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Counting local systems with principal unipotent monodromy

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Non-Abelian Fundamental Groups in Arithmetic Geometry

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. Let n>1 be an integer. Assume N is even if n is even and n>2. Fix a prime ell with (ell,q)=1. We compute, jointly with P. Deligne, in terms of the zeta function of the curves X_m=Xotimes_{F_q}F_{qm} over F_{qm} obtained from X, where m|n, the number of equivalence classes of irreducible rank n ell-adic local systems on (X-S)otimes_{F_q}ov{F_q}, namely n-dimensional ell-adic representations of pi_1(X^Sotimes_{F_q}ov{F_q}), invariant under the Frobenius, whose local monodromy at each point of S is a single Jordan block of rank n. This number is reduced to that of the nowhere ramified infinite dimensional cuspidal automorphic representations of the multiplicative group of a division algebra of degree n over F=F_q(X), which we compute using the trace formula. This number is shown to be the trace of Frobenius of a virtual motive on a suitable moduli stack over Z of curves X and sets S of points on X.

This talk is part of the Isaac Newton Institute Seminar Series series.

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