On the pro-semisimple completion of the fundamental group of a smooth variety over a finite field

2015/09/20 by Drinfeld, Vladimir · 1 citation
#11G35 (Primary) #14G15 #20G05 (Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1509.06059

Abstract

Let Π be the fundamental group of a smooth variety X over F_p. Given a non-Archimedean place λ of the field of algebraic numbers which is prime to p, consider the λ-adic pro-semisimple completion of Π as an object of the groupoid whose objects are pro-semisimple groups and whose morphisms are isomorphisms up to conjugation by elements of the neutral connected component. We prove that this object does not depend on λ. If dim X=1 we also prove a crystalline generalization of this fact. We deduce this from the Langlands conjecture for function fields (proved by L. Lafforgue) and its crystalline analog (proved by T. Abe) using a reconstruction theorem in the spirit of Kazhdan-Larsen-Varshavsky. We also formulate two related conjectures, each of which is a "reciprocity law" involving a sum over all l-adic cohomology theories (including the crystalline theory for l=p).

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• G-companions on algebraic stacks and applications to canonical ℓ-adic local systems on Shimura stacks

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