Several natural complex configuration spaces admit surprising uniformizations as arithmetic ball quotients, by identifying each parametrized object with the periods of some auxiliary object. In each case, the theory of canonical models of Shimura varieties gives the ball quotient the structure of a variety over the ring of integers of a cyclotomic field. I will show that these (transcendentally-defined) period maps respect these algebraic structures, and thus are actually arithmetic.

For the introductory slides on this topic, please see: http://www.fields.utoronto.ca/sites/default/files/uploads/achter_warmup0...