Counting Tame Local Systems by Anisotropic Tools

Abstract We compute explicitly the cardinality of a set of Galois-invariant isomorphism classes of irreducible rank two $$\overline{\mathbb {Q}}_\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>Q</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>ℓ</mml:mi> </mml:msub> </mml:math> -smooth sheaves on $$X-S$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>-</mml:mo> <mml:mi>S</mml:mi> </mml:mrow> </mml:math> , where X is a smooth projective absolutely irreducible curve of genus g over a finite field $$\mathbb {F}_q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:math> and S is a reduced divisor, with pre-specified tamely ramified ramification data at S , including at least two points $$S_1^{D^+}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> <mml:msup> <mml:mi>D</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:msubsup> </mml:math> where the monodromy is principal unipotent. Properties of this cardinality are studied. In particular we show this number is geometric, thus has the form $$\sum _jn_j\gamma _j^m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mo>∑</mml:mo> <mml:mi>j</mml:mi> </mml:msub> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:msubsup> <mml:mi>γ</mml:mi> <mml:mi>j</mml:mi> <mml:mi>m</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> as $$\mathbb {F}_q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:math> changes to $$\mathbb {F}_{q^m}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:msup> <mml:mi>q</mml:mi> <mml:mi>m</mml:mi> </mml:msup> </mml:msub> </mml:math> , $$m\in \mathbb {Z}_{\ge 1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>Z</mml:mi> <mml:mrow> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , for suitable “multiplicities” $$n_j$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:math> and “eigenvalues” $$\gamma _i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> . This is done when the cardinality of $$S_1^{D^+}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> <mml:msup> <mml:mi>D</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:msubsup> </mml:math> 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 $${\text {GL}}(2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>GL</mml:mtext> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , and using pseudo-coefficients of Steinberg, tamely ramified principal series and tamely ramified discrete series representations.