Complex Multiplication and Shimura Stacks

Abstract

We prove a variant of the reciprocity laws for CM abelian varieties, CM K3 surfaces, and CM points on Shimura varieties. Given a CM object over the complex numbers, our variation describes the set of all models over a given number field $F$ in terms of associated representations of the absolute Galois group of $F$. An essential feature is that we work with stacky Shimura varieties to deal with objects that have non-trivial automorphisms. To prove the result on K3 surfaces, we show that the stack of polarized K3 surfaces of given degree is an open substack of a certain Shimura stack. The precise statement of this folklore fact seems to be missing from the literature.

(v4: condition (0) removed from Theorem 1.4, implied by the other conditions)

References in corpus (1)

• The discriminant and the determinant of a hypersurface of even dimension