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Zeros of certain modular functions and an application. (English) Zbl 0907.11013
Let \(j(\tau)\) be the elliptic modular invariant, and let \(\varphi_n(j)=n(j-744)| T(n)\) be the monic polynomial obtained from \(j-744\) by the action of the Hecke operator \(T(n)\). Theorem: The zeros of \(\varphi_n(j)\) are simple and lie in the interval \(0<j<1728\); the zeros of \(\varphi_n (j(\tau))\) in the standard fundamental domain lie on the unit circle.
The method of proof is adapted from R. Rankin and H. P. F. Swinnerton-Dyer [Bull. Lond. Math. Soc. 2, 169-170 (1970; Zbl 0203.35504)], who proved an analogous result for the Eisenstein series \(E_k(\tau)\). Another theorem gives an expansion formula for certain Green’s kernel functions in the sense of M. Eichler [Lect. Notes Math. 320, 75-151 (1973; Zbl 0258.10013)]. An application of both results yields: Let \(1/j(\tau)=\sum^\infty_{n=1} r(n)q^n\), \(q=e^{2\pi i\tau}\); then \((-1)^{n-1}r(n)\) is a positive integer for every \(n\). The method provides similar results (positive resp. alternating coefficients) for some other meromorphic modular forms.

MSC:
11F03 Modular and automorphic functions
11F30 Fourier coefficients of automorphic forms
11F11 Holomorphic modular forms of integral weight
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