scholarly journals Point counting on K3 surfaces and an application concerning real and complex multiplication

2016 ◽  
Vol 19 (A) ◽  
pp. 12-28 ◽  
Author(s):  
Andreas-Stephan Elsenhans ◽  
Jörg Jahnel
Keyword(s):  
Finite Fields ◽  
Efficient Method ◽  
Complex Multiplication ◽  
K3 Surfaces ◽  
Point Counting

We report on our project to find explicit examples of K3 surfaces having real or complex multiplication. Our strategy is to search through the arithmetic consequences of RM and CM. In order to do this, an efficient method is needed for point counting on surfaces defined over finite fields. For this, we describe algorithms that are$p$-adic in nature.

The Tate conjecture for ordinary K3 surfaces over finite fields

1983 ◽  
Vol 74 (2) ◽  
pp. 213-237 ◽  
Author(s):  
N. O. Nygaard
Keyword(s):  
Finite Fields ◽  
K3 Surfaces ◽  
Tate Conjecture

Transcendental cycles on ordinary $K3$ surfaces over finite fields

1993 ◽  
Vol 72 (1) ◽  
pp. 65-83 ◽  
Author(s):  
Yuri G. Zarhin
Keyword(s):  
Finite Fields ◽  
K3 Surfaces

scholarly journals On the distribution of Atkin and Elkies primes for reductions of elliptic curves on average

2015 ◽  
Vol 18 (1) ◽  
pp. 308-322 ◽  
Author(s):  
Igor E. Shparlinski ◽  
Andrew V. Sutherland
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Riemann Hypothesis ◽  
Complex Multiplication ◽  
Log P ◽  
Point Counting ◽  
Generalized Riemann Hypothesis ◽  
Expected Time ◽  
Almost All ◽  
Counting Algorithm

For an elliptic curve$E/\mathbb{Q}$without complex multiplication we study the distribution of Atkin and Elkies primes$\ell$, on average, over all good reductions of$E$modulo primes$p$. We show that, under the generalized Riemann hypothesis, for almost all primes$p$there are enough small Elkies primes$\ell$to ensure that the Schoof–Elkies–Atkin point-counting algorithm runs in$(\log p)^{4+o(1)}$expected time.

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.

K3 surfaces with algebraic period ratios have complex multiplication

2015 ◽  
Vol 11 (05) ◽  
pp. 1709-1724
Author(s):  
Paula Tretkoff
Keyword(s):  
Number Theory ◽  
Vector Space ◽  
Hodge Structure ◽  
Complex Multiplication ◽  
K3 Surfaces ◽  
K3 Surface ◽  
Detailed Proof ◽  
Tate Group ◽  
Calabi Yau Manifolds

Let Ω be a non-zero holomorphic 2-form on a K3 surface S. Suppose that S is projective algebraic and is defined over [Formula: see text]. Let [Formula: see text] be the [Formula: see text]-vector space generated by the numbers given by all the periods ∫γ Ω, γ ∈ H2(S, ℤ). We show that, if [Formula: see text], then S has complex multiplication, meaning that the Mumford–Tate group of the rational Hodge structure on H2(S, ℚ) is abelian. This result was announced in [P. Tretkoff, Transcendence and CM on Borcea–Voisin towers of Calabi–Yau manifolds, J. Number Theory 152 (2015) 118–155], without a detailed proof. The converse is already well known.

scholarly journals Computing cardinalities of -curve reductions over finite fields

2016 ◽  
Vol 19 (A) ◽  
pp. 115-129
Author(s):  
François Morain ◽  
Charlotte Scribot ◽  
Benjamin Smith
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Lower Degree ◽  
Point Counting ◽  
Low Degree ◽  
Asymptotic Complexity ◽  
Specialized Point ◽  
Counting Algorithm

We present a specialized point-counting algorithm for a class of elliptic curves over $\mathbb{F}_{p^{2}}$ that includes reductions of quadratic $\mathbb{Q}$-curves modulo inert primes and, more generally, any elliptic curve over $\mathbb{F}_{p^{2}}$ with a low-degree isogeny to its Galois conjugate curve. These curves have interesting cryptographic applications. Our algorithm is a variant of the Schoof–Elkies–Atkin (SEA) algorithm, but with a new, lower-degree endomorphism in place of Frobenius. While it has the same asymptotic asymptotic complexity as SEA, our algorithm is much faster in practice.

Constructing Elliptic Curves over Ramanujan's Class Invariants

2014 ◽  
Vol 915-916 ◽  
pp. 1336-1340
Author(s):  
Jian Jun Hu
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Imaginary Quadratic Field ◽  
Hilbert Polynomials ◽  
Class Invariants ◽  
Curves Over Finite Fields ◽  
Class Polynomials

The Complex Multiplication (CM) method is a widely used technique for constructing elliptic curves over finite fields. The key point in this method is parameter generation of the elliptic curve and root compution of a special type of class polynomials. However, there are several class polynomials which can be used in the CM method, having much smaller coefficients, and fulfilling the prerequisite that their roots can be easily transformed to the roots of the corresponding Hilbert polynomials.In this paper, we provide a method which can construct elliptic curves by Ramanujan's class invariants. We described the algorithm for the construction of elliptic curves (ECs) over imaginary quadratic field and given the transformation from their roots to the roots of the corresponding Hilbert polynomials. We compared the efficiency in the use of this method and other methods.

scholarly journals CM liftings of surfaces over finite fields and their applications to the Tate conjecture

2021 ◽  
Vol 9 ◽  
Author(s):  
Kazuhiro Ito ◽  
Tetsushi Ito ◽  
Teruhisa Koshikawa
Keyword(s):  
Finite Field ◽  
Algebraic Group ◽  
Finite Fields ◽  
Complex Multiplication ◽  
Shimura Varieties ◽  
Canonical Models ◽  
Tate Conjecture ◽  
Finite Height ◽  
Generic Fibre

Abstract We give applications of integral canonical models of orthogonal Shimura varieties and the Kuga-Satake morphism to the arithmetic of $K3$ surfaces over finite fields. We prove that every $K3$ surface of finite height over a finite field admits a characteristic $0$ lifting whose generic fibre is a $K3$ surface with complex multiplication. Combined with the results of Mukai and Buskin, we prove the Tate conjecture for the square of a $K3$ surface over a finite field. To obtain these results, we construct an analogue of Kisin’s algebraic group for a $K3$ surface of finite height and construct characteristic $0$ liftings of the $K3$ surface preserving the action of tori in the algebraic group. We obtain these results for $K3$ surfaces over finite fields of any characteristics, including those of characteristic $2$ or $3$ .

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