Explicit local multiplicative convolution of \(\ell\)-adic sheaves (Q1989550)

Given two objects \(K, L \in D^b_c(\mathbb{G}_{m, \bar{\mathbb{F}}_q};\overline{\mathbb{Q}}_{\ell})\), their convolution is defined to be the object \[ K*L := R\mu_{!}(\text{pr}_1^*K\otimes^L\text{pr}_2^*L) \in D^b_c(\mathbb{G}_{m, \bar{\mathbb{F}}_q};\overline{\mathbb{Q}}_{\ell}), \] where \(\text{pr}_i: \mathbb{G}_{m, \bar{\mathbb{F}}_q} \times \mathbb{G}_{m, \bar{\mathbb{F}}_q} \to \mathbb{G}_{m, \bar{\mathbb{F}}_q} \) is the \(i\)-th projection and \(\mu: \mathbb{G}_{m, \bar{\mathbb{F}}_q} \times \mathbb{G}_{m, \bar{\mathbb{F}}_q} \to \mathbb{G}_{m, \bar{\mathbb{F}}_q} \) is the multiplication map. In [\textit{N. M. Katz}, Gauss sums, Kloosterman sums, and monodromy groups. Princeton: Princeton University Press (1988; Zbl 0675.14004)], Katz shows that, if \(K=\mathcal{F}[1]\) and \(L=\mathcal{G}[1]\) are smooth sheaves which are tamely ramified at 0 and totally wild at \(\infty\), then so is \(K*L\). Moreover, the local monodromy of \(K*L\) is completely determined by the local monodromies of \(K\) and \(L\). In this paper, the author gives an explicit formula for those local momodromies for a wide class of representations: twisting by Kummer sheaves and Artin-Schreier sheaves. As a corollary, this formula implies a formula of local Fourier transform, which recover a result of \textit{L. Fu} [Manuscripta Math. 133, No. 3--4, 409--464 (2010; Zbl 1206.14035)].