NEWFORMS OF HALF-INTEGRAL WEIGHT

THOMAS R. SHEMANSKE

May 14, 1996

Abstract. Using a characterization of half-integral weight newforms which originated with Serre and Stark, we consider the structure of the space of newforms and the multiplicity with which half-integral weight newforms (associated to a given integral weight newform) occur. On the full space (not the Kohnen subspace) of cusp forms of squarefree level, we show that half-integral weight newforms associated to a given integral weight newform occur with multiplicity one or two, and of those occurring with multiplicity two, one newform is associated with the Kohnen subspace. Moreover, we give an explicit decomposition of the space of half-integral weight newforms which demonstrates that newforms of squarefree level 4N can correspond to integral weight newforms of any level dividing 2N.

For nonsquarefree levels, we give a decomposition of Ueda’s spaces of newforms in terms of the spaces of newforms defined à la Serre and Stark. In particular, the decomposition shows how newforms defined by Ueda of level 4N can correspond to integral weight newforms of any level dividing 2N. More importantly, this decomposition explicitly connects the notions of half-integral weight newforms as defined by Serre and Stark with those defined by Kohnen and Ueda.

1. Background

Two very different definitions of a newform of half-integral weight are present and continued to be developed in the literature. The first definition originated with Serre and Stark for forms of weight 1/2 [5], and is analogous to the definition of newform for integral weight forms, which uses forms of lower level and shifts of such forms (f(τ) ↦ f | B_{d} = f(dτ)) to characterize the notion of oldforms. The second definition originated with Kohnen for half-integral weight forms of squarefree level [1], who used forms of lower level and their image under the U_{m^{2}} operator (f = ∑ a_{n}e^{2πinτ} ↦ f | U_{m^{2}} = ∑ a_{nm^{2}}e^{2πinτ}) to define the notion of oldforms. The choice of the U_{m^{2}} operator over the shift operator B_{d} seems a propitious one, since the U operator commutes with the action of the Shimura lift, while the shift operator B does not. More to the point, Kohnen was able to develop a newform theory on a distinguished subspace of the full space of cusp forms (now referred to as the Kohnen subspace), and obtained a multiplicity-one result (with respect to Hecke eigenvalues) for half-integral weight newforms in this subspace. Even nicer, the multiplicity-one result was established by showing that there is a one-to-one correspondence between newforms of level 4N in the subspace and the newforms of integral weight of level N.

1991 Mathematics Subject Classification. 11F37, 11F11.

Several others provided further generalizations of the newform theory using both definitions of half-integral weight newforms ([1], [3], [4], [6], [7],[12]). By far the most dramatic development of this theory was obtained recently by Ueda [12]. In this paper, he obtains for arbitrary level a multiplicity-one theorem on a certain subspace of the Kohnen subspace. The generalization from squarefree level to arbitrary level is not at all obvious, and requires very different accommodations for primes dividing the level with multiplicity greater than one. One curious phenomenon which Ueda mentions is that while he was successful in obtaining a multiplicity-one result, newforms of level 4N sometimes correspond (under the Shimura correspondence) to newforms of level a proper divisor of 2N. Ueda continues to refine his subspace in an attempt to remedy this issue.

In joint work with Walling [7], in which we take the alternate definition of newform and work in the full space of cusp forms and not the Kohnen subspace, we determine in two special cases precisely how it is possible for newforms of half-integral weight and squarefree level to correspond to integral weight newforms of any level dividing the expected level given by the Shimura lift. We conjectured that those special cases should generalize.

This paper has two main thrusts. The first is to continue the investigation initiated in [7]. While the desire to obtain multiplicity-one results has directed a great deal of research towards a careful development of the notion of newforms within the Kohnen subspace, a very natural question is to explore whether the multiplicities with which newforms occur on the full space can be determined. In this paper we answer this question for squarefree level. The rather pleasing answer (see Theorem 2.3) is that newforms of half-integral weight correspond to integral weight newforms with multiplicity one or two, and those that occur with multiplicity two can be identified with Kohnen subspaces in such a way that if the full space of newforms (of level 4N) is is decomposed into its Kohnen component and a non-Kohnen component, the newforms in each (corresponding to integral weight newforms of level N or 2N) occur with multiplicity one.

In the second part of the paper we consider nonsquarefree levels and demonstrate a relation between Ueda’s space of newforms and our own (see Theorem 3.1 and Corollary 3.7), by decomposing Ueda’s space of newforms into subspaces defined using the alternate characterization of newform. In particular, these results make clear why in certain circumstances, Ueda’s newforms often correspond to integral weight newforms whose level is lower than expected. It also brings closer together what were apparently two very different notions of a half-integral weight newform. These results are obtained for both the full space of cusp forms and the Kohnen subspace.

2. Decomposition Theorems — squarefree level

Let N be an odd positive integer and ψ an even Dirichlet character defined modulo 4N. For an odd integer k ≥ 3, we denote by S_{k/2}(4N, ψ) the space of cusp forms of weight k/2 and character ψ for Γ_{0}(4N). When k = 3, we take as usual S_{3/2}(4N, ψ) to be the orthogonal complement in the full space of cusp forms of the space generated by theta series attached to quadratic forms in one variable with spherical harmonic (see [12]). For a positive integer M, an integer ℓ ≥ 2, and a Dirichlet character φ defined modulo M, let 𝔖_{ℓ}(M, φ) denote the space of cusp forms of weight ℓ and character φ for Γ_{0}(M). The rest of the notation and the definition of standard operators like the Hecke operators can be found in [12] or [8].

2.1. An Internal Decomposition. We recall (see [6]) that the space generated by the oldforms, denoted S_{k/2}^{−}(4N, ψ), is defined as

(2.1) S_{k/2}^{−}(4N, ψ) = ∑_{\begin{matrix}q|N \\ cond(ψ)|4N/q \end{matrix}} S_{k/2}(4N/q, ψ) + ∑_{\begin{matrix}q|N \\ cond(ψ(\frac{q}{∗}))|4N/q \end{matrix}} S_{k/2}(4N/q, ψ (\frac{q}{∗})) | B_{q}

where B_{q} is the shift operator taking f(τ) to (f | B_{q})(τ) = f(qτ). The subspace generated by the newforms, denoted S_{k/2}^{+}(4N, ψ) is the orthogonal complement of S_{k/2}^{−}(4N, ψ) in the full space S_{k/2}(4N, ψ). Our first theorem provides an internal decomposition of the space S_{k/2}(4N, ψ) into sums of spaces of newforms when the level N is odd and squarefree.

It turns out to be easier to handle first the case where the character ψ is trivial and to generalize to nontrivial characters through another means.

Theorem 2.1. Let N be an odd squarefree integer, and ψ the trivial character modulo 4N. Then

S_{k/2}(4N, 1) = S_{k/2}(4N, ψ) = ⨁_{d|N} S_{k/2}^{+}(4d).

Remark 2.2. It is completely straightforward from the definitions of the spaces of oldforms and newforms to show that S_{k/2}(4N, 1) = ∑_{d|N} S_{k/2}^{+}(4d). Moreover, if these were spaces of integral weight newforms, such a sum would obviously be direct (in fact orthogonal) since newforms of distinct levels have infinitely many distinct eigenvalues under the Hecke operators. Unfortunately for half-integral weight forms, this is very far from true. In fact to describe this phenomenon more clearly is a major focus of this paper. The special cases where N is a prime or the product of two distinct primes were established in [7].

Proof. For notational ease within this proof, we suppress the notation for the weight and character, and write S(4d) for S_{k/2}(4d, 1) and S^{±}(4d) for S_{k/2}^{±}(4d, 1). Write N = p_{1}p_{2} ⋯ p_{r}, the product of distinct odd primes. We always have

S(4N) = S^{+}(4N) ⊕ S^{−}(4N),

and since ψ = 1,

S^{−}(4N) = ∑_{i=1}^{r} S(4p_{1} ⋯ p_{r}/p_{i}).

Thus we need to show that

S^{−}(4N) = ⨁_{\begin{matrix}d|N \\ d<N \end{matrix}} S_{k/2}^{+}(4d).

As we remarked above, a trivial induction shows that

S^{−}(4N) = ∑_{\begin{matrix}d|N \\ d<N \end{matrix}} S_{k/2}^{+}(4d).

