zbMATH — the first resource for mathematics

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.

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