COMPUTING WEIGHT 2 MODULAR FORMS OF LEVEL p^{2}

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 p^{2}. For p ≡ 3 mod 4 we define a special subspace V_{0} of V which contains the space of modular forms with CM by the ring of integers of ℚ(\sqrt{−p}) and describe the calculation of the corresponding Brandt matrices.

1. 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 T r(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 O_{l}(I) := {x ∈ B / xI ⊂ I}; similarly, its right order is O_{r}(I) := {x ∈ B / Ix ⊂ I}.

(4) For a lattice I and a prime q we let I_{q} := I ⊗ ℤ_{q}

(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 I_{q} = O_{q}a_{q} for some a_{q} ∈ (B ⊗ ℚ_{q})^{×}.

(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 N_{I} : I ⟶ ℤ as x ↦ N(x)/N(I).

(8) Given a lattice I its dual is I^{#} := {b ∈ B / T r(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 O_{l}(I) is maximal, then I is a left O_{l}(I)- ideal.

Proof: See [Vi, page 86]. □

Theorem 1. Let I be a left O-ideal and I^{#} its dual. Then I^{#} is a right O-ideal and I^{ι} := N(I)p\overline{I}^{#} is a left O-ideal contained in I with I/I^{ι} ≃ ℤ/pℤ ⊕ ℤ/pℤ as abelian groups and N(I^{ι}) = N(I)p. If I = O then O/O^{ι} ≃ 𝔽_{p^{2}} as rings.

The first named authors were supported in part by grants from TARP and NSF (DMS-99-70109); they would like to thank the Department of Mathematics at Harvard University, where part of this work was done, for its hospitality.

Proof: If O is an order then, by definition, O^{ι} is its different. Since B has only one ramified prime, P = O^{ι} is the unique maximal 2-sided prime over p. Since all ideals are locally principal, we have that if I_{q} = O_{q}a_{q} then \overline{I_{q}} = \overline{a_{q}}\overline{O_{q}} = \overline{a_{q}}O_{q} for all primes q; also, it is not hard to check that I_{q}^{ι} = O_{q}^{ι}a_{q}. By [Vi, Lemma 4.7, p. 24], the different is a bilateral O-ideal of norm p. It follows that O/O^{ι} ≃ O_{p}/O_{p}^{ι} ≃ 𝔽_{p^{2}} 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 O_{l}(I^{#}) = O_{r}(I) and O_{r}(I^{#}) = O_{l}(I).

Proof: It is clear that if α is in I^{#} and x is in O_{l}(I), then αx ∈ I^{#}, which implies that O_{l}(I) ⊂ O_{r}(I^{#}); using that \overline{I^{#}} = Ī^{#} and replacing I by Ī we get that O_{r}(I) ⊂ O_{i}(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 J_{q} = O_{q}α_{q} then N(J_{q}) = N(α_{q}). Since J is integral O_{q}α_{q} ⊂ O_{q}; its index is up to a unit in ℤ_{q}^{×} the determinant of multiplication by α_{q}, which equals N(α_{q})^{2}. □

Lemma 2. Let I be a left O-ideal and J ⊂ I a sublattice of index p, such that I^{ι} ⊂ J ⊂ I. Then O_{l}(J) ⊂ O with index p.

Proof: By theorem 1 we can find bases such that: I = ⟨v_{1}, v_{2}, v_{3}, v_{4}⟩, J = ⟨pv_{1}, v_{2}, v_{3}, v_{4}⟩ and I^{ι} = ⟨pv_{1}, pv_{2}, v_{3}, v_{4}⟩. Note that for a prime q ≠ p, we have that I_{q}^{ι} = J_{q} = I_{q} so it is clear that O_{l}(J)_{q} = O_{q}.

