# (Explicit) Basis for Kohnen's plus-space of modular forms of half integral weight

Sorry if this is trivial, but I could not find any reference.

Let $$k,a,b$$ be integers. The space of modular forms of integer weight $$M_k(\text{SL}_2(\mathbb{Z}))$$ admits a basis of the form $$\{ E_4^aE_6^b : 4a+6b = k \}$$ where $$E_4$$ and $$E_6$$ are normalized Eisenstein series of weight 4 and 6 respectively. Similarly, the space of modular forms of half integer weight $$M_{k+\frac{1}{2}}(4)$$ for $$\Gamma_0(4)$$ has a basis of the form $$\{ \theta^aF^b : \frac{a}{2}+2b = k+ \frac{1}{2} \},$$ where $$\theta = 1+\sum_{n\geq1}q^{n^2}$$ and $$F=\sum_{n\geq1, n \text{ odd}}\sigma_1(n)q^n$$ (see page 254 of Kohnen's paper).

Question: Does there exists similar 'explicit' basis for the Kohnen's plus subspace : $$M_{k+\frac{1}{2}}^+(4) = \{ f \in M_{k+\frac{1}{2}}(4) : a_f(n) = 0 \text{ if } (-1)^kn\equiv 2,3\pmod4 \,\} ?$$

Extras: What about the generalized cases of forms of level $$N$$ in respective spaces? For $$M_k(\Gamma_0(N))$$ this is answered here. What about $$M_{k+\frac{1}{2}}(4N)$$ and $$M_{k+\frac{1}{2}}^+(4N)$$ ?

## 1 Answer

There does exist an explicit basis when $$k$$ is EVEN: denote by $$E_{k,4}$$ the Eisenstein series $$E_k(4\tau)$$. Then the Rankin-Cohen brackets $$[\theta,E_{k-2j,4}]_j$$ for $$0\le j\le\lfloor k/6\rfloor$$ (with a small modification for $$k=2$$) form a basis of $$M^+_{k+1/2}(4)$$, where of course $$\theta$$ is the usual generator of $$M_{1/2}(4)$$.

The case $$k$$ odd is more difficult. The best I could come up with is to use brackets of $$\theta$$ with the two Eisenstein series of weight $$k$$ and quadratic character modulo 4, but the Kohnen space is only a subset of this.

• Thank you very much. So, at least the question is not trivial, and I think the case for general $N$ is trickier. - I will wait for some more time (no offence!) for other ideas and accept the answer if none comes. Apr 17 '20 at 18:56