A General Geometric Fourier Transform Convolution Theorem

Fourier transform and middle convolution for irregular D-modules

S.Block and H.Esnault constructed the local Fourier transform for D-modules. We present a different approach to the local Fourier transform, which makes its properties almost tautological. We apply the local Fourier transform to compute the local version of Katz's middle convolution.

On the $\ell$-adic Fourier transform and the determinant of the middle convolution

We study the relation of the middle convolution to the $\ell$-adic Fourier transformation in the étale context. Using Katz' work and Laumon's theory of local Fourier transformations we obtain a detailed description of the local monodromy and the determinant of Katz' middle convolution functor $\MC_χ$ in the tame case. The theory of local $ε$-constants then implies that the property of an étale sheaf of having an at most quadratic determinant is often preserved under $\MC_χ$ if $χ$ is quadratic.

Calculation of local Fourier transforms for formal connections

We calculate the local Fourier transforms for formal connections. In particular, we verify an analogous conjecture suggested in Laumon's paper: "Transformation de Fourier, constantes d'equations fonctionnelles et conjecture de Weil, 2.6.3".

有限域上的算术傅里叶变换：一般性消没、卷积与等分布

Arithmetic Fourier transforms over finite fields: generic vanishing, convolution, and equidistribution

本文研究一般连通交换代数群上迹函数的算术傅里叶变换，通过证明通用消没定理并构造Tannakian范畴，利用Deligne的黎曼假设导出等分布定理，推广了Katz的工作，并给出应用实例与开放问题。

Local Fourier transform and epsilon factors

Laumon introduced the local Fourier transform for $\ell$-adic Galois representations of local fields, of equal characteristic $p$ different from $\ell$, as a powerful tool to study the Fourier-Deligne transform of $\ell$-adic sheaves over the affine line. In this article, we compute explicitly the local Fourier transform of monomial representations satisfying a certain ramification condition, and deduce Laumon's formula relating the epsilon factor to the determinant of the local Fourier transform under the same condition.