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Zeros of certain weakly holomorphic modular forms for the Fricke group \(\varGamma_0^+(3)\). (English) Zbl 1465.11115

In this article, the authors determine the location of zeros of weakly holomorphic modular forms for the Fricke group \(\Gamma_0^+(3)\) of level \(3\) generated by the elements of the group \(\Gamma_0(3)\) and the Fricke involution \(W_3\). Hitherto, for example, the location of zeros was determined in the following cases: the Eisenstein series for \(\mathrm{SL}_2(\mathbb Z)\), weakly holomorphic modular forms with poles only at the cusps equivalent to the cusp \(i\infty\) for the congruence subgroups \(\Gamma_0(N)\) (\(1\le N\le 5\)), and weakly holomorphic modular forms for the Fricke group of level \(2\). A weakly holomorphic modular form is an automorphic form which is holomorphic outside the cusps. Since \(\Gamma_0^+(3)\) has only one cusp, the space of weakly holomorphic modular forms of weight \(k\in 2\mathbb Z\) has a natural basis consisted of the forms \(f_{k,m},m\ge-2\ell_k-e_k\) with the following \(q\)-expansion:\(f_{k,m}(z)=q^{-m}+O(q^{2\ell_k+e_k+1})\), where \(q=e^{2\pi iz}\),\(\ell_k=[k/12]\) and \(e_k=0,1\). The authors construct a natural basis \(\{f_{k,m}\}_m\) by using the eta function and the Eisenstein series explicitly and give an integral representation of \(f_{k,m}\). Let \(F^+(3)\) be the fundamental domain of \(\Gamma_0^+(3)\) such that \(S=\{e^{i\theta}/\sqrt{3}\mid \pi/2\le\theta\le 5\pi/6\}\) is the lower boundary arc. The authors show that all zeros of \(f_{k,m}\) in \(F^+_0(3)\) lie on \(S\), if \(m\ge 18|\ell_k|+23\), by estimating the difference of two functions \(e^{ik\theta/2-g(\theta)}f_{k,m}(e^{i\theta}/\sqrt{3})\) and \(2\cos(k\theta/2-g(\theta))\), where \(g(\theta)=(2\pi m \cos \theta)/\sqrt{3}\). This estimation is obtained from the integral representation of \(f_{k,m}\) with deliberate and detailed arguments.

MSC:

11F37 Forms of half-integer weight; nonholomorphic modular forms
11F03 Modular and automorphic functions
11F11 Holomorphic modular forms of integral weight
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References:

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