We rewrite this sum as

∑_{k=0}^{r−1} ∑_{\begin{matrix}0≤ν_{i}≤1 \\ ∑ ν_{i}=k \end{matrix}} S^{+}(4p_{1}^{ν_{1}}p_{2}^{ν_{2}} ⋯ p_{r}^{ν_{r}}) (2.2)

We first show that for fixed k, the inner sum is direct. There are two cases. Case 1. For k = 0, there is only one summand, so the result is clear. When k = 1,

r ≥ 2 and the sum becomes ∑_{i=1}^{r} S^{+}(4p_{i}). Suppose that ∑ f_{i} = 0 with f_{i} ∈ S^{+}(4p_{i}). Then we may assume there exists an index i so that f_{i} is not zero. Then

f_{i} = − ∑_{j≠i} f_{j} ∈ S^{+}(4p_{i}) ∩ S(4p_{1} ⋯ p_{r}/p_{i}) ⊂ S(4) ⊂ S^{−}(4p_{i}),

which implies f_{i} = 0, a contradiction.

Case 2. In the remaining case we have 2 ≤ k < r. This time we employ the trace operator (adjoint to the inclusion) to achieve our goal. For properties of the trace operator, see [6] or [5].

Suppose that ∑_{∑ ν_{i}=k} S^{+}(4p_{1}^{ν_{1}} ⋯ p_{r}^{ν_{r}}) is not direct. Then there exists a nontrivial dependence relation of the form ∑ f_{j_{1}···j_{k}} = 0 with f_{j_{1}···j_{k}} ∈ S^{+}(4p_{j_{1}} ⋯ p_{j_{k}}). Fix an index i_{1} ⋯ i_{k} for which f_{i_{1}···i_{k}} ≠ 0, and put Tr = Tr_{4p_{i_{1}}···p_{i_{k}}}^{4N}. We shall show that ∑ f_{j_{1}···j_{k}} | Tr = f_{i_{1}···i_{k}} = 0, providing our contradiction.

Certainly

f_{i_{1}···i_{k}} | Tr = f_{i_{1}···i_{k}} | Tr_{4p_{i_{1}}···p_{i_{k}}}^{4N} = f_{i_{1}···i_{k}}

since f_{i_{1}···i_{k}} has level 4p_{i_{1}} ⋯ p_{i_{k}}. Now consider f_{j_{1}···j_{k}} ≠ f_{i_{1}···i_{k}}. Let D = gcd(p_{i_{1}} ⋯ p_{i_{k}}, p_{j_{1}} ⋯ p_{j_{k}}), and write p_{i_{1}} ⋯ p_{i_{k}} = DI and p_{j_{1}} ⋯ p_{j_{k}} = DJ. Note that I and J are both greater than 1, since {i_{1}, …, i_{k}} ≠ {j_{1}, …, j_{k}}. Finally, write N = p_{1} ⋯ p_{r} = DIJK for some integer K.

Then

f_{j_{1}···j_{k}} | Tr_{4p_{i_{1}}···p_{i_{k}}}^{4N} = f_{j_{1}···j_{k}} | Tr_{4DI}^{4DIJK}
= f_{j_{1}···j_{k}} | Tr_{4D}^{4DJK} since gcd(I, JK) = 1

and letting p be any prime dividing J (recall J > 1)

= f_{j_{1}···j_{k}} | Tr_{4DJ}^{4DJK} | Tr_{4DJ/p}^{4DJ} | Tr_{4D}^{4DJ/p}
= f_{j_{1}···j_{k}} | Tr_{4DJ/p}^{4DJ} | Tr_{4D}^{4DJ/p} since f_{j_{1}···j_{k}} has level 4DJ
= 0 | Tr_{4D}^{4DJ/p} = 0 by Theorem 5.2 of [6].

So we now have that

S^{−}(4N) = ∑_{k=0}^{r−1} ⨁_{\begin{matrix}0≤ν_{i}≤1 \\ ∑ ν_{i}=k \end{matrix}} S^{+}(4p_{1}^{ν_{1}}p_{2}^{ν_{2}} ⋯ p_{r}^{ν_{r}}) (2.3)

We now show that the outer sum is direct. Suppose not. Then there exists a nontrivial dependence relation

f_{0} + f_{1} + ⋯ + f_{r−1} = 0 where f_{k} ∈ ⨁_{\begin{matrix}0≤ν_{i}≤1 \\ ∑ ν_{i}=k \end{matrix}} S^{+}(4p_{1}^{ν_{1}}p_{2}^{ν_{2}} ⋯ p_{r}^{ν_{r}})

Let ν^{∗} be the largest index so that f_{ν^{∗}} ≠ 0, and write

f_{ν^{∗}} = ∑ g_{i_{1}···i_{ν^{∗}}} where g_{i_{1}···i_{ν^{∗}}} ∈ S^{+}(4p_{i_{1}} ⋯ p_{i_{ν^{∗}}})

Choose any index k_{1} ⋯ k_{ν^{∗}} for which g_{k_{1}···k_{ν^{∗}}} ≠ 0, and consider the trace operator Tr = Tr_{4p_{k_{1}}···p_{k_{ν^{∗}}}}^{4N}. Exactly as in the first part of the proof, we see that

f_{ν^{∗}} | Tr = ∑ g_{i_{1}···i_{ν^{∗}}} | Tr_{4p_{k_{1}}···p_{k_{ν^{∗}}}}^{4N} = g_{k_{1}···k_{ν^{∗}}}

Now we claim that for ν < ν^{∗}, f_{ν} | Tr ∈ S^{−}(4p_{k_{1}} ⋯ p_{k_{ν^{∗}}}). This will imply that

f_{ν^{∗}} | Tr = g_{k_{1}···k_{ν^{∗}}} = −(f_{1} + ⋯ + f_{ν^{∗}−1}) | Tr ∈ S^{−}(4p_{k_{1}} ⋯ p_{k_{ν^{∗}}}),

forcing g_{k_{1}···k_{ν^{∗}}} = 0, a contradiction.

To establish the claim, we first note that essentially by definition of the trace operator, f_{ν} | Tr ∈ S(4p_{k_{1}} ⋯ p_{k_{ν^{∗}}}). For ν < ν^{∗}, write

f_{ν} = ∑ g_{ℓ_{1}···ℓ_{ν}} where g_{ℓ_{1}···ℓ_{ν}} ∈ S^{+}(4p_{ℓ_{1}} ⋯ p_{ℓ_{ν}})

Let D = gcd(p_{ℓ_{1}} ⋯ p_{ℓ_{ν}}, p_{k_{1}} ⋯ p_{k_{ν^{∗}}}), and write p_{k_{1}} ⋯ p_{k_{ν^{∗}}} = DK and p_{ℓ_{1}} ⋯ p_{ℓ_{ν}} = DL for integers K and L. Note that K > 1 since ν < ν^{∗}, but L could equal 1. Finally write N = p_{1} ⋯ p_{r} = DKLM for some integer M, and note that gcd(K, LM) = 1.

There are two cases. If L = 1, then D = p_{ℓ_{1}} ⋯ p_{ℓ_{ν}} is a proper divisor of p_{k_{1}} ⋯ p_{k_{ν^{∗}}}. This implies that g_{ℓ_{1}···ℓ_{ν}} | Tr = g_{ℓ_{1}···ℓ_{ν}} since g_{ℓ_{1}···ℓ_{ν}} has level dividing p_{k_{1}} ⋯ p_{k_{ν^{∗}}}. Moreover, since D is a proper divisor of p_{k_{1}} ⋯ p_{k_{ν^{∗}}}, g_{ℓ_{1}···ℓ_{ν}} ∈ S^{−}(4p_{k_{1}} ⋯ p_{k_{ν^{∗}}}).

If L > 1, then let p be any prime dividing L. Then

g_{ℓ_{1}···ℓ_{ν}} | Tr = g_{ℓ_{1}···ℓ_{ν}} | Tr_{4DK}^{4DLKM}
= g_{ℓ_{1}···ℓ_{ν}} | Tr_{4D}^{4DLM} since gcd(K, LM) = 1
= g_{ℓ_{1}···ℓ_{ν}} | Tr_{4DL}^{4DLM} | Tr_{4DL/p}^{4DL} | Tr_{4D}^{4DL/p}
= g_{ℓ_{1}···ℓ_{ν}} | Tr_{4DL/p}^{4DL} | Tr_{4D}^{4DL/p} since the level of g_{ℓ_{1}···ℓ_{ν}} is 4DL
= 0 | Tr_{4D}^{4DL/p} = 0 by Theorem 5.2 of [6].

Thus f_{ν} | Tr ∈ S^{−}(4p_{k_{1}} ⋯ p_{k_{ν^{∗}}}) which completes the proof of the claim, and of the theorem.

