Moduli spaces of sheaves on K3 surfaces and Galois representations

TL;DR

This paper shows that isomorphic Galois representations of K3 surfaces imply the same for their moduli spaces of stable sheaves, linking surface properties to their moduli spaces over arbitrary fields.

Contribution

It establishes a new relationship between Galois representations of K3 surfaces and their moduli spaces, extending known results to arbitrary fields and non-birational cases.

Findings

Galois representation isomorphism of surfaces implies the same for moduli spaces.

Equal zeta functions of K3 surfaces lead to equal zeta functions of moduli spaces over finite fields.

Results hold even when moduli spaces are not birational.

Abstract

We consider two K3 surfaces defined over an arbitrary field, together with a smooth proper moduli space of stable sheaves on each. When the moduli spaces have the same dimension, we prove that if the \'etale cohomology groups (with Q_ell coefficients) of the two surfaces are isomorphic as Galois representations, then the same is true of the two moduli spaces. In particular, if the field of definition is finite and the K3 surfaces have equal zeta functions, then so do the moduli spaces, even when the moduli spaces are not birational.