Observations on Fermat Motives ofK3-Type

✍ Scribed by Fernando Q Gouvêa; Noriko Yui

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 378 KB
• Volume: 71
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1998.2244

✦ Synopsis

In , Zarhin introduced the notion of varieties of K 3-type in even dimension over finite fields. Zarhin showed that ordinary abelian surfaces, ordinary K3 surfaces, and ordinary cubic fourfolds are examples of such varieties. As Zarhin already points out, it is easy to extend his method to define motives of K3-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 variety

of degree m and dimension n=2d. Such motives of K3-type need not be ordinary; on the other hand, they are never supersingular (essentially by definition see [1],

Chapter 3). Hence, our K3-type motives give rise to transcendental cycles. We show that being of K 3-type is (resp. is not) hereditary with respect to the type I inductive structure (resp. the type II inductive structure) of Fermat motives. When the degree m tends to infinity, the type I inductive structure gives rise to infinitely many Fermat motives of K 3-type (see Remark (3.9)). Following a suggestion of Zarhin, we also consider powers of Fermat motives of K3-type. Powers of motives of K3-type are no longer of K 3-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 of K 3-type. This is done in the Section 4, extending Theorem (6.1) of Zarhin . Finally, in Section 5 we discuss the Brauer numbers