# Local to global principle for the moduli space of K3 surfaces

@article{Baldi2018LocalTG, title={Local to global principle for the moduli space of K3 surfaces}, author={G. Baldi}, journal={Archiv der Mathematik}, year={2018}, volume={112}, pages={599-613} }

Recently S. Patrikis, J.F. Voloch, and Y. Zarhin have proven, assuming several well-known conjectures, that the finite descent obstruction holds on the moduli space of principally polarised abelian varieties. We show an analogous result for K3 surfaces, under some technical restrictions on the Picard rank. This is possible since abelian varieties and K3s are quite well described by ‘Hodge-theoretical’ results. In particular the theorem we present can be interpreted as follows: a family of… Expand

#### 3 Citations

Recognizing Galois representations of K3 surfaces

- Mathematics
- 2018

Under the assumption of the Hodge, Tate and Fontaine–Mazur conjectures we give a criterion for a compatible system of $$\ell $$ℓ-adic representations of the absolute Galois group of a number field to… Expand

Canonical models of K3 surfaces with complex multiplication

- Mathematics
- 2019

Let $X/ \mathbb{C}$ be a K3 surface with complex multiplication by the ring of integers of a CM number field $E$. Under some natural conditions on the discriminant of the quadratic form $T(X)$, we… Expand

Finite descent obstruction for Hilbert modular varieties

- Mathematics
- Canadian Mathematical Bulletin
- 2020

Abstract Let S be a finite set of primes. We prove that a form of finite Galois descent obstruction is the only obstruction to the existence of
$\mathbb {Z}_{S}$
-points on integral models of… Expand

#### References

SHOWING 1-10 OF 60 REFERENCES

Recognizing Galois representations of K3 surfaces

- Mathematics
- 2018

Under the assumption of the Hodge, Tate and Fontaine–Mazur conjectures we give a criterion for a compatible system of $$\ell $$ℓ-adic representations of the absolute Galois group of a number field to… Expand

INTEGRAL AND ADELIC ASPECTS OF THE MUMFORD–TATE CONJECTURE

- Mathematics
- Journal of the Institute of Mathematics of Jussieu
- 2018

Let $Y$ be an abelian variety over a subfield $k\subset \mathbb{C}$ that is of finite type over $\mathbb{Q}$. We prove that if the Mumford–Tate conjecture for $Y$ is true, then also some refined… Expand

Finiteness theorems for K3 surfaces and abelian varieties of CM type

- Mathematics
- Compositio Mathematica
- 2018

We study abelian varieties and K3 surfaces with complex multiplication defined over number fields of fixed degree. We show that these varieties fall into finitely many isomorphism classes over an… Expand

Anabelian geometry and descent obstructions on moduli spaces

- Mathematics
- 2015

We study the section conjecture of anabelian geometry and the sufficiency of the finite descent obstruction to the Hasse principle for the moduli spaces of principally polarized abelian varieties and… Expand

Hodge groups of K3 surfaces.

- Mathematics
- 1983

The purpose of this paper is to study the rational Hodge structure attached to the second rational cohomology group H(Y, 0) of a smooth irreducible projective surface Υ over the field C of complex… Expand

Lectures on K3 Surfaces

- Computer Science
- 2016

Famous open conjectures, for example the conjectures of Calabi, Weil, and Artin–Tate, are discussed in general and for K3 surfaces in particular and each chapter ends with questions and open problems. Expand

Algebraic Cycles and Motives: On the Transcendental Part of the Motive of a Surface

- Mathematics
- 2007

Bloch’s conjecture on surfaces [B1], which predicts the converse to Mumford’s famous necessary condition for finite-dimensionality of the Chow group of 0-cycles [Mum1], has been a source of… Expand

Kuga-Satake abelian varieties of K3 surfaces in mixed characteristic

- Mathematics
- 2010

Abstract Kuga and Satake associate with every polarized complex K3 surface (X, ℒ) a complex abelian variety called the Kuga-Satake abelian variety of (X, ℒ). We use this construction to define… Expand

Real multiplication on K3 surfaces and Kuga Satake varieties

- Mathematics
- 2006

The endomorphism algebra of a K3 type Hodge structure is a totally real field or a CM field. In this paper we give a low brow introduction to the case of a totally real field. We give existence… Expand

Abelian L-adic representation and elliptic curves

- Mathematics, Computer Science
- Advanced book classics
- 1989

This classic book contains an introduction to systems of l-adic representations, a topic of great importance in number theory and algebraic geometry, as reflected by the spectacular recent… Expand