Calculation of l-adic local Fourier transformations

TL;DR

This paper computes local Fourier transforms for certain sheaves, verifies conjectures by Laumon and Malgrange, and explores applications to hypergeometric sheaves and the Turrittin-Levelt Theorem in characteristic p.

Contribution

It provides explicit calculations of local Fourier transforms for $ar{Q}_ ext{ m iny $lacksquare$}}$-sheaves and verifies key conjectures, advancing understanding of $ ext{ m iny $lacksquare$}}$-sheaves and ramification.

Findings

Verified Laumon and Malgrange's conjecture.

Calculated local monodromy of $ ext{ m iny $lacksquare$}}$-sheaves.

Discussed characteristic p analogue of Turrittin-Levelt Theorem.

Abstract

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