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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