Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4584720 | Journal of Algebra | 2014 | 18 Pages |
Abstract
Let k be a finitely generated field of characteristic 0 and â a prime. Let S be a smooth, separated and geometrically connected scheme over k and let Ï:Ï1(S)âGLr(Zâ) be an â-adic representation of the étale fundamental group of S. Let G and G¯ denote the images of Ï1(S) and Ï1(Sk¯) respectively. Given a closed point sâS, we write Gs for the image of the absolute Galois group Îk(s) of the residue field k(s) viewed as a decomposition group at s in Ï1(S). By previous works of the authors, it is known that, when S is a curve, for all dâ¥1 and all but finitely many sâS with [k(s):k]â¤d, the codimension of Gs in G is at most 2. In this note, we improve this rigidity result as follows. Write g, g¯, gs for the Lie algebras of G, G¯, Gs respectively. Then for all but finitely many sâS with [k(s):k]â¤d, one of the following holds: (i) the codimension of Gs in G is at most 1 and gs contains D(g¯):=[g¯,g¯]; or (ii) the codimension of Gs in G is 2 and gs contains D2(g¯):=D(D(g¯)). We also obtain an arithmetic variant of this result, which involves the derived series of g.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Anna Cadoret, Akio Tamagawa,