For the ramified prime, let O_{p} = ⟨1, w_{2}, w_{3}, w_{4}⟩. It is clear that w_{i}pv_{1} ∈ J_{p}, w_{i}v_{3} ∈ J_{p} and w_{i}v_{4} ∈ J_{p} for all 1 ≤ i ≤ 4 (w_{1} := 1). So we just have to check which elements of O_{p} send v_{2} into J_{p}. Let w_{2}v_{2} = a_{1}v_{1} + a_{2}v_{2} + a_{3}v_{3} + a_{4}v_{4} and w_{3}v_{2} = b_{1}v_{1} + b_{2}v_{2} + b_{3}v_{3} + b_{4}v_{4}. Suppose p ∤ a_{1} and p ∤ b_{1}, then let w̃_{2} = a_{1}^{−1}w_{2} − b_{1}^{−1}w_{3}. Clearly w̃_{2} ∈ O_{p}, so another basis for O_{p} is {1, w̃_{2}, w_{3}, w_{4}}, with w̃_{2}v_{2} ∈ J_{p}. Proceeding in a similar way with w_{3} we see that we can pick a basis of O_{p} such that w_{2}J_{p} ⊂ J_{p} and w_{3}J_{p} ⊂ J_{p}, and clearly pw_{4}J_{p} ⊂ J_{p}. Then O_{l}(J) ⊂ O, and has index 1 or p. If O_{l}(J) = O by lemma 2 [I : J] = (N(J)/N(I))^{2} = p which is a contradiction. □

Definition. An order has level p^{2} if it has index p in some maximal order.

We denote by Õ = ℤ + P the unique suborder of level p^{2} in O (see [Pi, Lemma 1.4 p. 181]) and h, h̃ the class numbers of O, Õ respectively.

Proposition 3. Any lattice I with O_{l}(I) = Õ is an Õ-ideal.

Proof: Let I be such a lattice. By proposition 1, for all primes q ≠ p, I_{q} is principal, since Õ_{q} = O_{q}. For the ramified prime, since ℤ_{p} is a PID, there exists a_{p} ∈ I_{p} with (N(a_{p})) = N(I_{p}). Therefore, Õ_{p}a_{p} ⊂ I_{p} ⊂ O_{p}I_{p}. Since O_{p}I_{p} is an ideal for O_{p} of the same norm as I_{p} we have by lemma 1 that O_{p}I_{p} = O_{p}a_{p}. On the other hand, the index of Õ_{p} in O_{p} is p, hence, I_{p} = Õ_{p}a_{p}. □

COMPUTING WEIGTH 2 MODULAR FORMS OF LEVEL p^{2} 3

Proposition 4. Let I be a left Õ-ideal then

(1) If x ∈ I is such that p ∤ N_{I}(x), then (\frac{N_{I}(x)}{p}) is independent of x, where

(\frac{.}{.}) denotes the Kronecker symbol.
(2) (\frac{N_{I}(x)}{p}) only depends on the equivalence class of I.
(3) If Iis principal, then (\frac{N_{I}(x)}{p}) = 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 (\frac{N_{I}(x)}{p}). It is easy to check that χ(\overline{I}) = χ(I) where the bar denotes

conjugation. and χ(I^{−1}) = χ(I).

Corollary 2. Given two orders O_{j} of level p^{2} for j = 1, 2 and left O_{j}-ideals I_{j} for j = 1, 2 such that O_{r}(I_{1}) = O_{2} then χ(I_{1}I_{2}) = χ(I_{1})χ(I_{2}).

Proof: Pick x_{j} ∈ I_{j} for j = 1, 2 with p ∤ N_{I_{j}}(x_{j}) and take x_{1}x_{2} ∈ I_{1}I_{2}; note that N(I_{1}I_{2}) = N(I_{1})N(I_{2}). □

2. Computing left Õ-ideals representatives

Proposition 5. Let p be a prime and B = (a, b) be the quaternion algebra ramified at p and infinity with i^{2} = a and j^{2} = b. Then a Õ order is given by the basis:

• ⟨\frac{1}{2}(1 + j), \frac{1}{2}(pi + k), j, k⟩ with a = −1, b = −p if p ≡ 3 mod 4

• ⟨\frac{1}{3}(1 + j + k), \frac{1}{4}(pi + 2j + k), j, k⟩ with a = −2, b = −p if p ≡ 5 mod 8 • ⟨\frac{1}{2} + \frac{pj}{2} , \frac{i}{2} + \frac{k}{2} , k, \frac{pj}{q} + \frac{sk}{q} ⟩ with a = −p, b = −q if p ≡ 1 mod 8 where

q is a prime such that (\frac{p}{q}) = −1, q ≡ 3 mod 4 and s is an integer with

s^{2} ≡ −p mod q and s ≡ −q mod p

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 = ⟨\frac{1}{2} + \frac{j}{2}, \frac{i}{2} + \frac{k}{2} , k, \frac{j}{q} + \frac{(s+q)k}{pq} ⟩. The conditions on s makes this an order, and is easy to check 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 ∤ N_{I}(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 I_{p} = O_{p}a_{p}, then we saw that I_{p}^{ι} = O_{p}^{ι}a_{p}; since v ∈ I, v = ua_{p} with u ∈ O_{p}. If p|N_{I}(v), then p|N(u) hence u ∈ O_{p}^{ι} 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. Given a left O-ideal I an a p-subideal J we have OJ = I and N(J) = N(I).

