Abstract
We compute 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. Properties of this cardinality are studied. The approach is based on using a relatively elementary explicit form of the trace formula for \({\text {GL}}(2)\), and introducing new types of almost pseudo-coefficients of principal series and discrete series representations.
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Flicker, Y.Z. Counting local systems with tame ramification. manuscripta math. 173, 771–830 (2024). https://doi.org/10.1007/s00229-023-01477-4
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DOI: https://doi.org/10.1007/s00229-023-01477-4