Local systems with tame, and a unipotent, local monodromy

We compute the cardinality of a set of Galois-invariant isomorphism classes of irreducible rank two Q¯ _ℓ -smooth sheaves on X₁- S₁ , where X₁ is a smooth projective absolutely irreducible curve of genus g over a finite field F_q and S₁ is a reduced divisor, with pre-specified tamely ramified monodromy data at S, including precisely one point of principal unipotent monodromy, twisted by a tame character. Equivalently, we compute the number of the corresponding automorphic representations. The approach is based on using an explicit form of the trace formula for GL (2) , extending the work “Counting local systems with tame ramification” to include a Steinberg (= special) component, twisted by a tame character, by employing a pseudo-coefficient thereof.