Twisted \(L\)-functions and monodromy (Q2781439)

The book under review deals with \(L\)-functions of \(\ell\)-adic étale sheaves over finite fields. The ``twisted'' in the title refers to tensoring them with a Kummer-sheaf depending on a parameter which lies in an open subset of the affine space. For a finite field \(k\) one can average over all \(k\)-points of this parameter space and compute asymptotics as the order of \(k\) approaches infinity. By the fundamental result of Deligne these are determiend by the arithmetic and geometric monodromy groups of \(\ell\)-adic sheaves on the parameter space. Thus the bulk of the book computes these monodromies, or better shows that they are big. They lie either between some \(\text{SL}(d)\) and \(\text{GL}(d)\), or are equal (in case the sheaves are self-dual) to either the orthogonal groups \(\text{SO}(d)\) or \(\text{O}(d)\), or to the symplectic group \(\text{SP}(d)\). These results are derived from considerations about the actions of inertia at the boundary. The author computes many examples, the most interesting aspect being the subtle difference between \(\text{SO}(d)\) and \(\text{O}(d)\).