On the Ramanujan-Petersson conjecture for modular forms of half-integral weight (Q5919537)

Let \(k\) be an integer. The Ramanujan-Petersson Conjecture gives a nice arithmetic information about the growth of the Fourier coefficients of cusp forms of weight \(k\). This conjecture is proven by Pierre Deligne in 1973. The case \(k+1/2\), there is a similar conjecture but nothing has been proven yet. This kind of ``sharp'' inequalities provide some tools for progress of Sato-Tate like problems. Hence it is important to have some best possible bounds. In the paper under review, the authors show that the Ramanujan-Petersson bound is ``best possible'' for newforms of weight \(k+1/2\) in the Kohnen-plus space. More precisely, they show that if \(g \in S_{k+1/2} ^{+, \text{new}} (4N) \) with \(k \geq 1\) and \(N\) is odd and square-free and \(c(n)\) be the Fourier coefficients of \(g\) with \(n \geq 1\), then \[ \lim \sup_{n \to \infty} \frac{|c(n)|}{n^{k/2-1/4}} = \infty \] Using, the celebrated Sato-Tate Conjecture, they improve their main result in Theorem 3. Lastly, they obtain stronger results in Theorem 4 for the case \(g\) is an Hecke eigenform with real Fourier coefficients. The proofs are based on analytical number theory tools and they use the Sato-Tate conjecture efficiently on the calculations.