Calculation of \(\ell \)-adic local Fourier transformations
The global \(\ell\)-adic Fourier transform was first introduced by Deligne. To study the local behavior of the global Fourier transform, Laumon discovered the stationary phase principle and introduced local Fourier transformations. Laumon and Malgrange gave conjectural formulas of local Fourier transformations for a certain class of \(\overline{\mathbb Q}_{\ell}\)-sheaves. In this paper the local Fourier transformations are calculated for a more general class of \(\overline{\mathbb Q}_{\ell}\)-sheaves; in particular, the conjecture of Laumon and Malgrange (in fact: a slight correction of it) is proved. As an application, the local monodromy of \(\ell\)-adic hypergeometric sheaves as introduced by Katz is calculated. The methods used in this paper can also be used to show that the Fourier transform of an analytic sheaf with meromorphic ramification still has meromorphic ramification, cf. [\textit{L. Ramero}, J. Algebr. Geom. 7, No. 3, 405--504 (1998; Zbl 0964.14019)]. Some of the results in this paper, and in fact more general versions of them, have also been obtained by \textit{A. Abbes} and \textit{T. Saito} [Compos. Math. 146, No. 6, 1507--1551 (2010; Zbl 1206.14034)], but the methods in the present paper are more direct and global.
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