Iwasawa Theory for K3 Surfaces over Finite Fields

Abstract

In this paper, we initiate Iwasawa theory for K3 surfaces over finite fields. First, using the Artin-Tate conjecture, which is known to hold for K3 surfaces, we prove an analogue of Mazur's control theorem for elliptic curves over number fields. Second, we prove an analogue of Iwasawa's class number formula for Brauer groups in two different ways. We also give explicit examples in the case of Kummer surfaces. Finally, we establish an analogue of the Iwasawa main conjecture for Brauer groups.