Arithmetic of K3 Surfaces

Banff, November 30 -- December 5

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Abstracts and relevant papers:

Ekaterina Amerik
Title: Potential density of rational points on the variety of lines of a cubic fourfold.
Abstract: This is a joint work with Claire Voisin. Conjecturally, rational points must be potentially dense on any smooth projective variety with trivial canonical class, defined over a number field. This is well-known for abelian varieties, but the simply-connected case is much less understood. For instance, though potential density has been proved by Bogomolov and Tschinkel for certain families of K3 surfaces with special properties, it is unknown whether a "generic" K3 surface is potentially dense over a number field; in particular, there is no example of a potentially dense K3 surface with Picard number 1. Our purpose is to provide such an example in dimension 4. More precisely, we exhibit a family of irreducible holomorphic symplectic fourfolds, such that its "sufficiently general" member defined over a number field has cyclic Picard group and is potentially dense. To find explicit equations for such a variety is still an open problem.
Papers:

Potential density of rational points on the variety of lines of a cubic fourfold, to appear at Duke Math. J. (joint with C. Voisin).
A computation of invariants of a rational self-map, to appear at Ann. Fac. Sci. Toulouse.

Arthur Baragar
Title: Finding automorphisms on K3 surfaces with small Picard number.
Abstract: In this talk we will investigate, via an example, some techniques for finding the group of automorphisms for a K3 surface. Some of the techniques will be general, some will be specific to the example, and some will depend on the small Picard number.
Papers:

Orbits of points on certain K3 surfaces, to appear.
The ample cone for a K3 surface, to appear.
K3 surfaces, rational curves, and rational points, to appear, (joint with D. McKinnon).
Orbits of curves on certain K3 surfaces, Compositio Math, 137 (2), 115--134 (2003).
Rational Points on K3 Surfaces in PxPxP, Math. Ann., 305, 541--558 (1996).

Arnaud Beauville slides
Title: Algebraic cycles on K3 and derived equivalences.
Abstract: The Chow ring of a complex K3 surface is very large, but contains a finite-dimensional subring where most of the action takes place. I will describe this subring, and the recent result of Huybrechts proving that it is stable under derived equivalences.
Papers:

On the Chow ring of a K3 surface (joint with C. Voisin)
Counting rational curves on K3 surfaces

Martin Bright slides magma-demo
Title: Computing Brauer-Manin obstructions on diagonal quartic surfaces.
Abstract: I will describe the theoretical process of calculating the algebraic Brauer-Manin obstruction on a surface, and show practically how to do it in the case of a diagonal quartic surface.

Serge Cantat
Title: Dynamics of Automorphisms.
Abstract: Let f be an automorphism of a (projective) K3 surface. I will describe the distribution of periodic points of f and list a few open questions regarding the closure of the set of periodic points.
Papers:

Deux exemples concernant une conjecture de Serge Lang; C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 7, 581--586.
Dynamique du groupe d'automorphismes des surfaces K3; Transform. Groups 6 (2001), no. 3, 201--214.
Dynamique des automorphismes des surfaces K3; Acta Math. 187:1 (2001), 1--57. ( see also C. R. Acad. Sci. Paris Sé. I Math. 328:10 (1999), 901--906)
Caratérisation des exemples de Lattès et de Kummer; Compositio Mathematica, 144:5 (2008)

Thomas Dedieu
Title: Self-rational maps of K3 surfaces. Action on rational points and on nodal curves.
Abstract: I will explain how self-rational maps can be used to show that certain special algebraic K3 surfaces have potential density. It is however expected that generic algebraic K3 surfaces do not carry any self-rational map with degree >1. By looking at the action of such maps on nodal curves, I will relate this conjecture to another conjecture about the irreducibility of the universal Severi varieties for K3 surfaces.
Papers:

Severi varieties and self rational maps of K3 surfaces

Evis Ieronymou
Title: Transcendental Brauer group and counterexamples to weak approximation for diagonal quartic surfaces.
Abstract: Diagonal quartic surfaces are a special subclass of K3 sufaces. Following work of Swinnerton-Dyer and Martin Bright we have some control over the algebraic Brauer group of such surfaces. The role played by elements of the Brauer group which are not algebraic is quite mysterious. In this talk we discuss the construction of such elements, and applications to weak approximation.

Alessandra Sarti
Title: Automorphism Groups of K3 Surfaces.
Abstract: I will present recent progress in the study of prime order automorphisms of K3 surfaces. An automorphism is called (non-) symplectic if the induced operation on the global nowhere vanishing holomorphic two form is (non-) trivial. After a short survey on the problem, I will describe the topological structure of the fixed locus, the geometry of these K3 surfaces and their moduli spaces.
Papers:

Non-symplectic automorphisms of order 3 on K3 surfaces (joint with Michela Artebani)
Elliptic fibrations and symplectic automorphisms on K3 surfaces (joint with A. Garbagnati)
Symplectic automorphisms of prime order on K3 surfaces (joint with A. Garbagnati)

Chad Schoen
Title: Calabi-Yau threefolds with vanishing third Betti number.
Abstract: Algebraic varieties of the type described in the title do not exist in characteristic 0. In the past 10 years a number of examples have been constructed in positive characteristic. We will describe some of these constructions, several of which involve K3 surfaces and their moduli. We will also briefly touch on some natural questions about these varieties which generalize classical questions about K3 surfaces. This talk, by a non-expert in geometry in positive characteristic, is aimed at non-experts.

