We compute explicitly the cardinality of a set of Galois-invariant isomorphism classes of irreducible rank two \(\overline{\mathbb {Q}}_\ell \) -smooth sheaves on \(X-S\) , where X is a smooth projective absolutely irreducible curve of genus g over a finite field \(\mathbb {F}_q\) and S is a reduced divisor, with pre-specified tamely ramified ramification data at S, including at least two points \(S_1^{D^+}\) where the monodromy is principal unipotent. Properties of this cardinality are studied. In particular we show this number is geometric, thus has the form \(\sum _jn_j\gamma _j^m\) as \(\mathbb {F}_q\) changes to \(\mathbb {F}_{q^m}\) , \(m\in \mathbb {Z}_{\ge 1}\) , for suitable “multiplicities” \(n_j\) and “eigenvalues” \(\gamma _i\) . This is done when the cardinality of \(S_1^{D^+}\) is not only at least two – the case studied here – but also when it is at least one, and also zero, cases studied elsewhere. The approach is based on using the trace formula for an anisotropic form of \({\text {GL}}(2)\) , and using pseudo-coefficients of Steinberg, tamely ramified principal series and tamely ramified discrete series representations.
