# zbMATH — the first resource for mathematics

On the zeros and coefficients of certain weakly holomorphic modular forms. (English) Zbl 1200.11027
A weakly holomorphic modular form (say, $$f$$) of weight $$k\in 2\mathbb{Z}$$ for the full modular group $$\mathrm{PSL}_{2}(\mathbb{Z})$$ is defined the same way as holomorphic modular form, only $$f$$ is allowed to have a finite number of negative powers in its $$q$$-expansion. Write $$k=12\ell+k'$$ with uniquely determined $$\ell\in\mathbb{Z}$$ and $$k'\in\{0,4,6,8,10,14\}$$. Let $$\mathcal{M}_{k}$$ be (an infinite dimensional) linear space of such forms of weight $$k$$. Then for each integer $$m\geq-\ell$$, there exists a unique $$f_{k,m}\in\mathcal{M}_{k}$$ with $$q$$-expansion of the form $$f_{k,m}(\tau)=q^{-m}+O(q^{\ell+1})$$, $$q=e^{2\pi i\tau}$$. Therefore, let $$f_{k,m}(\tau)=q^{-m}+\sum_{n\geq\ell+1}a_{k}(m,n)q^{n}$$. The function $$f_{k,m}$$ can be constructed explicitly as $$f_{k,m}=\Delta^{\ell}E_{k'}F_{k,\ell+m}(j)$$; here $$\Delta$$ is the discriminant cusp form, $$E_{k'}$$ is the normalized Eisenstein series of weight $$k'$$, $$j$$ stands for the $$j-$$invariant, and $$F_{k,\ell+m}$$ is a monic polynomial of degree $$\ell+m$$.
Generalizing the work of F. K. C. Rankin and H. P. F. Swinnerton-Dyer [Bull. Lond. Math. Soc. 2, 169–170 (1970; Zbl 0203.35504)] in case of Eisenstein series and the work by R. A. Rankin [Compos. Math. 46, 255–272 (1982; Zbl 0493.10034)] in case of Poincaré series, the authors of the current paper prove that if $$m\geq|\ell|-\ell$$, then all of the zeros of $$f_{k,m}$$ in $$\mathcal{F}$$ (standard fundamental domain) lie on the unit circle. This result does not hold in general if we drop a restriction on $$m$$. Further, if we set $$f_{k}=\Delta^{\ell}E_{k'}$$, ($$k=12\ell+k'$$ is an even integer), the authors show that $$\sum_{m\geq-\ell}f_{k,m}(z)q^{m}=\frac{f_{k}(z)f_{2-k}(\tau)}{j(\tau)-j(z)}$$. This implies $$a_{k}(m,n)=-a_{2-k}(n,m)$$ for all integers $$m,n$$ and any even integer $$k$$. Finally, when $$k\in\{4,6,8,10,14\}$$ (when the Fourier coefficients $$a_{k}(m,n)$$ are integers), the authors present division results for these coefficients, which imply recursive congruences for the Fourier coefficients of the $$j$$-invariant.

##### MSC:
 11F11 Holomorphic modular forms of integral weight 11F30 Fourier coefficients of automorphic forms 11F33 Congruences for modular and $$p$$-adic modular forms
Full Text: