ℓ-adic local systems and Higgs bundles: the generic case

2023/04/13 by Yu, Hongjie
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2304.06637

Abstract

Let X be a projective smooth geometrically connected curve defined over a finite field \mathbbF_q of cardinality q. Let S be a finite set of closed points of X. Let X and S be the base change of X, S to an algebraic closure. We consider the set of ℓ-adic (ℓ\nmid q) local systems of rank n over X-S with prescribed tame regular semisimple and generic ramifications in S. The genericity ensures that such an ℓ-adic local system is automatically irreducible. We show that the number of these ℓ-adic local systems fixed by Frobenius endomorphism equals the number of stable logarithmic Higgs bundles of rank n and degree e coprime to n, with a fixed residue, up to a power of q. In the split case, this number is equal to the number of stable parabolic Higgs bundles (with full flag structures) fixed by \mathbbG_m-action with generic parabolic weights.

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