Abelian part of compatible system and geometry of roots

Chun Yin Hui

Published 2016-03-03Version 1

Let {V_l} be a strictly compatible rational system of Galois representations arising from geometry. Let V_l^ss be the semisimplification of V_l, V_l^ab the maximal abelian subrepresentation of V_l^ss, and G_l the algebraic monodromy group of V_l^ss. Conjecturally, the abelian part {V_l^ab} is again a strictly compatible rational system (Conj. 1.2). The conjecture has meaningful consequences to arithmetic geometry. The main theme of this paper is to investigate the conjecture via l-independence of G_l. More precisely, we prove that the conjecture follows if (i) G_l is connected for all l, (ii) the tautological representation of G_l^der is independent of l, and (iii) the roots of G_l meet a criteria (Thm. 1.3). These conditions are satisfied under some group theoretic assumptions of G_l (Thm. 1.5), for example, (ii) and (iii) hold if G_l^der(C) is connected and semisimple of type A_4+A_6+A_6 for some l; (iii) holds if G_l^der(C) is almost simple of type different form A_7,A_8,B_4,D_8.