Proof: Since J ⊂ I we have J ⊂ OJ ⊂ OI = I. On the other hand, J has index p in I so either OJ = I or OJ = J. But if OJ = J then O = O_{l}(OJ) = O_{l}(J) = Õ, which is a contradiction; hence OJ = I.

Since I^{ι} ⊂ J ⊂ I we have that N(I)|N(J)|N(I^{ι}), which means that either N(J) = N(I) or N(J) = pN(I). To see which is the case, we just need to see what happens at the ramified prime p. Let us say

(2)  I_{p}^{ι} = O_{p}c_{p} ⊂ J_{p} = Õ_{p}b_{p} ⊂ I_{p} = O_{p}a_{p}

We know N(I^{ι}) = pN(I) ⇒ N(c_{p}) = pN(a_{p}). Suppose that N(b_{p}) = N(c_{p}). Then N(b_{p}^{−1}c_{p}) = 1; in particular, b_{p}^{−1}c_{p} ∈ O_{p}^{×}. Because O_{p} is the only maximal order, we have that O_{r}(I_{p}^{ι}) = O_{p} and hence I_{p}^{ι}(b_{p}^{−1}c_{p}) = I_{p}^{ι} . Multiplying (2) by b_{p}^{−1}c_{p} on the right yields O_{p}c_{p} = I_{p}^{ι}b_{p}^{−1}c_{p} ⊂ Õ_{p}c_{p}, which is a contradiction. □

Corollary 3. Let I_{i} for i = 1, 2 be left O-ideals and J_{i} ⊂ I_{i} for i = 1, 2 corresponding p-subideals. If I_{1} and I_{2} are non-equivalent then so are J_{1} and J_{2}.

Proof: If J_{1} = J_{2}α for some α ∈ Õ then OJ_{1} = OJ_{2}α. By the proposition I_{1} = I_{2}α, 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 O_{q} = Õ_{q} then O_{q}J_{q} = J_{q} = I_{q}^{j}, so no primes other than p appear in the index. As for the ramified prime, let us say that J_{p}α^{−1} = Õ_{p}a_{p}, and I_{p}^{j} = O_{p}c_{p}. Since O_{p}J_{p} = I_{p}, we have

that O_{p}a_{p} = O_{p}c_{p} so I_{p}^{j} = O_{p}a_{p}; therefore |I_{p}^{j}/(J_{p}α^{−1})| = |O_{p}a_{p}/Õ_{p}a_{p}| = p.

Since Jα^{−1} ⊂ I^{j} with index p, to see that I^{ι} ⊂ Jα^{−1} it is enough to check locally p. Let J_{p} = Õ_{p}b_{p}, I_{p} = O_{p}a_{p}. Without loss of generality we may assume that b_{p}α^{−1} = a_{p}. By the proof of Theorem 1, we see that I_{p}^{ι} = O_{p}^{ι}a_{p}. Also O_{p}^{ι} ⊂ Õ_{p}, therefore O_{p}^{ι}a_{p} ⊂ Õ_{p}a_{p} = Õ_{p}b_{p}α^{−1} = J_{p}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 ∈ O_{r}(I)^{×}.

Corollary 4. Given a left O-ideal I the number of non-equivalent p-subideals J ⊂ I is (p + 1)|O_{r}(J)^{×}|/|O_{r}(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.

Proof: This is just proposition 5.12 of [Pi2]. □

For j = 1, ⋯, h we let O_{j} = O_{r}(I^{j}) and let Õ_{j} be its suborder of index p.

Corollary 5. We have

(3)  h̃ = (p + 1) ∑_{j=1}^{h} \frac{|Õ_{j}^{×}|}{|O_{j}^{×}|}.

And if p > 3 then h̃ = (p^{2} − 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 α_{q} of α are as follows: for q ≠ p we set α_{q} = 1 and for q = p we want α_{p} with zero trace such that

(\frac{a}{p}) = −1,

where a = N(α_{p})/p^{n} and n = v_{p}(N(α_{p})) with v_{p} the valuation at p. We then have that χ(α_{p}J) = −χ(J). We denote by δ the involution

(4)  δ : J ↦ αJ

Note that if J and J′ are equivalent then so are δJ and δJ′.

