Observations on Fermat Motives ofK3-Type

Abstract

In [8, 9], Zarhin introduced the notion of varieties ofK3-typein even dimension over finite fields. Zarhin showed that ordinary abelian surfaces, ordinaryK3 surfaces, and ordinary cubic fourfolds are examples of such varieties. As Zarhin already points out, it is easy to extend his method to definemotivesofK3-type. In this note, we extend Zarhin's work by giving this definition and then finding and studying a large number of examples of such motives. Our examples are Fermat motives, i.e., they arise from the Fermat varietyX:Xm0+Xm1+…+Xmn+1=0⊂Pnof degreemand dimensionn=2d. Such motives ofK3-type need not be ordinary; on the other hand, they are never supersingular (essentially by definition—see [1], Chapter 3). Hence, ourK3-type motives give rise to transcendental cycles. We show that being ofK3-type is (resp. is not) hereditary with respect to thetype I inductive structure(resp. thetype II inductive structure) of Fermat motives. When the degreemtends to infinity, the type I inductive structure gives rise to infinitely many Fermat motives ofK3-type (see Remark (3.9)). Following a suggestion of Zarhin, we also consider powers of Fermat motives ofK3-type. Powers of motives ofK3-type are no longer ofK3-type, and hence they can give rise to classes of algebraic cycles. Thus, it is of interest to establish the validity of Tate's conjecture for powers of Fermat motives ofK3-type. This is done in the Section 4, extending Theorem (6.1) of Zarhin [10]. Finally, in Section 5 we discuss theBrauer numbersof Fermat motives ofK3-type, especially, of even composite degreem. This is an attempt to extend the results about Brauer numbers of Fermat motives (of prime or odd prime power degree) in Gouvêa and Yui [1, 2], Miki [4], Shoida [5], and Yui [7]. Throughout, we retain Zarhin's assumptions that the dimensionnis even and that we are working over a finite field.