The Ample Cone for a K3 Surface

2011 ◽  
Vol 63 (3) ◽  
pp. 481-499 ◽  
Author(s):  
Arthur Baragar
Keyword(s):  
Hausdorff Dimension ◽  
Number Field ◽  
K3 Surface ◽  
Picard Number ◽  
Asymptotic Growth ◽  
Group Of Automorphisms ◽  
Line Parallel ◽  
Ample Cone

Abstract In this paper, we give several pictorial fractal representations of the ample or K¨ahler cone for surfaces in a certain class of K3 surfaces. The class includes surfaces described by smooth (2, 2, 2) forms in ℙ1 × ℙ1 × ℙ1 defined over a sufficiently large number field K that have a line parallel to one of the axes and have Picard number four. We relate the Hausdorff dimension of this fractal to the asymptotic growth of orbits of curves under the action of the surface's group of automorphisms. We experimentally estimate the Hausdorff dimension of the fractal to be 1.296 ± .010.

Canonical Vector Heights on Algebraic K3 Surfaces with Picard Number Two

2003 ◽  
Vol 46 (4) ◽  
pp. 495-508 ◽  
Author(s):  
Arthur Baragar
Keyword(s):  
Positive Integer ◽  
Number Field ◽  
Nonnegative Integer ◽  
Bounded Function ◽  
K3 Surface ◽  
Picard Number ◽  
Weil Height ◽  
Logarithmic Height ◽  
Real Quadratic ◽  
Canonical Vector

AbstractLet V be an algebraic K3 surface defined over a number field K. Suppose V has Picard number two and an infinite group of automorphisms A = Aut(V/K). In this paper, we introduce the notion of a vector height h: V → Pic(V) ⊗ and show the existence of a canonical vector height with the following properties:where σ ∈ A, σ* is the pushforward of σ (the pullback of σ−1), and hD is a Weil height associated to the divisor D. The bounded function implied by the O(1) does not depend on P. This allows us to attack some arithmetic problems. For example, we show that the number of rational points with bounded logarithmic height in an A-orbit satisfiesHere, μ(P) is a nonnegative integer, s is a positive integer, and ω is a real quadratic fundamental unit.

scholarly journals Ordinary reduction of K3 surfaces

2009 ◽  
Vol 7 (2) ◽  
Author(s):  
Fedor Bogomolov ◽  
Yuri Zarhin
Keyword(s):  
Number Field ◽  
Field Extension ◽  
K3 Surfaces ◽  
K3 Surface ◽  
Zero Set ◽  
Algebraic Field ◽  
Density Zero ◽  
Archimedean Place ◽  
Ordinary Reduction

AbstractLet X be a K3 surface over a number field K. We prove that there exists a finite algebraic field extension E/K such that X has ordinary reduction at every non-archimedean place of E outside a density zero set of places.

Fox formulas for twisted Alexander invariants associated to representations of knot groups over rings of S-integers

2018 ◽  
Vol 27 (05) ◽  
pp. 1850033 ◽  
Author(s):  
Ryoto Tange
Keyword(s):  
Number Field ◽  
Asymptotic Growth ◽  
Homology Groups ◽  
Finite Set ◽  
Twisted Homology ◽  
Alexander Invariants

We present a generalization of the Fox formula for twisted Alexander invariants associated to representations of knot groups over rings of [Formula: see text]-integers of [Formula: see text], where [Formula: see text] is a finite set of finite primes of a number field [Formula: see text]. As an application, we give the asymptotic growth of twisted homology groups.

scholarly journals On the Supersingular Reduction of K3 Surfaces with Complex Multiplication

2018 ◽  
Vol 2020 (20) ◽  
pp. 7306-7346
Author(s):  
Kazuhiro Ito
Keyword(s):  
Comparison Theorem ◽  
Complex Multiplication ◽  
K3 Surfaces ◽  
Explicit Description ◽  
Brauer Group ◽  
K3 Surface ◽  
Good Reduction ◽  
Picard Number ◽  
Reduction Modulo ◽  
Modulo P

Abstract We study the good reduction modulo $p$ of $K3$ surfaces with complex multiplication. If a $K3$ surface with complex multiplication has good reduction, we calculate the Picard number and the height of the formal Brauer group of the reduction. Moreover, if the reduction is supersingular, we calculate its Artin invariant under some assumptions. Our results generalize some results of Shimada for $K3$ surfaces with Picard number $20$. Our methods rely on the main theorem of complex multiplication for $K3$ surfaces by Rizov, an explicit description of the Breuil–Kisin modules associated with Lubin–Tate characters due to Andreatta, Goren, Howard, and Madapusi Pera, and the integral comparison theorem recently established by Bhatt, Morrow, and Scholze.

