Vol. 13, No. 6, 2019

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Algebraic monodromy groups of $l$-adic representations of Gal$(\overline{\mathbb{Q}} /\mathbb{Q})$

Shiang Tang

Vol. 13 (2019), No. 6, 1353–1394
DOI: 10.2140/ant.2019.13.1353
Abstract

In this paper we prove that for any connected reductive algebraic group G and a large enough prime l, there are continuous homomorphisms

Gal( ̄) G( ̄l)

with Zariski-dense image, in particular we produce the first such examples for SLn,Sp2n,Spinn,E6sc and E7sc. To do this, we start with a mod-l representation of Gal( ̄) related to the Weyl group of G and use a variation of Stefan Patrikis’ generalization of a method of Ravi Ramakrishna to deform it to characteristic zero.

Keywords
Galois representation, Galois deformation theory
Mathematical Subject Classification 2010
Primary: 11F80
Milestones
Received: 15 June 2018
Revised: 18 October 2018
Accepted: 20 November 2018
Published: 18 August 2019
Authors
Shiang Tang
Department of Mathematics
University of Utah
Salt Lake City, UT
United States