Plectic Galois action on CM points and connected components of Hilbert modular varieties

Abstract

We expand on Nekovář's construction of the plectic half transfer to define a plectic Galois action on Hilbert modular varieties. More precisely, we study in a unifying fashion Shimura varieties associated to groups that differ only in the centre from $R_{F/\mathbb{Q}}{\mathrm{GL}}_2$. We define plectic Galois actions on the CM points and on the set of connected components of these Shimura varieties, and show that these two actions are compatible. This extends the plectic conjecture of Nekovář--Scholl.

This is the version to appear in Bull. Lond. Math. Soc., https://doi.org/10.1112/blms.12692. The start of Section 2.3 has been rewritten to expand on the moduli space interpretation of Hilbert modular varieties. Comments welcome!

References in corpus (5)

• A Serre weight conjecture for geometric Hilbert modular forms in characteristic p
• The fundamental theorem of complex multiplication
• A plectic Taniyama group
• Generalized multiple zeta values over number fields I
• Galois conjugates of special points and special subvarieties in Shimura varieties