A G ] 1 3 Se p 20 07 K 3 Surfaces of Finite Height over Finite Fields ∗ † (original) (raw)

K3 surfaces of finite height over finite fields

arXiv (Cornell University), 2007

Arithmetic of K3 surfaces defined over finite fields is investigated. In particular, we show that any K3 surface X of finite height over a finite field k of characteristic p ≥ 5 has a quasi-canonical lifting Z to characteristic 0, and that for any such Z, the endormorphism algebra of the transcendental cycles V (Z), as a Hodge module, is a CM field over Q. The Tate conjecture for the product of certain two K3 surfaces is also proved. We illustrate by examples how to determine explicitly the formal Brauer group associated to a K3 surface over k. Examples discussed here are all of hypergeometric type.

Transcendental cycles on ordinary K3K3K3 surfaces over finite fields

Duke Mathematical Journal, 1993

Let Z be a complex algebraic K3 surface, and V(Z) the-lattice of transcendental cycles on Z. By definition, V(Z) is the orthogonal complement of NS(Z) (R) of the second rational cohomology group H2(Z,) with respect to the intersection pairing. Here NS(Z) is the Neron-Severi group of Z. It is well known that V(Z) carries a natural rational Hodge structure of weight 2. In [28] we have proven that this structure is irreducible and its endomorphism algebra is a number field. Now let Y be an ordinary K3 surface over a finite field k of characteristic p. We write Y for Y x k(a) where k(a) is an algebraic closure of k. For each rational prime different from p, let us consider the second twisted /-adic cohomology group n2(Ya, t)(1) of Y. The Galois group G(k) of k acts on n2(Ya, t)(1) in a natural way. One may identify NS(Y)t NS(Ya)(R) t with a certain Galois-invariant subspace of H2(y,)(1), and a theorem of Nygaard [12] asserts that this subspace coincides with G(k)-invariants H2(y,)(1) tk if k is "sufficiently large". (This theorem proves a special case of a general conjecture due to Tate 1-19].) We define the t-lattice V(Y) as the orthogonal complement of NS(Ya) in H2(y,)(1) with respect to the intersection pairing. Recall that this pairing and its restriction to NS(Ya) are nondegenerate. This gives us a canonical splitting H2(Ya, Qt)(1)= NS(Y)t) Vt(Y). Since the intersection pairing is Galois-invariant, Vt(Y) is a Galois-invariant subspace and the splitting above is also Galois-invariant. Recall that G(k) is procyclic and has a canonical generator, namely, the arithmetic Frobenius automorphism trk: k(a) k(a), x-. x where q is the number of elements of k. Clearly, q is an integral power of p. Another canonical generator of G(k) is the geometric Frobenius automorphism tPk tr-1. In this paper we examine the characteristic polynomial P,,tr(t) .'= det(id trPk, V(Y)).

Formal Brauer groups arising from certain weighted K3 surfaces

Journal of Pure and Applied Algebra, 1999

We compute the height of the formal Brauer groups, and the zeta-functions of certain K3 surfaces in weighted projective 3-spaces. In particular, we construct K3 surfaces with formal Brauer groups of height 1,2,3,4,6 and 10.

Formal Brauer groups arising from certain weighted surfaces

Journal of Pure and Applied Algebra, 1999

A finiteness theorem for the Brauer group of abelian varieties and K3 surfaces

2008

Let k be a field finitely generated over the field of rational numbers, and Br (k) the Brauer group of k. For an algebraic variety X over k we consider the cohomological Brauer-Grothendieck group Br (X). We prove that the quotient of Br (X) by the image of Br (k) is finite if X is a K3 surface. When X is an abelian variety over k, and X is the variety over an algebraic closure k of k obtained from X by the extension of the ground field, we prove that the image of Br (X) in Br (X) is finite.

The Arithmetic and Geometry of Algebraic Cycles

CRM proceedings & lecture notes, 2000

Regulators and characteristic classes of flat bundles Harris, Bruno and Wang, Bin, Height pairing asymptotics and Bott-Chem forms Kato, Kazuya and Usui, Sampei, Logarithmic Hodge structures and classifying spaces Chow Groups and Motives Asakura, Masanori, Motives and algebraic deRham cohomology Burgos Gil, Jose Ignacio, Hermitian vector bundles and characteristic classes de Jeu, Rob, A counterexample to a conjecture of Beilinson Hanamura, Masaki, The mixed motive of a projective variety Pedrini, Claudio, Bloch's conjecture and the K-theory of projective spaces Ramachandran, Niranjan, Jirom Jacobians to one-motives: Exposition of a conjecture of Deligne Saito, Shuji, Motives, algebraic cycles and Hodge theory Arithmetic Methods Doran, Charles, Picard-Fuchs uniformization: Modularity of the mifTOr maps and mifTOr-moonshine Goren, Eyal, Hilbert modular varieties in positive characteristic Goto, Yasuhiro, On the Neron-Severi groups of some K3 sur/aces van Hamel, Joost, Torsion zero-cycles and the Abel-Jacobi map over the real numbers Kimura, Ken-Ichiro, A remark on the Griffiths group of certain product varieties Nekovar, Jan, p-adic Abel-Jacobi maps and p-adic heights Shiho, Atsushi, Crystalline fundamental groups and p-adic Hodge theory vii Vlll Verrill, Helena and Yui, Noriko, Thompson series, and the mirror maps of certain pencils of K3 surfaces

Finiteness results for K3 surfaces over arbitrary fields

European Journal of Mathematics, 2019

Over an algebraically closed field, various finiteness results are known regarding the automorphism group of a K3 surface and the action of the automorphisms on the Picard lattice. We formulate and prove versions of these results over arbitrary base fields, and give examples illustrating how behaviour can differ from the algebraically closed case. Keywords K3 surfaces • Automorphism groups • Picard groups • Non-algebraically closed fields Mathematics Subject Classification 14J28 • 14J50 • 14G27 The second author would like to thank the Tutte Institute for Mathematics and Computation for its partial support for a visit to the University of Leiden during which much of this research was done.

A G ] 1 3 Se p 20 07 K 3 Surfaces of Finite Height over Finite Fields ∗ † (original) (raw)

Related papers