Signs of Fourier coefficients of half-integral weight modular forms (Q2664162)

Let \(g(z):= \sum_{n \geq 1} c(n) n^{\frac{k-1/2}{2}} e(nz) \) be a half-integral weight \(k+1/2\) modular form of level \(4\) in Kohnen-plus space and \(k \geq 2\). Suppose that \(g(z)\) is normalized for all \(n \geq 1\) such that it has real Fourier coefficients. The arithmetic of Fourier coefficients receives much attention and it is widely studied. For instance, as in the case of the paper under review, th ecelebrated Waldspurger's theorem relates \(|c(|d|)|^2\) with the central value of an \(L\)-function. That means the magnitude of the \(L\)-function essentially determines the size of the coefficient \(c(n)\). Of course, in this case, the signs of \(c(n)\) remains as an open question. In the paper under review, it is aimed to understand the sign of such coefficients at fundamental discriminants through examining the number of coefficients which are positive (respectively negative) as well as the number of sign changes in-between. Conditionally on the Generalized Riemann Hypothesis (GRH), the authors show that \(c(n) < 0\) and respectively \(c(n) > 0\) holds for a positive proportion of fundamental discriminants \(n\). (Theorem 1). There are two unconditional results in the paper which are weaker than above: (i) For any \(\varepsilon\) and all \(X\) sufficiently large the sequence \(\{ c(n) \}_{n \in \mathbb{N}_{g}^{\flat} (X) }\) has \(\gg X^{1-\varepsilon}\) sign changes where \(\mathbb{N}_{g}^{\flat} (X):=\{ n \in \mathbb{N}^{\flat} \cap [1, X] : c(n) \neq 0 \}\) and \(\mathbb{N}^\flat\) denotes the set of fundamental discriminants of the form \(8n\) with \(n > 0\) odd, square-free. (Theorem 2) (ii) For all sufficiently large \(X\), \[ \# \{ n \leq X : c(n) \lessgtr 0 \} \gg \frac{X}{\log X} .\] (Theorem 3) Finally they work the same problem for higher levels, more precisely they discuss the same problem for general half-integral weight forms \(g\) of level \(4N\) with \(N\) odd, square-free. The proof of Theorem 1 is based on effective use of Waldspurger's theorem in sense of \textit{W. Kohnen} and \textit{D. Zagier} [Invent. Math. 64, 175--198 (1981; Zbl 0468.10015)]. Classical analytical tools are used in the proof of Theorem 2 and 3.