On the ℓ\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 \'etale 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_\chi in the tame case. The theory of local $\epsilon$-constants then implies that the property of an \'etale sheaf of having an at most quadratic determinant is often preserved under \MC_\chi if $\chi$ is quadratic

On globally nilpotent differential equations

In a previous work of the authors, a middle convolution operation on the category of Fuchsian differential systems was introduced. In this note we show that the middle convolution of Fuchsian systems preserves the property of global nilpotence. This leads to a globally nilpotent Fuchsian system of rank two which does not belong to the known classes of globally nilpotent rank two systems. Moreover, we give a globally nilpotent Fuchsian system of rank seven whose differential Galois group is isomorphic to the exceptional simple algebraic group of type $G_2.

Rigid G2-Representations and motives of Type G2

We prove an effective Hilbert Irreducibility result for residual realizations of a family of motives with motivic Galois group G2