We calculate the local Fourier transformations for a class of $\bar{\mathbbQ}_\ell$-sheaves. In particular, we verify a conjecture of Laumon andMalgrange. As an application, we calculate the local monodromy of $\ell$-adichypergeometric sheaves introduced by Katz. We also discuss the characteristic$p$ analogue of the Turrittin-Levelt Theorem for $D$-modules. The method usedin this paper can be used to show a conjecture of Ramero which states that theFourier transformation of an analytic sheaf with meromorphic ramification stillhas meromorphic ramification.

Calculation of l-adic local Fourier transformations