Comptage des systèmes locaux ℓ-adiques sur une courbe

Abstract : Let X1 be a projective, smooth and geometrically connected curve over Fq with q = pn elements where p is a prime number, and let X be its base change to an algebraic closure of Fq.We give a formula for the number of irreducible ℓ-adic local systems (ℓ ≠ p) with a fixed rank over X fixed by the Frobenius endomorphism.We prove that this number behaves like a Lefschetz fixed point formula for a variety over Fq, which generalises a result of Drinfeld in rank 2 and proves a conjecture of Deligne. To do this, we pass to the automorphic side by Langlands correspondence, then use Arthur’s non-invariant trace formula and link this number to the number of Fq-points of the moduli space of stable Higgs bundles.
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Hongjie Yu. Comptage des systèmes locaux ℓ-adiques sur une courbe. Mathématiques générales [math.GM]. Université Sorbonne Paris Cité, 2018. Français. ⟨NNT : 2018USPCC057⟩. ⟨tel-02167864⟩

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