# Necessity of conditions $N$ odd, square-free and $\chi$ quadratic in Kohnen's plus space - modular forms of half-integral weight

Kohnen introduced the "plus" space as a subspace of the space of modular forms of half integral weight, first in his 1980 paper and then generalized the work in a later 1982 paper. Why is the condition $$N$$ odd, square-free and $$\chi$$ quadratic necessary?

To elaborate: Let $$S_{k+1/2}\left(4N,\left(\frac{4\chi(-1)}{.}\right)\chi\right)$$ denote the space of half-integral weight modular forms of level $$4N$$ and character $$\left(\frac{4\chi(-1)}{.}\right)\chi$$, where $$\chi$$ is a Dirichlet character of modulus $$N$$ and $$\left(\frac{a}{b}\right)$$ is the Kronecker symbol. Kohnen defines the plus sub-space $$S_{k+1/2}^+\left(4N,\left(\frac{4\chi(-1)}{.}\right)\chi\right)$$ by attaching certain conditions on the Fourier coefficients of the modular forms, and then develops nice theory analogous to the Atkin-Lehner-Li theory of newforms in the case of integral weight modular forms.

Question: Why are the conditions $$N$$ odd, sqaure-free and $$\chi$$ quadratic necessary in the second paper? Kohnen remarks that these are not necessary for few of the stated results, but I cannot figure out where exactly these are required.

The question is motivated by the fact that several authors (like Ueda, Yamana, Manickam-Ramakrishnan-Vasudevan) have generalized this work to other levels (like $$8N,16N,32N$$) and non-quadratic characters, and also to the full-space (a nice history can be found (but not restricted to) in the introduction of this paper and this paper); but this condition $$N$$ odd and square free is still there, and in some places $$\chi$$ is quadratic. I could not find the reason for these conditions.

This is not an answer and I cannot comment due to lack of reputation, but it seems to first pop up in his Lemma 4 on p. 50. This occurs after a long, very technical discussion about double coset operators. The goal is to show that two specific representations of the Hecke algebra are equivalent (one whose image lies in $$End_{\mathbb{C}}(S_{k+1/2}(N, \chi))$$ and the other in $$End_{\mathbb{C}}(S_{2k}(N))$$). The Lemma:

Lemma 4 (p. 50). Every elliptic or hyperbolic conjugacy class in $$\Gamma_{0}(4N)C_{n}\Gamma_{0}(4N)$$ contains an element $$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ with $$d > 0$$, $$(b, d) = 1$$, and $$\left( \frac{b}{f}, \frac{t^{2} - 64n^{2}}{f^{2}} \right) = 1$$. Here, $$t := a + d$$ and $$f := (d-a, b, c)$$.

Now, for squarefree $$M$$ set

$$\mu(t, f, n, M) := \prod_{p \mid (M, f)} (1+p)\cdot \#\{x \in \mathbb{Z} : 1 \leq x \leq M, (x, M) = 1, x^{2} - tx + n \equiv 0 \mod (fM, M^{2})\},$$

Lemma 5. Let $$A \in GL_{2}(\mathbb{Z})$$ be an elliptic or hyperbolic matrix with $$\det A = 16n^{2}$$, $$t \equiv 0 \mod{4}$$ and $$f$$ odd. Then for $$N$$ odd and squarefree there are $$\mu(\frac{t}{4}, f, n^{2}, N)$$ matrices $$B \in \Gamma(1)/\Gamma_{0}(4N)$$ with $$B^{-1}AB \in \Gamma_{0}(4N)C_{n}\Gamma_{0}(4N)$$.

So it seems that the definition of $$\mu$$ depends on $$M$$ being squarefree, and in turn this allows Kohnen the counting argument in Lemma 5. I don't know why squarefree-ness is needed from this, though. But maybe this information can help you or someone else determine the answer.

EDIT:

Answer: If $$N$$ is not squarefree, there exists a case such that all common eigen-subspaces of $$S_{k+1/2}^{+}(4N, \chi)$$ for Hecke operators have dimension at least 2, whence a strong multiplicity one theorem does not hold (see Ueda's On twisting operators and newforms of half-integral weight).

We may try to decompose our cusp space into eigen-subspaces of twisting operators (why??). Let $$M := \prod_{\substack{p \mid N \\ ord_{p}(N) \geq 2}} p^{ord_{p}(N)}, \quad \Pi := \{p \text{ prime} : p \mid M\}.$$ One can decompose the Kohnen space as

$$S_{k+1/2}^{+}(4N, \chi) = \left( \bigoplus_{\kappa: \Pi \rightarrow \{\pm 1\}} S^{0, \kappa} \right) \oplus Ker\left(R_{\Pi}; S_{k+1/2}^{+}(4N, \chi) \right),$$

where $$S^{0, \kappa} := \{f \in S_{k+1/2}^{+}(4N, \chi) : f \mid R_{l} = \kappa(l)f, \forall l \in \Pi\}$$ and $$R_{\Pi}, R_{l}$$ are the twisting operators of the characters $$\prod_{l \in \Pi} \left( \frac{*}{l} \right)$$ resp. $$\left( \frac{*}{l} \right)$$. The 'kernel' part consists of oldforms, and each $$S^{0, \kappa}$$ is stable under the Hecke operators $$T(n^{2})$$ for $$(n, 4N) = 1$$.

Why we need squarefree: There exists a case such that $$S^{0, \kappa} \cong S^{0, \kappa'}$$ as Hecke modules for distinct $$\kappa, \kappa': \Pi \rightarrow \{\pm 1\}$$. Hence when $$M \neq 1$$ Kohnen's theory does not work.