Around $\ell$-independence. (17th October 2017)

• Record Type: Journal Article
• Authors: Chiarellotto, Bruno; Lazda, Christopher
• Is Part Of: Compositio mathematica. Volume 154:Number 1(2018)
• Page Start: 223
• Page End: 248
• Publication Date: 2017-10-17
• Subjects: 11G20 (primary), -- 14F99 (secondary); local function fields, -- cohomology, -- L-functions, -- motives, -- unipotent fundamental groups; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X17007527
• Languages: English
• ISSNs: 0010-437X

Abstract

Abstract : In this article we study various forms of$\ell$ -independence (including the case$\ell =p$ ) for the cohomology and fundamental groups of varieties over finite fields and equicharacteristic local fields. Our first result is a strong form of$\ell$ -independence for the unipotent fundamental group of smooth and projective varieties over finite fields. By then proving a certain 'spreading out' result we are able to deduce a much weaker form of$\ell$ -independence for unipotent fundamental groups over equicharacteristic local fields, at least in the semistable case. In a similar vein, we can also use this to deduce$\ell$ -independence results for the cohomology of smooth and proper varieties over equicharacteristic local fields from the well-known results on$\ell$ -independence for smooth and proper varieties over finite fields. As another consequence of this 'spreading out' result we are able to deduce the existence of a Clemens–Schmid exact sequence for formal semistable families. Finally, by deforming to characteristic$p$, we show a similar weak version of$\ell$ -independence for the unipotent fundamental group of a semistable curve in mixed characteristic.