2.2. An External Decomposition. Next we give a correspondence between newforms of half-integral weight and newforms of integral weight. For squarefree level, we shall see that half-integral weight newforms occur with multiplicity 1 or 2, and those occurring with multiplicity 2 are precisely those with at least one member in the Kohnen subspace of newforms.

Let N be an odd positive squarefree integer, and k ≥ 3 and odd integer. Ueda [10, §3] defines an abstract Hecke algebra H(N) and two representations of it, one in the space of integral weight cusp forms, the other in the space of half-integral weight cusp forms. Isomorphisms (as H(N)-modules) between spaces of cusp forms of integral and half-integral weight have the property that an eigenform for a half-integral weight Hecke operator T̃_{k/2}(n^{2}) (with (n, 2N) = 1) corresponds to an eigenform for the integral weight Hecke operator T_{k−1}(n) which has the same eigenvalue as its halfintegral weight counterpart. Since the Hecke eigenvalues for T_{k−1}(p) for almost all primes p characterize integral weight newforms precisely, we obtain our multiplicity results as a consequence.

In particular, we have the

Theorem 2.3. Let N be an odd positive squarefree integer, and k ≥ 3 and odd integer. Then

S_{k/2}^{+}(4N, 1) ≅ ⨁_{d|N} [𝔖_{k−1}^{+}(2d) ⊕ 2𝔖_{k−1}^{+}(d)]

where the isomorphism is as modules for the Hecke algebra H(N).

Remark 2.4. Note that this theorem says that newforms of half-integral weight, “squarefree level” 4N and trivial character correspond to integral weight newforms of all possible levels dividing 2N, 2N being the level to which they correspond under any Shimura lift. It is also remarkable to note that viewed in the opposite direction, the theorem says that given any newform F of squarefree level d and trivial character, there exist half-integral weight newforms of level 2Md or 4Md having the same eigenvalues as F for any squarefree integer M prime to 2d. Note that while there obviously exist cusp forms of arbitrarily high levels with the same eigenvalues as F, it is surprising that newforms should exist with this behavior. In particular, such newforms cannot arise as character twists of half-integral weight newforms of level 4d. Moreover, we note that all newforms in S_{k/2}^{+}(4N, 1) which correspond to integral weight newforms of level 2d for any d | N, do in fact occur with multiplicity one.

Before proving the theorem, we state a corollary which connects the notion of newform developed by Kohnen with the one used in this paper which parallels the one for integral weight forms. With the notation as above, denote the Kohnen subspace by S_{k/2}(4N, 1)_{K} and the corresponding subspace of newforms (see [1] for definitions) by S_{k/2}^{new}(4N, 1)_{K}. By Theorem 2.4 of [7], we know that S_{k/2}^{new}(4N, 1)_{K} ⊂ S_{k/2}^{+}(4N, 1). Let S_{k/2}^{new}(4N, 1)_{NK} (NK meaning nonKohnen) be the orthogonal complement of S_{k/2}^{new}(4N, 1)_{K} in S_{k/2}^{+}(4N, 1). Then both S_{k/2}^{new}(4N, 1)_{K} and S_{k/2}^{new}(4N, 1)_{NK} are Hecke (i.e., H(N)) submodules of S_{k/2}^{+}(4N, 1). From Kohnen’s work [1], we know that as H(N)-modules S_{k/2}^{new}(4d, 1)_{K} ≅ 𝔖_{k−1}^{+}(d, 1) for any divisor d of N. In particular, we obtain the

Corollary 2.5. With the notation as above,

S_{k/2}^{+}(4N, 1) = S_{k/2}^{new}(4N, 1)_{K} ⊕ S_{k/2}^{new}(4N, 1)_{NK}

≅ ⨁_{d|2N} 𝔖_{k−1}^{+}(d) ⊕ ⨁_{e|N} 𝔖_{k−1}^{+}(e)

≅ ⨁_{d|2N} 𝔖_{k−1}^{+}(d) ⊕ ⨁_{e|N} S_{k/2}^{new}(4e, 1)_{K}

In particular,

S_{k/2}^{new}(4N, 1)_{NK} ≅ 𝔖_{k−1}^{+}(2N) ⊕ 𝔖_{k−1}^{+}(N) ⊕ ⨁_{\begin{matrix}d|N \\ d<N \end{matrix}} [𝔖_{k−1}^{+}(2d) ⊕ 2𝔖_{k−1}^{+}(d)]

Proof. (of Theorem 2.3) Once again for convenience we shall abbreviate S_{k/2}^{±}(4N, 1) to S^{±}(4N), and we shall also abbreviate 𝔖_{k−1}^{±}(M, 1) to 𝔖^{±}(M). The basic idea of the proof is quite straightforward. First, by the corollary in §3 of [10], we obtain

S_{k/2}(4N, 1) = S_{k/2}^{+}(4N, 1) ⊕ S_{k/2}^{−}(4N) ≅ 𝔖_{k−1}(2N, 1).

Then we use the integral weight newform theory to decompose 𝔖_{k−1}(2N, 1) and induction with Theorem 2.1 above to identify the oldform parts. Finally, using semisimplicity of the Hecke algebra, we cancel factors yielding the result.

Write N = p_{1} ⋯ p_{r} where the p_{i} are distinct odd primes. Let σ_{0}(n) denote the number of positive divisors of n. Then as modules for the Hecke algebra H(N), integral weight newform theory yields

\begin{aligned}𝔖(2p_{1} ⋯ p_{r}) & ≅ ⨁_{d|2p_{1}···p_{r}} σ_{0} (\frac{2p_{1} ⋯ p_{r}}{d} ) 𝔖^{+}(d) \\ ≅ & ⨁_{d|p_{1}···p_{r}} & [σ_{0} (\frac{p_{1} ⋯ p_{r}}{d} ) 𝔖^{+}(2d) ⊕ σ_{0} (\frac{2p_{1} ⋯ p_{r}}{d} ) 𝔖^{+}(d)] \\ & ≅ ⨁_{d|p_{1}···p_{r}} σ_{0} (\frac{p_{1} ⋯ p_{r}}{d} ) [𝔖^{+}(2d) ⊕ 2𝔖^{+}(d)] \\ & = ⨁_{k=0}^{r} ⨁_{\begin{matrix}0≤ν_{i}≤1 \\ ∑ ν_{i}=k \end{matrix}} 2^{r−k} [𝔖^{+}(2p_{1}^{ν_{1}} ⋯ p_{r}^{ν_{r}}) ⊕ 2𝔖^{+}(p_{1}^{ν_{1}} ⋯ p_{r}^{ν_{r}})] \end{aligned} (2.4)

On the other hand, by Theorem 2.1, we have

\begin{aligned}S(4p_{1}^{ν_{1}} ⋯ p_{r}^{ν_{r}}) & = ⨁_{d|p_{1}^{ν_{1}}···p_{r}^{νr}} S^{+}(4d) = ⨁_{ℓ=0}^{r} ⨁_{\begin{matrix}0≤μ_{i}≤1 \\ ∑ μ_{i}=ℓ \end{matrix}} S^{+}(4p_{1}^{μ_{1}} ⋯ p_{r}^{μ_{r}}) \\ & = S^{+}(4p_{1} ⋯ p_{r}) ⊕ ⨁_{ℓ=0}^{r−1} ⨁_{\begin{matrix}0≤μ_{i}≤1 \\ ∑ μ_{i}=ℓ \end{matrix}} S^{+}(4p_{1}^{μ_{1}} ⋯ p_{r}^{μ_{r}}) \end{aligned} (2.5)

Now we proceed by induction on r. The base case r = 0 being trivial, we assume that r ≥ 1 and

S^{+}(4p_{1}^{μ_{1}} ⋯ p_{r}^{μ_{r}}) ≅ ⨁_{d|p_{1}^{μ_{1}}···p_{r}^{μr}} [𝔖^{+}(2d) ⊕ 2𝔖^{+}(d)]

for any set of primes with 0 ≤ μ_{i} ≤ 1 and ∑ μ_{i} = ℓ < r. We rewrite the above as

S^{+}(4p_{1}^{μ_{1}} ⋯ p_{r}^{μ_{r}}) ≅ ⨁_{j=0}^{ℓ} ⨁_{\begin{matrix}0≤m_{i}≤μ_{i} \\ ∑ m_{i}=j \end{matrix}} [𝔖^{+}(2p_{1}^{m_{1}} ⋯ p_{r}^{m_{r}}) ⊕ 2𝔖^{+}(p_{1}^{m_{1}} ⋯ p_{r}^{m_{r}})] .

