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Counting Galois representations with Steinberg monodromy

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Let X be a smooth projective absolutely irreducible curve over Fq . Let S be a finite set of closed points of X of cardinality N > 1 . Put XS = X – S . Let pi1(XS) be the arithmetic fundamental group of the affine curve XS . It is Gal(FS/F) , where F is the function field of X over Fq and FS is the maximal extension of F unramified at each closed point of XS 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(XS), 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.

This talk is part of the Number Theory Seminar series.

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