Arithmetic Properties of the Derived Category for Calabi-Yau Varieties

Abstract

This thesis develops a theory of arithmetic Fourier-Mukai transforms in order to obtain results about equivalences between the derived category of Calabi-Yau varieties over non-algebraically closed fields. We obtain answers to classical questions from number theory and arithmetic geometry using these results. The main results of this thesis come in three types. The first concerns classifying moduli of vector bundles on genus one curves. Fourier-Mukai equivalences of genus one curves allow us to produce examples of non-isomorphic moduli spaces when a genus one curve has large period. The next result extends the result of Lieblich-Olsson which says that derived equivalent K3 surfaces are moduli spaces of sheaves over algebraically closed fields. We get the same result over arbitrary fields of characteristic different from 2. The last class of results are about finding properties that are preserved under derived equivalence for Calabi-Yau threefolds. Examples of such arithmetic invariants include local Zeta functions, modularity, L-series, and the a-number. We then prove a Serre-Tate theory for liftable, ordinary Calabi-Yau threefolds in positive characteristic in order to show that the derived equivalence induces an isomorphism of their deformation functors that sends the canonical lift of one to the canonical lift of the other.