Galois realizations of classical groups and the middle convolution

Abstract

We study the middle convolution of local systems on the punctured affine line in the setting of singular cohomology and in the setting of \'etale 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 \'etale 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 ${\bf Q}(t).$