Arithmetic occult period maps. (English) Zbl 07262979

Achter, Jeffrey D.

Algebr. Geom. 7, No. 5, 581-606 (2020).

Summary: Several natural complex configuration spaces admit surprising uniformizations as arithmetic ball quotients, obtained 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. We show that the (transcendentally defined) period map actually respects these algebraic structures, and thus that occult period maps are arithmetic. As an intermediate tool, we develop an arithmetic theory of lattice-polarized K3 surfaces.

MSC:

14J10	Families, moduli, classification: algebraic theory
11G18	Arithmetic aspects of modular and Shimura varieties
14D05	Structure of families (Picard-Lefschetz, monodromy, etc.)

Keywords:

K3 surface; Shimura variety; period map