Recall that a Mumford–Tate domain DM is an open set in its compact dual ĎM — the latter is a rational homogeneous variety defined over Q (think of the upper half plane H ⊂ P1). Thus a polarized Hodge structure (PHS) φ has an “upstairs” field of definition k(φ). If H•,• ⊂ T •,• is a subalgebra then the Noether-Lefschetz locus NLH = {φ : Hg•,• φ ⊇ H} is defined over Q. Its components are defined ...