Complex Multiplication and Shimura Stacks

TL;DR

This paper extends reciprocity laws to CM abelian varieties, K3 surfaces, and points on Shimura varieties, using stacky Shimura varieties to handle automorphisms, and characterizes models over number fields via Galois representations.

Contribution

It introduces a new approach to reciprocity laws for CM objects using stacky Shimura varieties, including a novel proof that the stack of polarized K3 surfaces is an open substack of a Shimura stack.

Findings

Reciprocity laws for CM abelian varieties, K3 surfaces, and Shimura points are established.

The stack of polarized K3 surfaces of given degree is shown to be an open substack of a Shimura stack.

The work provides a new perspective on models over number fields via Galois representations.

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.