2.1. Construction of α_{p}. 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 α_{p} to be one of i or j.

(2) If p ≡ 3 mod 4 then −1 is a non-square and we look for α_{p} with norm −p. If α = x_{1}i + x_{2}j + x_{3}k, with i^{2} = −1, j^{2} = −p = k^{2}, then N(α_{p}) = x_{1}^{2} + p(x_{2}^{2} + x_{3}^{3}). We can take x_{1} = 0, and look for a solution to the equation x_{2}^{2} + x_{3}^{2} = −1 in ℤ_{p}, which is achieved by finding a solution to x_{2}^{2} + x_{3}^{2} ≡ −1 mod p, and then lifting the solution using Hensel’s lemma.

2.2. Action of α_{p} on I. We will follow [Ei, Theorem 7, p. 34]. First we need to compute an r such that α_{p}J ⊃ Jp^{r}.

Lemma 4. Let n = v_{p}(N(α_{p})) be the p-valuation of the norm of α_{p}. Then α_{p}J ⊃ Jp^{⌈n/2⌉+1}.

Proof: In order that α_{p}J ⊃ Jp^{s} we must have α_{p}^{−1}p^{s} ∈ Õ. Note that if β ∈ O then pβ ∈ Õ; hence it is enough to check when α_{p}^{−1}p^{s−1} ∈ O or, equivalently, when v_{p}(N(α_{p}^{−1}p^{s−1})) ≥ 0. It is now straightforward to verify that is enough to take

s ≥ ⌈\frac{v_{p}(N(α_{p}))}{2}⌉ + 1. □

Set r = ⌈n/2⌉ + 1. Starting with a global basis for Jp^{r} we start adjoining elements until we find a generating set for αJ. Say J = ⟨u_{1}, u_{2}, u_{3}, u_{4}⟩ so that α_{p}J = ⟨α_{p}u_{1}, α_{p}u_{2}, α_{p}u_{3}, α_{p}u_{4}⟩. It is not hard to see that we have

α_{p}u_{j} ≡ v_{j} mod p^{r}J_{p}, j = 1, …, 4,

with p^{s}v_{j} ∈ J for some s. We set J′ = ⟨Jp^{r}, v_{1}, v_{2}, v_{3}, v_{4}⟩. Clearly α_{p}J_{p} = J′_{p} and for a prime q ≠ p we have v_{i} ∈ J_{q} for i = 1, …, 4 and hence J′_{q} = J_{q}.

Having computed representatives for some maximal order (respectively, of level p^{2}) 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 α_{p} to use.

Lemma 5. Given a left Õ-ideal J let α_{p} be as constructed above. In order to compute α_{p}J it is enough to know α_{p} to order O(p^{r+1}), where r = ⌈v_{p}(N(α_{p}))/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, {piu_{t}, pju_{t}, pku_{t}} ⊂ I for 1 ≤ t ≤ 4; hence, p^{r+1}α_{p}u_{t} ∈ p^{r}I and the denominator of the x_{j} is at most r + 1. □

Note that with our choice of α_{p} 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 α_{p}.

2.3. 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 O_{p}/O_{p}^{ι} is isomorphic to 𝔽_{p^{2}} and Õ_{p}/O_{p}^{ι} to 𝔽_{p}. Let S := (O_{p}/ O_{p}^{ι})^{×}, a cyclic group of order p^{2} − 1. Given a Õ-ideal J and u ∈ S we define uJ, with some notation abuse, by regarding O_{p} 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 (Õ_{p})^{×}/(O_{p}^{ι})^{×}. It is also easy to check that S acts on the set of p-subideals making it a principal homogeneous space for G := (O_{p}/O_{p}^{ι})^{×}/(Õ_{p}/O_{p}^{ι})^{×}, a cyclic group of order p + 1.

Let u be a generator of G and let J be some p-subideal of I. Then {u^{i}J}_{i=0}^{p} are all the p-subideals of I. By proposition 5.6 of [Pi] we know that if J_{p} = Õ_{p}α_{p} then χ(J) is the quadratic symbol of N(α_{p})/N(J) modulo p. Since the norm map from 𝔽_{p^{2}} to 𝔽_{p} is surjective we must have that N(u) is a non-square modulo p and hence χ(u^{i}J) = (−1)^{i}χ(J).

