Plectic arithmetic of Hilbert modular varieties
We introduce plectic Galois actions on the set of CM points, on the set of connected components, and on the set of cocharacters of Shimura varieties that differ in the centre from the Hilbert modular variety. By allowing the centre to vary, we extend the plectic framework of Nekovář--Scholl to include such Shimura varieties, thereby also bridging the gap to earlier work of Nekovář. Our main result is that the map that sends a point on the Shimura variety to its connected component is equivariant under the plectic action.
To achieve this, we define a generalisation of the plectic Taniyama element, describe the points of the Shimura varieties in question in terms of abelian varieties with extra structure, and orient ourselves by the main theorem of complex multiplication over the rationals to define the plectic action on CM points. Moreover, we use a description of the set of connected components as a zero-dimensional Shimura variety and then employ class field theoretic techniques to prove the main result.