par Deligne, Pierre ;Flicker, Yuval Z.
Référence Annals of mathematics, 178, 3, page (921-982)
Publication Publié, 2013
Article révisé par les pairs
Résumé : Let X1 be a curve of genus g, projective and smooth over Fq. Let S1⊂X1 be a reduced divisor consisting of N1 closed points of X1. Let (X,S) be obtained from (X1,S1) by extension of scalars to an algebraic closure F of Fq. Fix a prime l not dividing q. The pullback by the Frobenius endomorphism Fr of X induces a permutation Fr∗ of the set of isomorphism classes of rank n irreducible Q¯¯¯¯l-local systems on X−S. It maps to itself the subset of those classes for which the local monodromy at each s∈S is unipotent, with a single Jordan block. Let T(X1,S1,n,m) be the number of fixed points of Fr∗m acting on this subset. Under the assumption that N1≥2, we show that T(X1,S1,n,m) is given by a formula reminiscent of a Lefschetz fixed point formula: the function m↦T(X1,S1,n,m) is of the form ∑niγmi for suitable integers ni and “eigenvalues” γi. We use Lafforgue to reduce the computation of T(X1,S1,n,m) to counting automorphic representations of GL(n), and the assumption N1≥2 to move the counting to the multiplicative group of a division algebra, where the trace formula is easier to use.