## Abstract

Let X_{1} be a curve of genus g, projective and smooth over F_{q}. Let S_{1} sub; 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 Ql-local systems on X - S. It maps to itself the subset of those classes for which the local monodromy at each s ε 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 Σn_{i}γ^{m}_{i} for suitable integers n_{i} and "eigenvalues" γ_{i}. We use Lafforgue to reduce the computation of T(X_{1}; S_{1}; n,m) to countingv 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.

Original language | English |
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Pages (from-to) | 921-982 |

Number of pages | 62 |

Journal | Annals of Mathematics |

Volume | 178 |

Issue number | 3 |

DOIs | |

State | Published - 2013 |