Counting local systems with tame ramification

We compute the cardinality of a set of Galois-invariant isomorphism classes of irreducible rank two Q¯_ℓ-smooth sheaves on X-S, where X is a smooth projective absolutely irreducible curve of genus g over a finite field F_q and S is a reduced divisor, with pre-specified tamely ramified ramification data at S. Properties of this cardinality are studied. The approach is based on using a relatively elementary explicit form of the trace formula for GL(2), and introducing new types of almost pseudo-coefficients of principal series and discrete series representations.