What kind of geometric object do the conjectures talk about really? Even in the classical formulation, we always need to take motivic cohomology of an integral model. Scholbach’s thesis, discussion of motives over Z in old and new sense. Why not just work with schemes over Z? (However, note that Scholbach can spread out a motive over Q)
Periods: Do we need to mention Kontsevich-Zagier???
Beilinson ideas on compatifying Z, various cohomology functors (maybe ass. to primes??), Arakelov motives, l.e.s. involving regulator.
Possibly mention arithmetic Chow groups, and the idea of covolume being the actual special value.
nLab page on Beilinson conjectures KEY IDEAS