Abelian varieties, l-adic representations, and l-independence

In the case that K is a number field, this conjecture goes back almost thirty years. The Lie algebra analogue has been formulated by J. Tate [18], D. Mumford [10], and J.-P. Serre [13]. A more precise conjecture involves comparison with the singular homology group H1(A(C),Q) for a fixed embedding K ⊂ C. If G∞ denotes the associated Hodge group (cf. §4), the “Mumford-Tate” conjecture states that the comparison isomorphism induces an isomorphism G` ∼= G∞ ×Q` for every `. Serre’s conjecture [14] C.3.3, which is phrased in the language of algebraic groups, is even more precise.