We form a set of inequivalent p-subideals J = {J, uJ, …, u^{r−1}J} where r is the smallest positive integer such that u^{r}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 \overline{u} is its inverse since u\overline{u} = N(u) and N(u) ∈ Õ_{1}.

3. 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, however, that the paper does not treat the case of level p^{2} though the routines in Magma do).

We pick a maximal order O and we calculate representatives {I^{1}, …, I^{h(O)}} of left O-ideals classes (using Pizer’s algorithm for the level p case) and 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 J_{0} with χ(J_{0}) = 1 and consider

(5)  J_{0}, J_{2}, ⋯, J_{r−2}, J_{1}, J_{3}, ⋯, J_{r−1}

where J_{1} = δJ_{0} and J_{i+2j} = u^{2j}J_{i} for i = 0, 1, r is the number of non-equivalent p-subideals of I^{k} and δ is the involution defined in (4). Note that by construction χ(J_{i}) = (−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 follows

(6)  J_{0}^{1}, J_{2}^{1}, ⋯, J_{r_{1}−2}^{1}, J_{0}^{2}, J_{2}^{2}, ⋯, J_{r_{2}−2}^{2}, ⋯ J_{0}^{h}, J_{2}^{h}, ⋯, J_{r_{h}−2}^{h},

followed by those with χ = −1

(7)  J_{1}^{1}, J_{3}^{1}, ⋯, J_{r_{1}−1}^{1}, J_{1}^{2}, J_{3}^{2}, ⋯, J_{r_{2}−1}^{2}, ⋯ J_{1}^{h}, J_{3}^{h}, ⋯, J_{r_{h}−1}^{h},

where h = h(O) and J_{j}^{i} are the representatives for the p-subideals of I^{i} as described in (5).

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

B(q) = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ,

where each A, B, C, D has size h(Õ)/2 × h(Õ)/2. Then the following holds.

(1) If (\frac{q}{p}) = 1 then B = C = 0, and A = D;
(2) If (\frac{q}{p}) = −1 then A = D = 0, and B = C.

Proof: This is just a specific 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 (\frac{q}{p}) = 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 to restrict our attention to the matrix A. We let A_{i,j} be the r_{i}/2 × r_{j}/2 submatrix of A corresponding to the columns J_{l}^{j} and the rows J_{m}^{i} with l = 0, 2, ⋯, r_{j} and m = 0, 2, ⋯, r_{i}.

We index the rows and columns of A_{i,j} by indices l, m modulo r_{i}/2 and r_{j}/2 respectively.

Proposition 11. The matrix A_{i,j} has the following properties:

Let r = gcd(r_{i}/2, r_{j}/2). Then there exists coefficients c(k) indexed by k mod r such that the l, m entry of A_{i,j} equals c(m − l).

In practice this fact means, in particular, that the successive rows of A_{i,j} are obtained from the first by a shift of 1 step to the right.

Lemma 6. For v ∈ G we have vÕ_{p} = Õ_{p}v Proof: The order vÕ_{p}v^{−1} is a suborder of O_{p} of index p hence vÕ_{p}v^{−1} = Õ_{p} □

Proof of Prop: The entry [l, m] of the matrix A_{i,j} corresponds to the ideal (J_{l}^{j})^{−1}J_{m}^{i}. The p-subideal (J_{1}^{j})_{p} = Õ_{p}α_{p} for some element α_{p} ∈ O_{p} and since we assume that p does not divide the norm of the ideal class representatives, α_{p} determines an element u^{a} ∈ G. Hence (J_{1}^{j})_{p} = u^{a}Õ_{p} and similarly (J_{1}^{i})_{p} = u^{b}Õ_{p} for some 0 ≤ a, b < p + 1. Therefore, (J_{l}^{j})_{p} = u^{a+2l}Õ_{p} and (J_{m}^{i})_{p} = u^{b+2m}Õ_{p}. It follows that the p-subideal ((J_{l}^{j})^{−1}J_{m}^{i})_{p} equals u^{b−a+2m−2l}Õ_{p}, by lemma 6. We have then that (J_{l}^{j})^{−1}J_{m}^{i} = u^{2(m−l)}((J_{1}^{j})^{−1}J_{1}^{i}). Since, by definition, u^{r_{i}} sends J_{1}^{i} to an equivalent p-subideal and analogously for u^{r_{j}} and J_{1}^{j} the [l, m] entry of A_{i,j} depends only on the residue of m − l modulo r. □

4. The subspace V_{0}

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

γf(J) := f(γ^{−1}J), γ ∈ D.

