On the signs of Fourier coefficients of cusp forms
Let \(\Gamma\) be a finitely generated Fuchsian group of the first kind and assume that both \(i\infty\) and \(0\) are parabolic fixed points of \(\Gamma\). Theorem 1: If \(f\) is a non-zero cusp form (of arbitrary weight and multiplier system) on \(\Gamma\) whose Fourier coefficients \(a(n)\) are real for all \(n\), then the sequence of coefficients has infinitely many sign changes. A stronger result holds for a restricted class of groups. Namely, suppose that \(-r\) is a parabolic fixed point of \(\Gamma\) if and only if \(r\) is. This holds true if \(\Gamma\) is a subgroup of finite index in \(\text{SL}_2(\mathbb Z)\) or if \(\Gamma\) is invariant with respect to the involution \[ \begin{pmatrix} a & b\\ c & d\end{pmatrix}\mapsto \begin{pmatrix} a & -b\\ -c & d\end{pmatrix}. \] Theorem 2: Let \(a(n)\) be the Fourier coefficients of a non-zero cusp form \(f\) on a group \(\Gamma\) as specified. Then for any real number \(\varphi\), the sequence of real parts of \(e^{-i\varphi}a(n)\) either is identically 0, or it has infinitely many sign changes. The authors give transparent proofs of their results, with some lucid comments, and they add interesting remarks on related results which shed some light on well-known facts. For example, the classical Eisenstein series \(G_k\) of weight \(k\) have only positive coefficients if \(k\equiv 0\pmod 4\), and there is exactly one sign change if \(k\equiv 2\pmod 4\).
scientific article; zbMATH DE number 2022032