Combining the above with equations (2.4) and (2.5), we obtain

(2.6) S^{+}(4p_{1} ⋯ p_{r}) ⊕ ⨁_{ℓ=0}^{r−1} ⨁_{\begin{matrix}0≤μ_{i}≤1 \\ ∑ μ_{i}=ℓ \end{matrix}} ⨁_{j=0}^{ℓ} ⨁_{\begin{matrix}0≤m_{i}≤μ_{i} \\ ∑ m_{i}=j \end{matrix}} [𝔖^{+}(2p_{1}^{m_{1}} ⋯ p_{r}^{m_{r}}) ⊕ 2𝔖^{+}(p_{1}^{m_{1}} ⋯ p_{r}^{m_{r}})]

≅ ⨁_{k=0}^{r} ⨁_{\begin{matrix}0≤ν_{i}≤1 \\ ∑ ν_{i}=k \end{matrix}} 2^{r−k} [𝔖^{+}(2p_{1}^{ν_{1}} ⋯ p_{r}^{ν_{r}}) ⊕ 2𝔖^{+}(p_{1}^{ν_{1}} ⋯ p_{r}^{ν_{r}})]

Fix a sequence ν_{1}, …, ν_{r} with 0 ≤ ν_{i} ≤ 1 and ∑ ν_{i} = k < r. Note that the summand [𝔖^{+}(2p_{1}^{ν_{1}} ⋯ p_{r}^{ν_{r}}) ⊕ 2𝔖^{+}(p_{1}^{ν_{1}} ⋯ p_{r}^{ν_{r}})] occurs with multiplicity 2^{r−k} on the RHS of equation (2.6). We shall show that it occurs with multiplicity 2^{r−k} − 1 on the LHS. Then using semisimplicity of the Hecke algebra, we cancel like terms on both sides of (2.6) to obtain

S^{+}(4p_{1} ⋯ p_{r}) ≅ ⨁_{k=0}^{r} ⨁_{\begin{matrix}0≤ν_{i}≤1 \\ ∑ ν_{i}=k \end{matrix}} [𝔖^{+}(2p_{1}^{ν_{1}} ⋯ p_{r}^{ν_{r}}) ⊕ 2𝔖^{+}(p_{1}^{ν_{1}} ⋯ p_{r}^{ν_{r}})]

= ⨁_{d|N} [𝔖^{+}(2d) ⊕ 2𝔖^{+}(d)]

as required. We need only establish the claim concerning the multiplicity.

To make an accurate count we need only count the number of times the r-tuple (m_{1}, …, m_{r}) = (ν_{1}, …, ν_{r}) occurs on the LHS. In the notation of (2.6), we make two simple observations: For 1 ≤ i ≤ r, we have ν_{i} = m_{i} ≤ μ_{i} and j = k = ∑_{i=1}^{r} ν_{i} = ∑_{i=1}^{r} m_{i} ≤ ∑_{i=1}^{r} ν_{i} = ℓ. Also recall that all entries in the r-tuples are zeros or ones. We need to count the number of r-tuples μ = (μ_{1}, …, μ_{r}) which can give rise to the r-tuple m = (m_{1}, …, m_{r}) subject to the above constraints. We count as a function of ℓ.

From the inequality above, we see that ℓ ≥ k. If ℓ = k, k = ∑_{i=1}^{r} m_{i} = ∑_{i=1}^{r} ν_{i} = ℓ which says there is only one possible r-tuple μ which admits m. For ℓ = k + 1, we know that any admissible μ has ones everywhere that m does, plus exactly one more. Since k = ∑_{i=1}^{r} m_{i}, there are n − k possible indices for the extra one, and hence n − k possible r-tuples μ which admit m. In general, when ℓ = k + s (0 ≤ s ≤ n − k − 1), any admissible μ has s additional ones where m has zeros, and there are n−k indices from which to choose where to place them. Of course, this means that there are \binom{n−k}{s} r-tuples μ which admit m. Thus in all there are

1 + (n − k) + \binom{n − k}{2} + ⋯ + \binom{n − k}{n − k − 1} = ∑_{s=0}^{n−k−1} \binom{n − k}{s}

admissible μ. Since it is well known that

∑_{s=0}^{m} \binom{m}{s} = (1 + 1)^{m} = 2^{m},

the number of admissible μ is clearly 2^{n−k} − 1, as claimed.

2.3. An Extension to Quadratic Character. The purpose of this section is to deduce an analog of Theorem 2.3 which holds when the character on the space of half-integral weight forms is an arbitrary quadratic character.

To describe the situation further, we set some notation. As above, let N be an odd positive squarefree integer, and k ≥ 3 an odd integer. Let ψ be an even quadratic Dirichlet character defined modulo 4N. Rather than attempt to derive another internal decomposition for S_{k/2}(4N, ψ) as in Theorem 2.1, we shall show that S_{k/2}^{+}(4N, ψ) and S_{k/2}^{+}(4N, 1) are isomorphic as modules for the Hecke algebra H(N), from which we shall derive the

Theorem 2.6. Let N be an odd positive squarefree integer, and k ≥ 3 and odd integer. Let ψ be an even quadratic Dirichlet character defined modulo 4N. Then

S_{k/2}^{+}(4N, ψ) ≅ ⨁_{d|N} [𝔖_{k−1}^{+}(2d) ⊕ 2𝔖_{k−1}^{+}(d)]

where the isomorphism is as modules for the Hecke algebra H(N).

The idea behind the desired isomorphism is simple. Since N is squarefree, the character ψ is equal to (\frac{Q}{∗}) for some divisor Q of N with (necessarily, since N is squarefree) gcd(Q, 4N/Q) = 1. A generalized Atkin-Lehner involution W_{Q} will act as a “hermitian involution” which commutes with the action of the Hecke operators, and hence provides an isomorphism (as modules for the Hecke algebra) between S_{k/2}^{+}(4N, 1) and S_{k/2}^{+}(4N, (\frac{Q}{∗})).

While W_{Q} operators have been defined in [1], [7], and [12], none of these operators is sufficiently general to give a commutation relation with the shift operators (B_{d}) which are fundamental to our definition of the space of oldforms (and hence of newforms). So we introduce a slightly more general operator with the desired properties. The desired properties (commutation relations with various other operators, hermitian properties, etc.) are completely analogous to properties for the integral weight analog. The proofs are more tedious because of the fussiness of automorphy factors, although there are no significant surprises. Therefore we state the properties and give only the barest of outlines of the proofs.

In defining the W_{Q} operator, there is no reason to assume that N is squarefree or that the character is quadratic. In fact is is quite revealing not to do so, since we shall see why the situation of quadratic character is special.

Let N be a positive integer, and Q an odd divisor of N with gcd(Q, 4N/Q) = 1. Let ψ be an even Dirichlet character defined modulo 4N, and let

γ_{Q} = (\begin{matrix}x & y \\ \frac{4N}{Q} z & Qw \end{matrix}) ∈ Γ_{0}(\frac{4N}{Q}), δ_{Q} = (\begin{matrix}Q & 0 \\ 0 & 1 \end{matrix}) , W_{Q} = γ_{Q}δ_{Q}.

For f ∈ S_{k/2}(4N, ψ) define W̃_{Q} by

f | W̃_{Q} = f | W̃_{Q}(ψ) = ψ_{Q}(y)ψ_{\frac{4N}{Q}} (x)f | γ_{Q}^{∗}δ̃_{Q}

where γ_{Q}^{∗} = {γ_{Q}, j(γ_{Q}, z)} and δ̃_{q} = {δ_{Q}, Q^{−1/4}} are elements in the metaplectic cover G of GL_{2}^{+}(ℚ) (see Shimura [8]), and ψ_{Q} and ψ_{\frac{4N}{Q}} are the Qth and \frac{4N}{Q} th parts of ψ. Here for γ ∈ Γ_{0}(4), j(γ, z) is the standard automorphy factor

j(γ, z) = \frac{θ(γ(z))}{θ(z)}, θ(z) = ∑_{n∈ℤ}e^{2πin^{2}z}.

It is trivial to check that W̃_{Q} is well-defined and that Ueda’s W̃_{Q} is a special case of this operator.

Proposition 2.7. The operator W̃_{Q} maps S_{k/2}(4N, ψ) to S_{k/2}(4N, \overline{ψ}_{Q}ψ_{\frac{4N}{Q}} (\frac{Q}{∗})).

This follows easily from the observation that for γ ∈ Γ_{0}(4N),

W̃_{Q}(ψ)γ^{∗}W̃_{Q}(ψ)^{−1} = (W_{Q}γW_{Q}^{−1})^{∗}\{1, (\frac{Q}{d})\}.