We consider the subspace V_{0} of V of functions f_{0} satisfying

f_{0}(u^{2}J) = −f_{0}(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 V_{0} 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 V_{0} into two subspaces V_{0}^{±} where δ acts as ±1. It is easy to verify that any generator u of G takes V_{0}^{+} isomorphically into V_{0}^{−} and viceversa.

Theorem 2. The subspaces V_{0}^{±} are stable under the action of all Brandt matrices B(q).

Proof: We first prove that V_{0} is stable under the Brandt matrices. Let v_{i} = (1, −1, …, −1) of length r_{i}/2 and similarly v_{j} = (1, −1, …, −1) of length r_{j}/2. We consider the case where (\frac{q}{p}) = 1, the other case is completely analogous. Using the choice of basis above it is enough to prove that A_{i,j}v_{j} = λv_{i} for some λ ∈ ℤ and this is clear from the form of the matrix A_{i,j} given by proposition (11). It is also easy to see that λ = 0 if r_{i}/2 is odd.

Since δ commutes with B(q) (see proposition 10) the subspaces V_{0}^{±} are also stable under the action of the Brandt matrices. □

We let B_{0}(q) be the matrix B(q) restricted to V_{0}^{+}. One of the main motivations to consider this subspace is that it contains, for p > 3, a copy of the space of modular forms of weight 2 and level p^{2} with CM by the ring of integers of ℚ(\sqrt{−p}). The proof of this fact is given by Benedict Gross in an appendix, and uses the local and global Jacquet-Langlands correspondence. Concretely, it is the subspace V_{CM}^{+} ⊂ V_{0}^{+} characterized by the vanishing of B_{0}(q) for all primes q with (\frac{q}{p}) = −1;

clearly, V_{CM}^{+} is stable under the Hecke algebra. For p > 3, V_{CM}^{+} has dimension h(−p), the class number of K = ℚ(\sqrt{−p}), and it can be identified with the tangent space of an abelian variety B(p)/ℚ obtained as the restriction of scalars of an certain elliptic curve A(p) with CM by the ring of integers of K (see [Gr]). For p = 3 both V_{CM}^{+} and V_{0}^{+} are zero.

We now obtain a formula for the dimension of V_{0}^{+}.

Proposition 12. for a prime p > 3 and congruent to 3 modulo 4 the dimension of V_{0}^{+} is given by:

dim(V_{0}^{+}) = \frac{1}{12} (p + 5) + \frac{1}{3} (1 − (\frac{−3}{p})) − \frac{1}{2} (1 − (\frac{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 4 to compute the number of nonequivalent p-subideals of a given ideal I = I_{j} 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 maximal order is determined by the quadratic symbols (\frac{−3}{p}) and (\frac{−4}{p}) respectively. It is

known that ℤ[i] and ℤ[(1 + \sqrt{−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 4 r_{j} = (p + 1)/w′ hence if w′ = 3 then r_{j} is always even and if w′ = 2 then r_{j} is even if and only if p ≡ 7 mod 8. The formula now follows. □

5. Tables

The following calculations were made with PARI-GP [GP] (check the website [PRV] for the corresponding routines).

p dim V_{0}^{+} dim V_{CM}^{+}
7  1  1
11  1  1
19  1  1
23  3  3
31  3  3
43  3  1
47  5  5
59  5  3
67  5  1
71  7  7
79  7  5
83  7  3
103  9  5

Example 1. Let p = 11. In this case the class number for maximal orders is 2, hence the matrix A_{i,j} will have four blocks. The first Brandt matrices are:

B(2) = [ \begin{matrix}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 \end{matrix} ]

B(3) = [ \begin{matrix}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 \end{matrix} ]

Example 2. Let p = 47. In this case V_{0}^{+} = V_{CM}^{+} is of dimension 5. We give some examples of the matrices B_{0}(q) for q with (\frac{q}{p}) = 1; since V_{CM}^{+} = V_{0}^{+} we know that B_{0}(q) vanishes for (\frac{q}{p}) = −1.

