## Abstract

Let F_{q} be a finite field with q elements, F = F_{q} an algebraic closure of F_{q}, X_{1} an absolutely irreducible projective smooth curve over F_{q}, and S_{1} a finite set of closed points of X_{1}. Let N_{1} be the cardinality of S_{1}. Erasing the index 1 indicates extension of scalars to F. Replacing it by m indicates extension of scalars to F_{q}^{m} ⊂ F. Let Fr be the Frobenius endomorphism of the curve X over F, deduced from the F_{q}-form X_{1} of X. For l a prime not dividing q, the pullback by Fr is an autoequivalence of the category of smooth l-adic sheaves (=ℚ_{l}-local systems) on X - S. It induces a permutation Fr∗ of the set of isomorphism classes of irreducible smooth l-adic sheaves of rank n on X - S, whose local monodromy at each s ϵ S is unipotent with a single Jordan block (= principal unipotent). Let T (X_{1}, S_{1}, n) be the number of fixed points of Fr∗ acting on this set. We will also consider the number of fixed points T (X_{1}, S_{1}, n, m) of the iterate Fr^{∗,m}; of Fr∗. As Fr^{∗,m}; : X → X is the Frobenius, for X_{1}=F_{q} replaced by X_{m}/F_{qm}, one has T (X_{1}, S_{1}, n, m) = T (X_{m}, S_{m}, n). The number T (X_{1}, S_{1}, n) is computed when N_{1} ≥ 2 in [4] where it is shown that the T (X_{1}, S_{1}, n, m) are given by a formula reminiscent of a Lefschetz fixed point formula. The automorphic-Galois reciprocity is used there to reduce the computation of T (X_{1}, S_{1}, n) to counting automorphic representations on GL(n), and the assumption N_{1} ≥ 2 to move the counting to a division algebra, where the trace formula is easier to use. The trace formula used in Section 4 of [4] is based on a new, categorical approach to computing the trace formula in the very special case of counting that is needed there. It is not yet known if it extends to the use of Hecke correspondences and in the non-compact quotient case. Our aim here is to give an alternative proof of Section 4 of [4] using the usual trace formula in the compact quotient case. As stated in [4], after Lemma 4.9, the equivalence between the two approaches seems to us interesting.

Original language | English |
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Journal | Journal fur die Reine und Angewandte Mathematik |

Volume | 2017 |

DOIs | |

State | Published - Oct 2017 |