Next we have the

Proposition 2.8. Let f ∈ S_{k/2}(4N, ψ) then

1. f | W̃_{Q}(ψ) | W̃_{Q}(\overline{ψ}_{Q}ψ_{\frac{4N}{Q}} (\frac{Q}{∗})) = (\frac{−1}{Q})^{k/2} \overline{ψ}_{\frac{4N}{Q}} (Q)ψ_{Q}(−1)f.

2. If Q′ | 4N with gcd(Q, \frac{4N}{Q}) = gcd(Q′, \frac{4N}{Q′}) = gcd(Q, Q′) = 1, then

f | W̃_{Q}(ψ) | W̃_{Q′}(\overline{ψ}_{Q}ψ_{\frac{4N}{Q}} (\frac{Q}{∗})) = \overline{ψ_{Q′}(Q)}f | W̃_{QQ′}(ψ).

Proof. The proof proceeds along a standard line, but to check the automorphy factors, one must exercise a little care. First note that if χ = (\frac{Q}{∗}), then the Q and 4N/Q-parts of χ are

χ_{Q} = \begin{cases}(\frac{Q}{∗}) & if Q ≡ 1 (mod 4) \\ (\frac{−Q}{∗}) & if Q ≡ 3 (mod 4) \end{cases} χ_{4N/Q} = \begin{cases}1 & if Q ≡ 1 (mod 4) \\ (\frac{−1}{∗}) & if Q ≡ 3 (mod 4) \end{cases}

Then, as one considers the cases Q ≡ ±1 (mod 4), one must recall that

(\frac{−1}{Qw})^{1/2} = (\frac{−1}{Q})^{1/2} (\frac{−1}{w})^{1/2} ε(Q, w)

where

ε(Q, w) = \begin{cases}−1 & if Q ≡ w ≡ 3 (mod 4) \\ 1 & otherwise \end{cases}

Finally one uses the quadratic reciprocity law for Jacobi symbols.

Next we have the

Proposition 2.9. Let f ∈ S_{k/2}(4N, ψ), with Q as above, and let p be a prime, p ∤ N. Then

f | T̃_{k/2,4N,ψ}(p^{2}) | W̃_{Q}(ψ) = ψ_{Q}(p^{2})f | W̃_{Q}(ψ) | T̃_{k/2,4N,\overline{ψ}_{Q}ψ_{\frac{4N}{Q}} (\frac{Q}{∗})}(p^{2})

Proof. We sketch the basic idea. Let τ̃ = \{τ, p^{1/2}\} ∈ G, where τ = ( \begin{matrix}1 & 0 \\ 0 & p^{2} \end{matrix} ). The Hecke operator T̃_{k/2}(p^{2}) is described by the action of the double coset p^{(k/4)−1}Δ_{0}(4N)τ̃Δ_{0}(4N). If we write

Δ_{0}(4N)τ̃Δ_{0}(4N) = ⋃_{ν} Δ_{0}(4N)τ̃_{ν},

then we need to show that

f | T̃_{k/2,4N,ψ}(p^{2}) | W̃_{Q}(ψ) = ∑_{ν} ψ(a_{ν})f | τ̃_{ν} | W̃_{Q}(ψ)

= ψ_{Q}(p^{2}) ∑_{ν′} \overline{ψ}_{Q}(a_{ν′})ψ_{\frac{4N}{Q}} (a_{ν′}) (\frac{Q}{a_{ν′}} ) f | W̃_{Q}(ψ) | τ̃_{ν′}

= ψ_{Q}(p^{2})f | W̃_{Q}(ψ) | T̃_{k/2,4N,\overline{ψ}_{Q}ψ_{\frac{4N}{Q}} (\frac{Q}{∗})}(p^{2})

where if τ̃_{ν} = {τ_{ν}, φ_{ν}} ∈ G, a_{ν} is the (1, 1)-entry of τ_{ν}.

In fact, we show that for each ν there is a unique ν′ so that

ψ(a_{ν})f | τ̃_{ν} | W̃_{Q} = ψ_{Q}(p^{2})\overline{ψ}_{Q}(a_{ν′})ψ_{\frac{4N}{Q}} (a_{ν′}) (\frac{Q}{a_{ν′}} ) f | W̃_{Q} | τ̃_{ν′}

Using Shimura’s notation [8] for the expression of the double coset as a union of right cosets,

Δ_{0}(4N)τ̃Δ_{0}(4N) = ⋃_{b=0}^{p^{2}−1} Δ_{0}(4N)α_{b}^{∗} ∪ ⋃_{h=0}^{p−1} Δ_{0}(4N)β_{h}^{∗} ∪ Δ_{0}(4N)σ^{∗},

we show that in the correspondence τ̃_{ν} ↔ τ̃_{ν′}, we can relate α_{b}^{∗} ↔ α_{b′}^{∗}, β_{h}^{∗} ↔ β_{h′}^{∗} and σ^{∗} with itself. Finally, since p ∤ 4N, we may assume that in the definition of γ_{Q} (W_{Q}), we have z ≡ 0 (mod p^{2}).

Next we describe the commutation relations of the shift operator and the W̃_{Q}- operator.

Proposition 2.10. Let M be a positive integer, and let Q be an odd divisor of MN with gcd(Q, 4MN/Q) = 1. Then for f ∈ S_{k/2}(4N, ψ) we have

f | B_{M} | W̃_{Q}^{4MN}(ψ (\frac{M}{∗} )) =

\begin{cases}\overline{ψ}_{Q}(M)f | W̃_{Q}(ψ) | B_{M} & if gcd(Q, M) = 1 \\ (\frac{−1}{M})^{k/2} \overline{ψ}_{4NM/Q}(M)M^{−k/2}f | W̃_{Q/M}^{4N}(ψ) & if M | Q \end{cases}

where the superscripts on the W̃-operators indicate the levels of the forms on which they operate.

Proof. This proof is analogous to the others, however it does require a case analysis based upon the congruence class of M modulo 4.

With these preliminary results in hand, we now address the interaction of the W̃_{Q}-operator with oldforms and newforms.

Proposition 2.11. Let N be a positive integer, Q an odd divisor of N with gcd(Q, 4N/Q) = 1, and ψ an even Dirichlet character defined modulo 4N. Then W̃_{Q}(ψ) takes S_{k/2}^{−}(4N, ψ) to S_{k/2}^{−}(4N, \overline{ψ}_{Q}ψ_{4N/Q} (\frac{Q}{∗})), and in particular takes oldforms to oldforms.

Proof. It is clear from Proposition 2.9, that W̃_{Q} takes Hecke eigenforms to Hecke eigenforms. Let f ∈ S_{k/2}^{−}(4N, ψ). By linearity, it is enough to suppose that for some prime q | N either f ∈ S_{k/2}(4N/q, ψ) (provided ψ is defined modulo 4N/q), or f = g | B_{q} for some g ∈ S_{k/2}(4N/q, ψ (\frac{q}{∗})) (provided ψ (\frac{q}{∗}) is defined modulo 4N/q).

For the first case, write

W_{Q} = W_{Q}^{4N} = γ_{Q}^{4N}δ_{Q} = (\begin{matrix}x & y \\ \frac{4N}{Q} z & Qw \end{matrix}) δ_{Q}

We have two subcases depending upon whether q | Q or not. Note that if q ∤ Q, then Q | 4N/q. We now write

γ_{Q}^{4N} = \begin{cases}(\begin{matrix}x & y \\ \frac{4N/q}{Q}(qz) & Qw \end{matrix}) & if q ∤ Q \\ (\begin{matrix}x & y \\ \frac{4N/q}{Q/q}z & \frac{Q}{q}(qw) \end{matrix}) & if q | Q \end{cases}

Notice that if q ∤ Q, then γ_{Q}^{4N} is a valid representative for γ_{Q}^{4N/q} (needed to define W̃_{Q/q}^{4N/q}), while if q | Q, then γ_{Q}^{4N} is a valid representative for γ_{Q/q}^{4N/q}. Thus we have

f | W̃_{Q}^{4N}(ψ) = \begin{cases}κf | W̃_{Q}^{4N/q}(ψ) & if q ∤ Q \\ κ′f | W̃_{Q/q}^{4N/q} | B_{q} & if q | Q \end{cases}

for constants κ and κ′. In either (sub)case, f | W̃_{Q}^{4N}(ψ) ∈ S_{k/2}^{−}(4N, \overline{ψ}_{Q}ψ_{4N/Q} (\frac{Q}{∗})). In the case that f = g | B_{q}, we have by Proposition 2.10 that

f | W̃_{Q}^{4N}(ψ) = g | B_{q} | W̃_{Q}^{4N}(ψ)

= \begin{cases}\overline{ψ}_{Q}(q)g | W̃_{Q}^{4N/q}(ψ (\frac{q}{∗})) | B_{q} & if q ∤ Q \\ (\frac{−1}{q})^{k/2} \overline{ψ}_{4Nq/Q}(q)q^{−k/2}g | W̃_{Q/q}^{4N/q}(ψ (\frac{q}{∗})) & if q | Q, \end{cases}

which again shows that f | W̃_{Q}^{4N}(ψ) ∈ S_{k/2}^{−}(4N, \overline{ψ}_{Q}ψ_{4N/Q} (\frac{Q}{∗})) as required.

