Counting Tame Local Systems by Anisotropic Tools: In memory of Heini Halberstam, 1926-2014

We compute explicitly 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 Fq and S is a reduced divisor, with pre-specified tamely ramified ramification data at S, including at least two points S1D+ where the monodromy is principal unipotent. Properties of this cardinality are studied. In particular we show this number is geometric, thus has the form ∑jnjγjm as Fq changes to Fqm, m∈Z≥1, for suitable “multiplicities” nj and “eigenvalues” γi. This is done when the cardinality of S1D+ 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 GL(2), and using pseudo-coefficients of Steinberg, tamely ramified principal series and tamely ramified discrete series representations.