B_{0}(2) = ( \begin{matrix}1 & 2 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 1 & −2 & 0 \\ 0 & 3 & 0 & 0 & 0 \end{matrix} )

B_{0}(3) = ( \begin{matrix}0 & 0 & 2 & 2 & 0 \\ 0 & 1 & 1 & −2 & 0 \\ 1 & 1 & −2 & 0 & 0 \\ 1 & −2 & 0 & 0 & 1 \\ 0 & 0 & 0 & 3 & 1 \end{matrix} )

Here is a table with the abelian varieties B(p) corresponding to V_{CM}^{+} for small p labeled as in William Stein’s list.

p Label dim
7 49A  1
11 121A 1
19 361A 1
23 529F 3
31 961G 3
43 1849A 1
47 2209F 5
59 3481C 3
67 4489A 1
71 5041F 7
79  5
83  3

Here is the corresponding table for subspaces of V_{0}^{+} stable by the Hecke algebra in the complement of V_{CM}^{+}.

p Label dim
43 1849E 2
59 3481A 2
67 4489E 4
79 6241A 1
79 6241B 1
83  4

The case of p = 79 is interesting. Is the only case with p ≤ 400 where the complement of V_{CM}^{+} in V_{0}^{+} contains one dimensional Hecke stable subspaces. By calculations of Cremona the two subspaces correspond to the elliptic curve of equation

y^{2} + xy = x^{3} − x^{2} − 64x − 179

and its quadratic twist by ℚ(\sqrt{−79}).

Appendix A. by B. Gross

We will prove that the space of CM modular forms of weight 2 and level p^{2} injects into the space V_{0}. Let Õ_{p}^{×} (respectively O_{p}^{×}) be the group of invertible elements of Õ_{p} (respectively of O_{p}). Then the quotient O_{p}^{×}/Õ_{p}^{×} is isomorphic to the group G hence O_{p}^{×} contains a unique subgroup K_{p} such that O_{p}^{×}/K_{p} is cyclic of order 4. Note that the group Õ_{p}^{×} is equal to ℤ_{p}^{×}(1 + P) where P is the unique integral order of norm p in B_{p}. Define:

(8)  M := {f : K_{p} × ∏_{l≠p} O_{l}^{×}\B̂^{×}/B^{×} → ℂ}.

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 u^{4} acts trivially with u a generator of G.

Recall that we have defined

(9)  V_{CM} := \{f ∈ M : f|T_{l} = 0 for all (\frac{l}{p}) = −1\}

where T_{l} is the l-th Hecke operator. Recall the involution δ defined in (4); it acts on M commuting with the T_{l} and hence also gives an involution of M_{CM}. We may therefore decompose M and M_{CM} into their eigenspaces M^{±}, M_{CM}^{±} with respect to δ.

Let [u] be a generator of O_{p}^{×}/K_{p}, then we can identify the space V_{0} with the functions f ∈ M^{+} such that f|[u^{2}] = −f.

Theorem 3. M_{CM}^{±} ⊂ V_{0}^{±} and has dimension h(−p) if p ≥ 7.

Proof: We know that the space of cusp forms F of weight 2 for Γ_{0}(p^{2}) with complex multiplication by ℚ(\sqrt{−p}) has dimension h(−p) (see [Gr]). By a theorem of Serre (see Thm 17, [Se]) this space is characterized by the condition that F |T_{l} = 0 for all primes l with (\frac{l}{p}) = −1. This gives h(−p) automorphic representations:

π = π_{∞} ⊗ π_{p} ⊗ ⊗_{l≠p}π_{l}

of P Gl_{2}(𝔸) with:

• π_{∞} a discrete series of weight 2 for P Gl_{2}(ℝ)

• π_{p} an irreducible representation of P Gl_{2}(ℚ_{p}) of conductor p^{2}.

• π_{l} an irreducible unramified representation of P Gl_{2}(ℚ_{l}) with Hecke eigenvalues a_{l} = 0 if (\frac{l}{p}) = −1.

The local Jaquet-Langlands correspondence gives a bijection between irreducible, square-integrable, representations π_{v} of P Gl_{2}(ℚ_{v}) and finite dimensional, irreducible representations π′_{v} of B_{v}^{×}/ℚ_{v}^{×}, where B_{v} is the quaternion division algebra over ℚ_{v}. The local correspondence is characterized by the identity T r(t|π_{v}) + T r(t|π′_{v}) = 0 for all regular elliptic conjugacy classes t.

If π_{∞} is the weight 2 discrete series of P Gl_{2}(ℝ) then π′_{∞} is the trivial representation of ℍ^{×}/ℝ^{×} = SO_{3}.

