Algorithm for Computing Shimura’s Decomposition

We recall Shimura’s decomposition (Theorem 3.5.3). Fix positive integer k,

N with k odd and 4 | N. Let χ be an even Dirichlet character of modulus

N. Let N0 = N/2. For M | N0 such that Cond(χ2) | M and a newform

φ∈Snew

k−1(M, χ2) define

here Tp(φ) = λp(φ)φ. Theorem 3.5.3 states that

S_k/⊥₂(N, χ) =M

φ

Sk/2(N, χ, φ) (3.16) where φ runs through all newforms φ ∈ Snew

k−1(M, χ2) with level M | N

0 _and

Cond(χ2)|M. The following lemma is obvious.

Lemma 3.6.1. Each Sk/2(N, χ, φ) is contained in a single Tp2-eigenspace for

every primep_-N.

The following theorem gives our algorithm for computing the Shimura decomposition.

Theorem 3.6.2. Let φ1, . . . , φm be the newforms of weight k −1, character

χ2 _{and level dividing N}0_{. For prime p, and φ one of these newforms, write}

Tp(φ) = λp(φ)φ. Let p1, . . . , pn - N be primes such that the m vectors of eigenvalues(λp1(φ), . . . , λpn(φ)), with φ=φ1, . . . , φm, are pairwise distinct. If

f ∈S_k/⊥₂(N, χ) is an eigenform for Tp2

i for i = 1, . . . , n then f belongs to one

of the summands Sk/2(N, χ, φ).

Proof. Suppose f ∈S_k/⊥₂(N, χ) is an eigenform for T_p2

i for i = 1, . . . , n. Write

T_p2

if =µif. By Shimura’s decomposition, we can write

f =X

φ

fφ

for some uniquefφ∈Sk/2(N, χ, φ); here φ varies over φi, 1≤i≤m. Thus

X φ λpi(φ)fφ=Tp2_if =µi X φ fφ.

As the decomposition is a direct sum, we find that (λpi(φ)−µi)fφ = 0, i= 1, . . . , n.

We will show that at most one fφ is non-zero. This will force f to be in one

therefore thatfφ1 6= 0 and fφ2 6= 0. Then

λpi(φ1) =µi =λpi(φ2), i= 1,2, . . . , n.

This contradicts the assumption that the vectors of eigenvalues are distinct, and completes the proof.

An Alternative Proof of Theorem 3.6.2. This proof is inspired by a similar ar- gument in [2, page 18] (however there is a certain step in that paper that we were unable to follow).

Let _T0 be the subalgebra of the Hecke algebra of S_k/⊥₂(N, χ) generated byTp2 for p6=p_i such that p_-N. Let

V = Span{T f :T ∈_T0}.

We note the following:

(i) We claim thatV is fixed under the action of the Hecke operatorsTp2 for

p _- N. If Tp2 ∈ _T0 then this is clear. If p =p_i, then T_p2

i commutes with

every T ∈ _T0_{. But f is an eigenform for T} p2

i, which proves the claim.

Hence, we can write an eigenbasis g1, . . . , gr for V with respect to the

Hecke operators Tp2 for p_-N.

(ii) Every element of V is an eigenfunction for T_p2

i having the same eigen-

values as f. This again follows from the fact that each T_p2

i commutes

with each T ∈ _T0_{. Thus for each i, the eigenfunctions g}

1, . . . , gr share

the same T_p2

i-eigenvalue.

Let g be one of the gj. Consider Sht(g). This is an eigenfunction for

all the Hecke operators Tp with p - N acting on Sk−1(N0, χ2). By Proposi- tion 2.2.13, there is a unique φi such that Sht(g) and φi share the same Tp-

eigenvalues for all p_-N. If g, g0 ∈V are two elements of the eigenbasis then it follows from (ii) and the hypothesis about the vectors of eigenvalues that Sht(g), Sht(g0) correspond to the same φi. By the properties of the Shimura

lift,g, g0 will have precisely the same Tp2-eigenvalues for all p_-N. Because f

is a linear combination of these eigenbasis elements, it is an eigenform forTp2

Remark. Our first proof is not only simpler but it also gives a good idea of the strategy that we will use to compute the summands in (3.16).

We can reframe Theorem 3.6.2 as follows.

Corollary 3.6.3. Letφ be a newform of weightk−1, levelM dividingN0, and characterχ2_{. Let p}

1, . . . , pn be primes not dividing N satisfying the following: for every newform φ0 6=φ of weightk−1, level dividing N0 and character χ2_,

there is some pi such that λpi(φ

0_)6=λ pi(φ), where Tpi(φ) =λpi(φ)·φ. Then Sk/2(N, χ, φ) = n f ∈S_k/⊥₂(N, χ) : T_p2 i(f) =λpi(φ)f for i= 1, . . . , n o .

Recall that S_k/⊥₂(N, χ) = Sk/2(N, χ) except possibly when k = 3. We have the following refinement of the above corollary which takes care of the case when S_k/⊥₂(N, χ)₍Sk/2(N, χ), that is, S0(N, χ)6= 0.

Corollary 3.6.4. Assuming the notation in the above corollary, the following stronger statement holds:

Sk/2(N, χ, φ) = n f ∈Sk/2(N, χ) : Tp2 i(f) =λpi(φ)f for i= 1, . . . , n o .

Proof. Letf1, . . . fr be the basis of eigenforms for S0(N, χ) as stated in The- orem 3.1.1. Recall that fi = V(ti)hψi where ψi is primitive odd character of

conductor rψi such that 4r

2

ψiti | N and χ =

−4ti

.

ψi. Let q = pi for some

fixed i. We claim that Tq2(f_i) 6= λ_q(φ)f_i for any 1 ≤ i ≤ r. Since φ is

a newform of weight 2 we know by Deligne’s work on Weil conjectures that

|λq(φ)| ≤ 2 √

q. By Lemma 3.1.2, Tq2(f_i) = ψ_i(q)(1 +q)f_i as q _- N. Clearly

|ψi(q)(1 +q)|=|1 +q|>2 √

q. Hence the claim follows. Let g ∈ Sk/2(N, χ) such that Tp2

i(g) = λpi(φ)g for 1 ≤ i ≤ n. We can

write g = g1 +g2 where g1 ∈ S0(N, χ) and g2 ∈ Sk/⊥2(N, χ). Since g1 and

g2 are linearly independent we get that Tp2

i(gj) = λpi(φ)gj for all 1 ≤ i ≤ n

and j = 1,2. Thus by the above corollary g2 ∈ Sk/2(N, χ, φ). We show that

g1 = 0. Let g1 =

Pr

i=1aifi. In particular for the prime q we must have

aiTq2(f_i) = a_iλ_q(φ)f_i. The above claim implies that a_i = 0 for all 1 ≤ i≤ r.

3.7

An Example of Non-Injectivity of Shimura