Galois Realizations of Classical Groups and the Middle Convolution

Up to date, the middle convolution is the most powerful tool for realizing classical groups as Galois groups over Q(t). We study the middle convolution of local systems on the punctured affine line in the setting of singular cohomology and in the setting of étale cohomology. We derive a formula to compute the topological monodromy of the middle convolution in the general case and use it to deduce some irreducibility criteria. Then we give a geometric interpretation of the middle convolution in the étale setting. This geometric approach to the convolution and the theory of Hecke characters yields information on the occurring arithmetic determinants. We employ these methods to realize special linear groups regularly as Galois groups over Q(t). The geometric theory of the middle convolution can also be used to compute Frobenius elements for many of the known Galois realizations of classical groups. We illustrate this by investigating specializations of PGL2(Fl)extensions of Q(t) which are associated to a family of K3-surfaces of Picard number 19.