Counting local systems via the trace formula

Let F_q be a finite field with q elements, F = F_q an algebraic closure of F_q, X₁ an absolutely irreducible projective smooth curve over F_q, and S₁ a finite set of closed points of X₁. Let N₁ be the cardinality of S₁. 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 X₁ 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 (X₁, S₁, n) be the number of fixed points of Fr∗ acting on this set. We will also consider the number of fixed points T (X₁, S₁, n, m) of the iterate Fr^∗,m; of Fr∗. As Fr^∗,m; : X → X is the Frobenius, for X₁=F_q replaced by X_m/F_qm, one has T (X₁, S₁, n, m) = T (X_m, S_m, n). The number T (X₁, S₁, n) is computed when N₁ ≥ 2 in [4] where it is shown that the T (X₁, S₁, 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 (X₁, S₁, n) to counting automorphic representations on GL(n), and the assumption N₁ ≥ 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.