Construction of relative motives with interesting étale realization using the middle convolution

We study the middle convolution of local systems on the punctured affine line in the singular and the étale case. We give a motivic interpretation of the middle convolution which yields information on the occurring determinants. Finally, we use these methods to realize special linear groups regularly as Galois groups over Q(t). Introduction If K is a field, then we set GK := Gal(K/K), where K denotes a separable closure of K. Let k be a number field. A profinite group G occurs regularly as Galois group over k(t), if one has a surjection φ : Gk(t) → G such that the restriction of φ to Gk̄(t) is still surjective. If G is a finite group, then φ corresponds to a Galois cover of Pk with Galois group G. The regular inverse Galois problem asks, whether every finite group occurs regularly as Galois group over Q(t). Our main motivation for studying the convolution stems from the fact that most of the regular Galois realizations for classical groups are obtained with methods, derived from Katz’ middle convolution functor MCχ, see [8], [5]. The functor MCχ can be viewed (under some restrictions, see [4]) as a functor from the category of local systems on the punctured affine line to itself. Using the properties of the Katz’ functor, one was able to obtain many new families of classical groups regularly as Galois groups over the rational function field ([5], [9]). In fact, all previously known examples can be explained using MCλ, except a few exceptional cases in low dimensions. Nevertheless, this method has its limitations: For example, It fails to produce groups like SLn(Fq) as Galois groups over Q(t). In this talk, we outline a more general approach to the middle convolution, in order to overcome some of the limitations of the