Our final proposition provides the last tools needed to complete the proof of Theorem 2.6.

Proposition 2.12. Let N be a positive integer, Q an odd divisor of N with (Q, 4N/Q) = 1, and ψ an even Dirichlet character defined modulo 4N.

1. For f, g ∈ S_{k/2}(4N, ψ), ⟨f, g⟩ = ⟨f | W̃_{Q}, g | W̃_{Q}⟩.

2. The operator W̃_{Q}(ψ) takes S_{k/2}^{+}(4N, ψ) to S_{k/2}^{+}(4N, \overline{ψ}_{Q}ψ_{4N/Q} (\frac{Q}{∗})), and in particular takes newforms to newforms.

3. If ψ is quadratic and N odd and squarefree, then S_{k/2}^{+}(4N, 1) and S_{k/2}^{+}(4N, ψ) are isomorphic as modules for the Hecke algebra.

Proof. The first statement is obvious, since for all ρ̃ ∈ G, ⟨f | ρ̃, g⟩ = ⟨f, g | ρ̃^{−1}⟩. For the second statement, let f ∈ S_{k/2}^{+}(4N, ψ) be a newform. By Proposition 2.9,

f | W̃_{Q}(ψ) is again a simultaneous Hecke eigenform. We need only show that it lies in S_{k/2}^{+}(4N, \overline{ψ}_{Q}ψ_{4N/Q} (\frac{Q}{∗})). Let g ∈ S_{k/2}^{−}(4N, \overline{ψ}_{Q}ψ_{4N/Q} (\frac{Q}{∗})). Then by Proposition 2.11,

g | W̃_{Q}(\overline{ψ}_{Q}ψ_{4N/Q} (\frac{Q}{∗})) ∈ S_{k/2}^{−}(4N, ψ), and g | W̃_{Q}(\overline{ψ}_{Q}ψ_{4N/Q} (\frac{Q}{∗})) | W̃_{Q}(ψ) = κg with |κ| = 1. Thus

⟨f | W̃_{Q}(ψ), g⟩ = κ⟨f | W̃_{Q}(ψ), g | W̃_{Q}(\overline{ψ}_{Q}ψ_{4N/Q} (\frac{Q}{∗})) | W̃_{Q}(ψ)⟩
= κ⟨f, g | W̃_{Q}(\overline{ψ}_{Q}ψ_{4N/Q} (\frac{Q}{∗}))⟩ = 0

as required. For the third statement, since N is odd and squarefree, we may take ψ = (\frac{Q}{∗}) for some divisor Q of N (necessarily satisfying (Q, 4N/Q) = 1). Then W̃_{Q}(1) maps S_{k/2}^{+}(4N, 1) to S_{k/2}^{+}(4N, ψ). Moreover, by Proposition 2.9, f and f | W̃_{Q}(1)

have the same Hecke eigenvalues, and since W̃_{Q}(1) is an “involution”, the proof is complete.

3. Decomposition Theorems - Prime Power Level

In this section, we decompose the distinguished subspace S^{∅}(4q^{m}, ψ) defined by Ueda [12] in terms of the space of newforms as defined in this paper. We give a decomposition of both the full space of cusp forms S_{k/2}^{∅}(4q^{m}, ψ) and of the Kohnen subspace S_{k/2}^{∅}(4q^{m}, ψ)_{K} where q is an odd prime, m ≥ 2, and ψ an even quadratic Dirichlet character defined modulo 4q^{m}. We begin with Ueda’s definition of S^{∅} using twisting operators. We remind the reader of Ueda’s notation [12].

Write N = 4q^{m} = M_{1}M_{2}^{+} with M_{1} = 1 and M_{2}^{+} = q^{m}. Let Π = {q}. Write S(4q^{m}, ψ) to represent either S_{k/2}(4q^{m}, ψ) or S_{k/2}(4q^{m}, ψ)_{K}. While we have defined the subspace of oldforms S_{k/2}^{−}(4q^{m}, ψ) in equation (2.1), we need to make sure the same definition makes sense in the Kohnen subspace. That is, we want to make the definition

S_{k/2}^{−}(4N, ψ)_{K} = ∑_{\begin{matrix}q|N \\ cond(ψ)|4N/q \end{matrix}} S_{k/2}(4N/q, ψ)_{K} + ∑_{\begin{matrix}q|N \\ cond(ψ(\frac{q}{∗}))|4N/q \end{matrix}} S_{k/2}(4N/q, ψ (\frac{q}{∗}))_{K} | B_{q} (3.1)

We need only check that S_{k/2}(4N/q, ψ (\frac{q}{∗}))_{K} | B_{q} ⊂ S_{k/2}(4N, ψ)_{K}. Clearly there is no issue with the level and character. The only condition to check is that the constraints on the Fourier coefficients which define the Kohnen subspace are met.

Let g = ∑ a(n)x^{n} ∈ S_{k/2}(4N/q, ψ (\frac{q}{∗}))_{K}, and write g | B_{q} = ∑ b(n)x^{n} ∈ S_{k/2}(4N, ψ). Since ψ is quadratic, we may write ψ = (\frac{N_{0}}{∗} ) for some positive divisor

N_{0} of N. Let ε = (\frac{−1}{N_{0}}). Then g | B_{q} ∈ S_{k/2}(4N, ψ)_{K} provided b(n) = 0 for

ε(−1)^{\frac{k−1}{2}} n ≡ 2, 3 (mod 4).

Since g ∈ S_{k/2}(4N/q, ψ (\frac{q}{∗}))_{K}, and ψ (\frac{q}{∗}) = (\frac{qN_{0}}{∗} ), we know that a(n) = 0 if

(\frac{−1}{q}) ε(−1)^{\frac{k−1}{2}} n ≡ 2, 3 (mod 4). Now b(n) = a(n/q) = 0 unless q | n, so write

n = qn_{0}. Then b(n) = a(n_{0}). Since (\frac{−1}{q}) q ≡ 1 (mod 4), if ε(−1)^{\frac{k−1}{2}} n ≡ 2, 3 (mod 4), then

ε(−1)^{\frac{k−1}{2}} n ≡ ε(−1)^{\frac{k−1}{2}} n_{0}q ≡ (\frac{−1}{q}) ε(−1)^{\frac{k−1}{2}} n_{0} ≡ 2, 3 (mod 4),

so that a(n_{0}) = b(n) = 0 as required.

We define the subspace of oldforms in the Kohnen subspace by equation (3.1), and define the subspace of newforms S_{k/2}^{+}(4N, ψ)_{K} as the orthogonal complement of S_{k/2}^{−}(4N, ψ)_{K} in S_{k/2}(4N, ψ)_{K}. Note that these subspaces are different than the ones Ueda defines, but it is precisely the point of this section to compare these subspaces by decomposing one in terms of the other.

So with S(4q^{m}, ψ) representing either the full space of cusp forms or the Kohnen subspace, we let S^{±}(4q^{m}, ψ) represent the corresponding subspaces of old or newforms as defined in and immediately following equations (2.1) and (3.1).

We begin our comparison of the decomposition of S(4q^{m}, ψ) using Ueda’s twisting operators and our notion of newforms. It follows from Propositions 1.5, 1.10, and 1.11 of [12] that for m ≥ 2,

S(4q^{m}, ψ) = S^{∅}(4q^{m}, ψ) ⊥ S(4q^{m−1}, ψ (\frac{q}{∗})) | B_{q} (3.2)

where ⊥ denotes orthogonal sum. Moreover, by definition, we know that

S(4q^{m}, ψ) = S^{+}(4q^{m}, ψ) ⊥ S^{−}(4q^{m}, ψ),

and from the definitions, S(4q^{m−1}, ψ (\frac{q}{∗})) | B_{q} ⊂ S^{−}(4q^{m}, ψ). Thus S^{+}(4q^{m}, ψ) ⊂ S^{∅}(4q^{m}, ψ).

Now by definition,

S^{−}(4q^{m}, ψ) = S(4q^{m−1}, ψ) + S(4q^{m−1}, ψ (\frac{q}{∗})) | B_{q}.

