Resumen de Counting local systems with principal unipotent local monodromy

Pierre Deligne, Yuval Z. Flicker

• Let X 1 be a curve of genus g , projective and smooth over F q . Let S 1 ?X 1 be a reduced divisor consisting of N 1 closed points of X 1 . Let (X,S) be obtained from (X 1 ,S 1 ) by extension of scalars to an algebraic closure F of F q . 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(X 1 ,S 1 ,n,m) be the number of fixed points of Fr *m acting on this subset. Under the assumption that N 1 =2 , we show that T(X 1 ,S 1 ,n,m) is given by a formula reminiscent of a Lefschetz fixed point formula: the function m?T(X 1 ,S 1 ,n,m) is of the form ?n i ? m i for suitable integers n i and �eigenvalues� ? i . We use Lafforgue to reduce the computation of T(X 1 ,S 1 ,n,m) to counting automorphic representations of GL(n) , and the assumption N 1 =2 to move the counting to the multiplicative group of a division algebra, where the trace formula is easier to use.