If π_{p} has conductor p^{n+1}, then π′_{p} is trivial on the subgroup 1 + π_{p}^{n}O_{p} of B_{p}^{×}. We want to apply this to the local component π_{p} of our CM forms. First,

we must check that π_{p} is square-integrable. In fact we will show it is a cuspidal representation, by checking that its Langlands parameter σ(π_{p}) : W (ℚ_{p}) → Gl_{2}(ℂ) gives an irreducible 2-dimension representation of the local Weil group.

By construction of the CM forms, we have

(10)  σ(π_{p}) = Ind_{W (k_{p})}^{W (ℚ_{p})} χ_{p}

where k_{p} = ℚ_{p}(η_{p}), with η_{p} = \sqrt{−p}, and χ_{p} is the local component of our Hecke characters of conductor (η_{p}). Since (\frac{−1}{p}) = −1, we have χ_{p}(−1) = −1. Hence if τ is the non-trivial automorphism of k_{p} over ℚ_{p}:

(11)  χ_{p}^{τ}(η_{p}) = χ_{p}(η_{p}^{τ}) = χ_{p}(−η_{p}) = −χ_{p}(η_{p})

and χ_{p}^{τ} ≠ χ_{p}. This shows that σ(π_{p}) is irreducible by Mackey’s criterion for induced representations.

We will now determine the corresponding irreducible representations π′_{p} of D = B_{p}^{×}/(1 + η_{p}O_{p})ℚ_{p}^{×}. D is a dihedral group of order 2(p + 1), with normal subgroup G = O_{p}^{×}/(1 + η_{p}O_{p})ℤ_{p}^{×} ≃ 𝔽_{p^{2}}^{×}/𝔽_{p}^{×}. Hence any irreducible representation of D has dimension 1 or 2.

Since π_{p} satisfies π_{p} ⊗ ϵ_{p}(det) ≃ π_{p}, where ϵ_{p} is the quadratic character of ℚ_{p} associated to the extension k_{p} = ℚ_{p}(\sqrt{−p}), the same holds for π′_{p}: π′_{p} ⊗ϵ_{p}(G) ≃ π′_{p}.

This is false if π′_{p} is 1-dimensional, so we must have:

(12)  π′_{p} = Ind_{G}^{D}(γ) = Ind_{G}^{D}(γ^{−1})

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 ϵ(G) on 𝔽_{p^{2}}^{×} 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 K_{p} of index 4 in O_{p}^{×} acts trivially on π′_{p}. Let [u] be a generator of G. Since the action of G on Ind_{G}^{D}(γ)

is given by ( \begin{matrix}γ & 0 \\ 0 & γ^{−1} \end{matrix} ) in an appropriate basis [u^{2}] acts as −1. Therefore, the

CM modular forms are actually in the space V_{0}.

Any D-subrepresentation W ⊂ V_{0} splits as a sum W = W^{+} ⊕ W^{−} of spaces W^{±} of half the dimension where δ acts by ±1.

To recapitulate, the local representation π_{∞} and π_{p} occur in the local Jaquet-Langlands correspondence, and we have identified π′_{∞} and π′_{p}. By the global correspondence if π = π_{∞} ⊗ π_{p} ⊗ ⊗_{l≠p}π_{l} is an automorphic cuspidal representation of P Gl_{2}(𝔸) then π′ = π′_{∞} ⊗ π′_{p} ⊗ ⊗_{l≠p}π_{l} is an automorphic cuspidal representation of B_{𝔸}^{×}/𝔸^{×} which appears with multiplicity one in the space of automorphic forms. Since we have h(−p) such irreducible π′, and each contributes a 2-dimensional space to M_{CM}, we get a space of dimension 2h(−p). Taking ±-eigenspaces under δ we conclude that V_{CM}^{±} ⊂ V_{0}^{±} with V_{CM}^{±} of dimension h(−p) as claimed. □

References

[Ei] 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.

[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.

[Pi] A. Pizer, Theta Series and Modular Forms of Level p^{2}M Compositio Mathematica, Vol. 40 Fasc. 2, 1980, pag. 177–241.

[Pi2] A. Pizer, An Algorithm for Computing Modular Forms on Γ_{0}(N)^{∗} Journal of Algebra 64, 1980, 340–390.

[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.

[Vi] M. F. Vigneras, Arithmetique des algebres de quaternions Lecture Notes in mathematics 800.

University of Texas at Austin, TX 78712, USA

   E-mail address: apacetti@math.utexas.edu villegas@math.utexas.edu