We compute that

S(4q^{m−1}, ψ) ∩ S(4q^{m−1}, ψ (\frac{q}{∗})) | B_{q} = S(4q^{m−2}, ψ (\frac{q}{∗})) | B_{q}

as follows. First it is obvious that the RHS is contained in the LHS. Now let f = g | B_{q} ∈ S(4q^{m−1}, ψ) with g ∈ S(4q^{m−1}, ψ (\frac{q}{∗})). Since f has level 4q^{m−1}, by Propositions 1.10 and 1.11, we have that g ∈ S(4q^{m−2}, ψ (\frac{q}{∗})) or g = 0, which is what we required.

It follows that

S^{−}(4q^{m}, ψ) = S(4q^{m−1}, ψ (\frac{q}{∗})) | B_{q} ⊕ [(S(4q^{m−2}, ψ (\frac{q}{∗})) | B_{q})^{⊥} in S(4q^{m−1}, ψ)]

On the other hand, just as in (3.2) for m ≥ 3,

S(4q^{m−1}, ψ) = S^{∅}(4q^{m−1}, ψ) ⊥ S(4q^{m−2}, ψ (\frac{q}{∗})) | B_{q}, (3.3)

hence

S^{−}(4q^{m}, ψ) = S^{∅}(4q^{m−1}, ψ) ⊕ S(4q^{m−1}, ψ (\frac{q}{∗})) | B_{q}.

Thus for m ≥ 3 we have

\begin{aligned}S(4q^{m}, ψ) & = S^{∅}(4q^{m}, ψ) ⊥ S(4q^{m−1}, ψ (\frac{q}{∗})) | B_{q} \\ & = S^{+}(4q^{m}, ψ) ⊥ [S^{∅}(4q^{m−1}, ψ) ⊕ S(4q^{m−1}, ψ (\frac{q}{∗})) | B_{q}] \end{aligned} (3.4)

If only the last sum in the above expression were an orthogonal sum, then by the uniqueness of orthogonal complements, we could easily deduce the decomposition:

S^{∅}(4q^{m}, ψ) = S^{+}(4q^{m}, ψ) ⊥ S^{∅}(4q^{m−1}, ψ).

However, the sum is not necessarily orthogonal. Nonetheless, we will eventually deduce exactly this identity with ⊥ replaced by ⊕. Unfortunately, the proof is a bit more indirect.

We begin by establishing the above identity as an isomorphism as modules for the Hecke algebra H(q) generated by all Hecke operators T̃_{k/2}(p^{2}) with (p, 2q) = 1.

Theorem 3.1. Let q be an odd prime, m ≥ 2 an integer, and ψ an even quadratic Dirichlet character defined modulo 4q^{m}. Then as H(q)-submodules of S(4q^{m}, ψ),

S^{∅}(4q^{m}, ψ) ≅ \begin{cases}S^{+}(4q^{m}, ψ) ⊕ S^{∅}(4q^{m−1}, ψ) & if m ≥ 3 or m = 2, ψ = 1 \\ S^{+}(4q^{2}, ψ) ⊕ S^{+}(4q, ψ) & if m = 2 and ψ = (\frac{q}{∗}). \end{cases}

Proof. We must separate the cases m ≥ 3 and m = 2. First consider m ≥ 3. Recall that H(q) is a commutative semisimple algebra. Since ψ is quadratic, by Lemma 5 of [9] T̃_{k/2}(p^{2}) = T̃_{k/2,4q^{m},ψ}(p^{2}) is hermitian, and by easy computations (see [12]), we can verify that all of the summands in equation (3.4) are H(q)-submodules of S(4q^{m}, ψ). Thus the equalities in equation (3.4) also hold as isomorphisms as H(q)-modules:

\begin{aligned}S(4q^{m}, ψ) & ≅ S^{∅}(4q^{m}, ψ) ⊕ S(4q^{m−1}, ψ (\frac{q}{∗})) | B_{q} \\ & ≅ S^{+}(4q^{m}, ψ) ⊕ S^{∅}(4q^{m−1}, ψ) ⊕ S(4q^{m−1}, ψ (\frac{q}{∗})) | B_{q} \end{aligned}

Using the semisimplicity of H(q) and canceling like terms yields the result in this case. For m = 2, equation (3.3) is not valid. By the conventions in Ueda [12], S(4q, ψ) = S^{∅}(4q, ψ), and S(4, ψ (\frac{q}{∗})) | B_{q} is not necessarily zero.

As above, we do have that

\begin{aligned}S(4q^{2}, ψ) & = S^{∅}(4q^{2}, ψ) ⊕ S(4q, ψ (\frac{q}{∗})) | B_{q} \\ & = S^{+}(4q^{2}, ψ) ⊕ S(4q, ψ (\frac{q}{∗})) | B_{q} ⊕ [(S(4, ψ (\frac{q}{∗})) | B_{q})^{⊥} in S(4q, ψ)] . \end{aligned} (3.5)

Moreover, these equalities also hold as H(q)-module isomorphisms.

If ψ = 1, then S(4, ψ (\frac{q}{∗})) = 0, so equation (3.5) and semisimplicity of H(q) yield

S^{∅}(4q^{2}, ψ) ≅ S^{+}(4q^{2}, ψ) ⊕ S(4q, ψ) = S^{+}(4q^{2}, ψ) ⊕ S^{∅}(4q, ψ)

where S^{∅}(4q, ψ) = S(4q, ψ) by Ueda’s conventions.

If ψ ≠ 1 (and hence ψ = (\frac{q}{∗})), we need to compute [(S(4, 1) | B_{q})^{⊥} in S(4q, (\frac{q}{∗}))]. But

S(4q, (\frac{q}{∗})) = S^{+}(4q, (\frac{q}{∗})) ⊥ S^{−}(4q, (\frac{q}{∗})) = S^{+}(4q, (\frac{q}{∗})) ⊥ S(4, 1) | B_{q},

so [(S(4, 1) | B_{q})^{⊥} in S(4q, (\frac{q}{∗}))] = S^{+}(4q, (\frac{q}{∗})).

Thus

S(4q^{2}, (\frac{q}{∗})) ≅ S^{∅}(4q^{2}, (\frac{q}{∗})) ⊕ S(4q, 1) | B_{q}
≅ S^{+}(4q^{2}, (\frac{q}{∗})) ⊕ S(4q, 1) | B_{q} ⊕ S^{+}(4q, (\frac{q}{∗})),

which yields S^{∅}(4q^{2}, (\frac{q}{∗})) ≅ S^{+}(4q^{2}, (\frac{q}{∗})) ⊕ S^{+}(4q, (\frac{q}{∗})) as required.

As a corollary we have Corollary 3.2. Let the notation be as above, and put μ = ord_{q}(cond(ψ)) (= 0 or 1), then

S^{∅}(4q^{m}, ψ) ≅ ⨁_{ℓ=μ}^{m} S^{+}(4q^{ℓ}, ψ).

Proof. It is immediate from Theorem 3.1 that

S^{∅}(4q^{m}, ψ) ≅ \begin{cases}⨁_{ℓ=2}^{m} S^{+}(4q^{ℓ}, ψ) ⊕ S^{∅}(4q, ψ) & if ψ = 1 \\ ⨁_{ℓ=2}^{m} S^{+}(4q^{ℓ}, ψ) ⊕ S^{+}(4q, ψ) & if ψ = (\frac{q}{∗}) \end{cases}

If ψ = (\frac{q}{∗}), we are done. To go further, we recall Ueda’s convention [12] that S^{∅}(4q, ψ) = S(4q, ψ). If ψ = 1, then S^{−}(4q, 1) = S(4, 1) = S^{+}(4, 1), so S(4q, 1) = S^{+}(4q, 1) ⊕ S^{−}(4q, 1) = S^{+}(4q, 1) ⊕ S^{+}(4, 1), as required.

Remark 3.3. Multiplicities of Newforms. When S(4q^{m}, ψ) = S_{k/2}(4q^{m}, ψ)_{K}, Ueda [12] has established a newform theory and strong multiplicity-one theorem. First, he decomposes the space S^{∅} using twisting operators. In our case, this decomposition is quite simple: S^{∅} = S^{∅,+} ⊕ S^{∅,−}, where S^{∅,±} = {f ∈ S^{∅} : f | R_{q} = ±f} and R_{q} is the twisting operator associated to (\frac{∗}{q}). He then decomposes each subspace S^{∅,±}

into a direct sum of a space of oldforms 𝔖̂^{∅,±} and its orthogonal complement 𝔖^{∅,±} (newforms). In our case S^{∅,±} = 𝔖^{∅,±}. By Theorem 3.11 of [12], each summand 𝔖^{∅,±} satisfies a strong multiplicity-one theorem. In particular, since

S^{∅} = S^{∅,+} ⊕ S^{∅,−} = 𝔖^{∅,+} ⊕ 𝔖^{∅,−} ≅ ⨁_{ℓ=μ}^{m} S^{+}(4q^{ℓ}, ψ),

we must have that newforms in the spaces S^{+}(4q^{ℓ}, ψ) occur with multiplicity one or two. This is analogous to our Theorem 2.3 which states a more precise result for the full space of cusp forms having squarefree level.

