Abstract

Let X be a smooth projective absolutely irreducible curve over F_q . Let S be a finite set of closed points of X of cardinality N > 1 . Put X^S = X – S . Let pi₁(X^S) be the arithmetic fundamental group of the affine curve X^S . It is Gal(F^S/F) , where F is the function field of X over F_q and F^S is the maximal extension of F unramified at each closed point of X^S inside a fixed separable closure \ov{F} of F. We compute, in terms of the zeta function of X, the number of equivalence classes of irreducible n-dimensional ell-adic representations of pi₁(X^S), whose local monodromy at each point of S is a single Jordan block of rank n, assuming N is even if n = 2, that n is a prime and (n,q) = 1. This number is reduced to that of the nowhere ramified cuspidal automorphic representations of the multiplicative group of a division algebra of degree n over F, which we compute using the trace formula.