On a conjecture of Igusa. II (Q2800426)

Let \(\displaystyle P\in \mathbb{Z}\left[X_1, X_2, X_3\right]\) be a normal ternary form of degree at least \(3\). Define the normalized complete exponential sum modulo a prime power \(p^r\) as follows: NEWLINE\[NEWLINE \mathcal{F}\left(w/p^r\right):=p^{-nr}\sum_{\mathbf{x}\in\left(\mathbb{Z}/p^r\right)^n} \text{e}^{\frac{2\pi iwP(\mathbf{x})}{p^r}}, NEWLINE\]NEWLINE where \(w\) is coprime to \(p\).NEWLINENEWLINEThe author proves that there exist a constant \(C\) and a finite set \(\mathcal{B}\) of primes such that \(p\not\in \mathcal{B}\) implies NEWLINE\[NEWLINE \left|\mathcal{F}\left(w/p^r\right)\right|< Cr^{\kappa}p^{-r\vartheta_P}, \qquad \text{ for all }r\geq 1, NEWLINE\]NEWLINE where \(\kappa\in \{0,1,2\}\) depends upon \((p,P)\), \(\vartheta_P\) depends only on \(P\).NEWLINENEWLINENEWLINEFor Part I, see [the author, Mathematika 59, No. 2, 399--425 (2013; Zbl 1335.11067)].