We now show that S^{∅}(4q^{m−1}, ψ) ⊂ S^{∅}(4q^{m}, ψ) (for m ≥ 3). To do so, we first need to compute the adjoint of the operator U_{q}. The proposition below is stated for the full space of cusp forms. The action on the Kohnen subspace is the same, as we discuss following the proof.

Proposition 3.4. Let q be an odd prime and N a positive integer. Suppose that q | N, and ψ is an even Dirichlet character defined modulo 4N. Let U_{q}^{∗} : S_{k/2}(4N, ψ (\frac{q}{∗})) → S_{k/2}(4N, ψ) be the adjoint of the U_{q} operator. Then for g ∈ S_{k/2}(4N, ψ (\frac{q}{∗})),

g | U_{q}^{∗} = q^{k/2}g | B_{q} | Tr_{4N}^{4Nq}(ψ)

where B_{q} is the shift operator and Tr is the trace operator.

Proof. Let Γ_{1} = Γ_{1}(4N), Δ_{1} = Δ_{1}(4N), α = ( \begin{matrix}1 & 0 \\ 0 & q \end{matrix} ) and ξ = [α, q^{1/4}] ∈ G. Then U_{q} = Δ_{1}ξΔ_{1}. By Lemma 1.9 of [11] (using Proposition 1.1 and equation (1.14) of

[8]), we have U_{q}^{∗} = Δ_{1}ξ′Δ_{1}, where ξ′ = [α′, q^{1/4}] ∈ G, α′ = ( \begin{matrix}q & 0 \\ 0 & 1 \end{matrix} ).

If α_{μ} = ( \begin{matrix}1 & 0 \\ 4Nμ & 1 \end{matrix} ), then since q | N, we have

Γ_{1} = Γ_{1}(4N) = ∪_{μ=1}^{q}Γ_{1}(4Nq)α_{μ} and Γ_{1}αΓ_{1} = ∪_{μ=1}^{q}Γ_{1}αα_{μ}.

Again by Proposition 1.1 and equation (1.14) of Shimura [8]

Δ_{1}(4N)ξ′Δ_{1}(4N) = ∪_{μ=1}^{q}Δ_{1}(4N)ξ′α_{μ}^{∗}

where α_{μ}^{∗} = [α_{μ}, j(α_{μ}, τ)] ∈ Δ_{1} ⊂ G.

So for g ∈ S_{k/2}(4N, ψ (\frac{q}{∗})), we have

g | U_{q}^{∗} = g | Δ_{1}ξ′Δ_{1} = det(α′)^{k/4−1} ∑_{μ=1}^{q} g | ξ′α_{μ}^{∗}

= q^{k/2−1} ∑_{μ=1}^{q} g | B_{q} | α_{μ}^{∗} = q^{k/2−1} ∑_{μ=1}^{q} \overline{ψ}(d_{μ})(g | B_{q}) | α_{μ}^{∗}

where d_{μ} = 1 is the d-entry of α_{μ}

= q^{k/2−1}[Γ_{1}(4N) : Γ_{1}(4Nq)]g | B_{q} | Tr_{4N}^{4Nq}(ψ) = q^{k/2}g | B_{q} | Tr_{4N}^{4Nq}(ψ)

as required.

We note that the same expression makes sense as well for the Kohnen subspace, provided N is odd so the space is defined. Apriori, the adjoint of the U_{q} operator restricted to the Kohnen subspace is the orthogonal projection of the adjoint U_{q}^{∗} into the Kohnen subspace, but we have already observed that the B_{q} operator preserves the Kohnen subspace, and since the trace operator is simply the adjoint of the inclusion map, it does as well.

Proposition 3.5. Let m ≥ 2, and f ∈ S^{∅} = S^{∅}(4q^{m}, ψ). Then f | U_{q} = 0.

Proof. Recall that S^{∅} = S^{∅,+} ⊕ S^{∅,−} where S^{∅,±} = {f ∈ S^{∅} : f | R_{q} = ±f}, and R_{q} is the twisting operator associated to (\frac{∗}{q}). Clearly, it is enough to verify the result for

f ∈ S^{∅,±}. For such an f = ∑ a_{n}x^{n}, f | R_{q} = ±f means that a_{n} = 0 whenever q | n. Thus f | U_{q} = ∑ a_{nq}x^{n} = 0.

Proposition 3.6. For m ≥ 3, S^{∅}(4q^{m−1}, ψ) ⊂ S^{∅}(4q^{m}, ψ).

Proof. Let f ∈ S^{∅}(4q^{m−1}, ψ) ⊂ S(4q^{m}, ψ). Then

f ∈ S^{∅}(4q^{m}, ψ) ⟺ ⟨f, g | B_{q}⟩_{4q^{m},ψ} = 0 for all g ∈ S(4q^{m−1}, ψ (\frac{q}{∗}))

⟺ ⟨f, g | B_{q} | Tr_{4q^{m−1}}^{4q^{m}}⟩_{4q^{m−1},ψ} = 0 for all g ∈ S(4q^{m−1}, ψ (\frac{q}{∗}))

⟺ ⟨f, g | U_{q}^{∗}⟩ = 0 for all g
⟺ ⟨f | U_{q}, g⟩ = 0 for all g

which is true by the previous proposition.

Corollary 3.7. For m ≥ 3, S^{∅}(4q^{m}, ψ) = S_{k/2}^{+}(4q^{m}, ψ) ⊕ S^{∅}(4q^{m−1}, ψ).

Proof. We have already verified that both summands are contained in the RHS. Moreover, S_{k/2}^{+}(4q^{m}, ψ) ∩ S^{∅}(4q^{m−1}, ψ) = 0 since S^{∅}(4q^{m−1}, ψ) ⊂ S(4q^{m−1}, ψ) ⊂ S_{k/2}^{−}(4q^{m}, ψ). Thus the RHS is a summand of the LHS. But by Theorem 3.1, both sides are isomorphic as H(q)-modules, and hence as finite dimensional vector spaces from which the assertion follows.

References

[1] W. Kohnen, Newforms of Half-Integral Weight, J. reine angew. Math., 333 (1982), 32–72.

[2] W. Li, Newforms and Functional Equations, Math. Annalen, 212 (1975), 285–315.

[3] M. Manickam, B. Ramakrishnan, T. Vasudevan, On the Theory of Newforms of Half-Integral Weight, J. Number Theory, 34 (1990), 210-224.

[4] S. Niwa, On Shimura’s Trace Formula, Nagoya Math J.,34 (1977), 183–202.

[5] J.-P. Serre and H. Stark, Modular Forms of Weight 1/2, In Lecture Notes in Math. 627, Springer-Verlag, Berlin and New York (1977), 27–67.

[6] T. Shemanske, Cuspidal Newforms and Character Twists, J. reine angew. Math., 328 (1981), 59–71.

[7] T. Shemanske and L. Walling, Determining Multiplicities of Half-Integral Weight Newforms, to appear Pacific J. Math.

[8] G. Shimura, On Modular Forms of Half-Integral Weight, Annals of Math., 97 (1973), 440–481.

[9] G. Shimura, The Critical Values of Certain Zeta Functions Associated with Modular Forms of Half-Integral Weight, J. Math. Soc. Japan, 33 (1981), 649–672.

[10] M. Ueda, The Decomposition of the Spaces of Cusp Forms of Half-Integral Weight and the Trace Formula of Hecke Operators, J. Math. Kyoto U., 28 (1988), 505–555.

[11] M. Ueda, The Trace Formulae of Twisting Operators on the Spaces of Cusp Forms of Half-Integral Weight and Some Trace Relations, Japanese J. of Math., 17 (1991), 83–135.

[12] M. Ueda, On Twisting Operators and Newforms of Half-Integral Weight, Nagoya Math J., 131 (1993), 135 – 205.

Department of Mathematics, 6188 Bradley Hall, Dartmouth College, Hanover, New Hampshire 03755-3551

 E-mail address: Thomas.Shemanske@dartmouth.edu