Euler-Poincaré functions and local systems

We develop here a new proof – using the trace formula – of a crucial reduction step done first in [7, section 5] using the geometric mass of a category technique. The result that we reprove is a count of the equivalence classes of local systems of rank n on a curve X1 over a finite field Fq, fixed by the Frobenius, with principal unipotent monodromy at least at two points; or rather the corresponding automorphic representations of GL(n,A) over the function field F of X1 that have unramified components except for a Steinberg twisted by an unramified character at least at two places. The case where there is a supporting division algebra is treated in [7, section 4] by the same geometric technique; a trace formula proof of that case is carried out in [11]. The case that there is no supporting division algebra is reduced to the case where there is such a division algebra. The trace formula is used then to reduce the general case (only principal unipotent monodromy, at least at two places) to complete the special case of [11], also done by the trace formula. The technique employs the Euler-Poincaré function of Serre, which is a pseudo-coefficient of the Steinberg representation, in view of computations of continuous cohomology of admissible representations. For completeness we review the result and the steps in its proof, anticipating the use of the technique in cases of other types of ramification. In an appendix we draw attention to an open global rank two problem from [12, section 8], give a shorter proof of adetail and discuss a number theoretic analogue.