On the Ramanujan-Petersson conjecture for modular forms of half-integral weight (Q5906568)

Let \(\Gamma_0 (4N)\) \((N \in \mathbb{N})\) be a congruence subgroup defined by \[ \left \{{ a\;b \choose c\;d} \in M_2 (\mathbb{Z}) \mid c \equiv 0 \text{mod} 4N \right\}. \] Let \(f\) be a cusp form of half-integral weight \(k + 1/2 \geq 3/2\) on the group \(\Gamma_0 (4N)\) and denote by \(a(n)\) \((n \in \mathbb{N})\) its Fourier coefficients. The paper begins with a conjecture: \[ a(n) \ll_{f, \varepsilon} n^{(k - 1/2)/2 + \varepsilon} (\varepsilon > 0). \tag{1} \] Estimate (1) is known as the Ramanujan- Petersson conjecture for modular forms of half-integral weight. The paper remarks that it is sufficient to prove (1) for \(n\) square free. The author illustrates that (1) is equivalent to an inequality (2) \(s_{n,k} \ll_{k, \varepsilon} n^\varepsilon\) \((\varepsilon > 0\); \(n\) squarefree), where \(s_{n,k}\) is a certain infinite sum. Next the author proves the estimate: \[ s^*_{n,k} \ll_{k, \varepsilon} n^\varepsilon\;(\varepsilon > 0;\;n \text{ squarefree)}, \tag{3} \] where \(s^*_{n,k}\) is defined by a similar manner to \(s_{n,k}\) but not identical. The estimate (3) is a very approximate (mentally) one to (2). The author comments that (3) can be deduced from Deligne's theorem. The arguments are restricted to the case where \(N = 1\) and in a narrowed situation, for the sake of simplicity.