Generalised Iwasawa invariants and the growth of class numbers

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Sören Kleine
Keyword(s):  
Number Field ◽  
Fixed Number ◽  
Class Numbers ◽  
Open Neighbourhood ◽  
Finitely Generated ◽  
Asymptotic Growth ◽  
Iwasawa Invariants ◽  
Locally Maximal ◽  
Global Boundedness ◽  
The Impact

AbstractWe study the generalised Iwasawa invariants of {\mathbb{Z}_{p}^{d}}-extensions of a fixed number field K. Based on an inequality between ranks of finitely generated torsion {\mathbb{Z}_{p}[\kern-2.133957pt[T_{1},\dots,T_{d}]\kern-2.133957pt]}-modules and their corresponding elementary modules, we prove that these invariants are locally maximal with respect to a suitable topology on the set of {\mathbb{Z}_{p}^{d}}-extensions of K, i.e., that the generalised Iwasawa invariants of a {\mathbb{Z}_{p}^{d}}-extension {\mathbb{K}} of K bound the invariants of all {\mathbb{Z}_{p}^{d}}-extensions of K in an open neighbourhood of {\mathbb{K}}. Moreover, we prove an asymptotic growth formula for the class numbers of the intermediate fields in certain {\mathbb{Z}_{p}^{2}}-extensions, which improves former results of Cuoco and Monsky. We also briefly discuss the impact of generalised Iwasawa invariants on the global boundedness of Iwasawa λ-invariants.

scholarly journals On Cox rings of K3 surfaces

2010 ◽  
Vol 146 (4) ◽  
pp. 964-998 ◽  
Author(s):  
Michela Artebani ◽  
Jürgen Hausen ◽  
Antonio Laface
Keyword(s):  
K3 Surfaces ◽  
Del Pezzo Surfaces ◽  
K3 Surface ◽  
Picard Number ◽  
Finitely Generated ◽  
Generators And Relations ◽  
Cox Rings ◽  
Double Covers ◽  
Del Pezzo ◽  
Effective Cone

AbstractWe study Cox rings of K3 surfaces. A first result is that a K3 surface has a finitely generated Cox ring if and only if its effective cone is rational polyhedral. Moreover, we investigate degrees of generators and relations for Cox rings of K3 surfaces of Picard number two, and explicitly compute the Cox rings of generic K3 surfaces with a non-symplectic involution that have Picard number 2 to 5 or occur as double covers of del Pezzo surfaces.

scholarly journals ABELIAN FIBRATIONS AND RATIONAL POINTS ON SYMMETRIC PRODUCTS

2000 ◽  
Vol 11 (09) ◽  
pp. 1163-1176 ◽  
Author(s):  
BRENDAN HASSETT ◽  
YURI TSCHINKEL
Keyword(s):  
Number Field ◽  
Finite Extension ◽  
Rational Points ◽  
Symmetric Product ◽  
Symmetric Power ◽  
K3 Surface ◽  
Potential Density ◽  
Symmetric Products ◽  
Geometric Problem ◽  
Basic Construction

Given a variety over a number field, are its rational points potentially dense, i.e. does there exist a finite extension over which rational points are Zariski dense? We study the question of potential density for symmetric products of surfaces. Contrary to the situation for curves, rational points are not necessarily potentially dense on a sufficiently high symmetric product. Our main result is that rational points are potentially dense for the Nth symmetric product of a K3 surface, where N is explicitly determined by the geometry of the surface. The basic construction is that for some N, the Nth symmetric power of a K3 surface is birational to an Abelian fibration over ℙN. It is an interesting geometric problem to find the smallest N with this property.

K-STRATUM LIMIT AND K-STRATUM INTEGRAL ON THE GENERALIZED NUMBER FIELD

1982 ◽  
Vol 2 (4) ◽  
pp. 375-388
Author(s):  
Jiwu Wang ◽  
Tai Kang
Keyword(s):  
Number Field
Sign in / Sign up

Export Citation Format

Share Document