Year: 2025
Type: article
Abstract: We compute explicitly the cardinality of a set of Galois-invariant isomorphism classes of irreducible rank two $$\overline{\mathbb {Q}}_\ell $$ Q ¯ ℓ -smooth sheaves on $$X-S$$ X - S , where X is a smooth projective absolutely irreducible curve of genus g over a finite field $$\mathbb {F}_q$$ F q and S is a reduced divisor, with pre-specified tamely ramified ramification data at S , including at least two points $$S_1^{D^+}$$ 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 $$\sum _jn_j\gamma _j^m$$ ∑ j n j γ j m as $$\mathbb {F}_q$$ F q changes to $$\mathbb {F}_{q^m}$$ F q m , $$m\in \mathbb {Z}_{\ge 1}$$ m ∈ Z ≥ 1 , for suitable “multiplicities” $$n_j$$ n j and “eigenvalues” $$\gamma _i$$ γ i . This is done when the cardinality of $$S_1^{D^+}$$ 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 $${\text {GL}}(2)$$ GL ( 2 ) , and using pseudo-coefficients of Steinberg, tamely ramified principal series and tamely ramified discrete series representations.
Source: Transformation Groups
Author: Yuval Z. Flicker
Institutions: Ariel University, The Ohio State University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
OpenAlex ID: W4409966016