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 , D. Mumford , and J.-P. Serre . 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  C.3.3, which is phrased in the language of algebraic groups, is even more precise.