1980, p. 177–241. MR81k:10040
[Pi2] A. Pizer, An Algorithm for Computing Modular Forms on Γ0 (N )∗ , Journal of Algebra 64,
1980, 340–390. MR83g:10020
[PRV] A. Pacetti and F. Rodriguez-Villegas, www.ma.utexas.edu/users/villegas/cnt/cnt.html.
[Se]
J.-P., Serre, Quelques applications du théoreème de Chebotarev, Publ. Math. IHES, 54
(1981), 123–201. MR83k:12011
[Vi]
M. F. Vigneras, Arithmetique des algebres de quaternions, Lecture Notes in Mathematics,
800. MR82i:12016
[Ei]
Department of Mathematics, University of Texas at Austin, Texas 78712
E-mail address: apacetti@math.utexas.edu
Department of Mathematics, University of Texas at Austin, Texas 78712
E-mail address: villegas@math.utexas.edu
Department of Mathematics, Harvard University, Cambridge, Massacusetts 02138
E-mail address: gross@math.harvard.edu
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