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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 PoincarĂ© 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
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