Matthias Schuett
Title: K3 surfaces and modular forms.
Abstract: A classical construction of Shimura associates every Hecke eigenform of weight 2 with rational coefficients to an elliptic curve over Q. The converse statement that every elliptic curve over Q is modular, is the Taniyama-Shimura-Weil conjecture, proven by Wiles et al.
For higher weight, however, the opposite situation applies: Nowadays we know the modularity for wide classes of varieties, but it is an open problem whether all newforms of fixed weight with rational coefficients can be realised in a single class of varieties.
I will present joint work with N. Elkies that provides the first solution to the realisation problem in higher weight: We show that every newform of weight 3 with rational coefficients is associated to a singular K3 surface over Q.
Papers:

K3 surfaces with non-symplectic automorphism of 2-power order
K3 surfaces of Picard rank 20 over Q
On the uniqueness of K3 surfaces with maximal singular fibre (joint with A. Schweizer)
Generalised Kummer constructions and Weil restrictions (joint with S. Cynk)
CM newforms with rational coefficients, to appear in: Ramanujan Journal
Arithmetic of K3 surfaces, To appear in: Jahresbericht der DMV
Arithmetic of a singular K3 surface, to appear in: Michigan Mathematical Journal
Fields of definition of singular K3 surfaces, Communications in Number Theory and Physics 1, 2 (2007), 307-321
An interesting elliptic surface over an elliptic curve, Proc. Jap. Acad. 83, 3 (2007), 40-45, (joint with T. Shioda)
Elliptic fibrations of some extremal K3 surfaces, Rocky Mountain Journal of Mathematics 37, 2 (2007), 609-652.
The maximal singular fibres of elliptic K3 surfaces, Archiv der Mathematik 79, 4 (2006), 309-319.
Modularity of Calabi-Yau varieties, in: Catanese et al. (eds.) - Global Aspects of Complex Geometry, Springer (2006), (with K. Hulek and R. Kloosterman)
Arithmetic of the [19,1,1,1,1,1] fibration, Commentarii Mathematici Universitatis Sancti Pauli 55, 1 (2006), 9-16, (joint with J. Top)

Tetsuji Shioda
Title: Mordell-Weil Lattices of Certain Elliptic K3 Surfaces.
Abstract: We discuss the rank formula and lattice structure of the Mordell-Weil lattice of certain elliptic surfaces, especially, the elliptic K3 surfaces of Inose-Kuwata, defined by an explicit Weierstrass equation.
Papers (some are available through T. Shioda's homepage):

K3 surfaces and sphere packings, Preprint MPIM, 2007; JMSJ (to appear).
A note on K3 surfaces and sphere packings, Proc. Japan Acad. 76A, 68--72 (2000).
Elliptic parameters and defining equations for elliptic fibrations on a Kummer surface, (joint with M. Kuwata), in: Algebraic Geometry in East Asia--Hanoi 2005, 177--215, Advanced Studies in Pure Mathematics 50 (2008).
The Mordell-Weil lattice of y²=x³ + t⁵ - 1/t⁵ -11, Comment. Math. Univ. St. Pauli 56, 45--70 (2007).
Correspondence of elliptic curves and Mordell-Weil lattices of certain elliptic K3 surfaces, in: Algebraic Cycles and Motives, vol. 2, 319--339, Cambridge Univ. Press (2007).
On the Mordell-Weil lattices, Comment. Math. Univ. St. Pauli 39, 211-- 240 (1990).

Joe Silverman slides
Title: Dynamics and canonical heights on K3 surfaces with noncommuting involutions.
Abstract: K3 surfaces embedded in P² x P² admit a pair of noncommuting involutions. In this talk I will discuss the orbits of (rational) points under the action of these involutions and construct a pair of canonical height functions that are useful for studying the arithmetic properties of orbits. As time permits, I will describe K3 dynamical analogues of various classical conjectures and theorems, and will discuss analogous results on K3 surfaces embedded in P¹ x P¹ x P¹ that admit three noncommuting involutions.
Papers:

Rational points on K3 surfaces: a new canonical height. Invent. Math. 105 (1991), no. 2, 347-373.
Computing the canonical height on K3 surfaces. Math. Comp. 65 (1996), no. 213, 259-290. (joitn with G. Call)

Michael Stoll slides
Title: Searching for rational points on genus 2 Jacobians.
Abstract: Let C be a curve of genus 2 over Q, with Jacobian J. In many situations, we need to know generators of the Mordell-Weil group J(Q) (or at least of a finite index subgroup). As is the case with elliptic curves, these generators can be exponentially large in terms of the coefficients in the curve equation. I will describe a method that allows us to find comparatively large points in J(Q). This method amounts to searching for points on 2-covering spaces X of J, but in practice, we search for points on a certain quotient Y of X that lift to X. This quotient is a Kummer Surface. We can also use a variant that works with 2-covering spaces X of the principal homogeneous space Pic_C¹ of J; in this case, we replace the quotient Y by a P¹-bundle over Y that has a nice explicit description. This amounts to a partial 4-descent on J and has allowed me to find some points of logarithmic height close to 100.

Jaap Top notes
Title: Arithmetic of a family of K3's with Picard number 19.
Abstract: This will be an expository talk. The aim is to present some of the techniques (Nikulin involutions, Shioda-Inose structures, recognizing a genus 2 curve from the Kummer surface of its jacobian, correspondences) which Bert van Geemen and I used in a joint paper (2006), to establish a relation between two families of K3 surfaces.
Papers:

An isogeny of K3 surfaces (with B. van Geemen, Bull. London Math. Soc. 38 (2006), 209--223).
More references to papers on the arithmetic of K3 surfaces.

Ronald van Luijk slides
Title: The analogue of the Batyrev-Manin conjecture for K3 surfaces.
Abstract: The Batyrev-Manin conjecture predicts for any Fano variety the asymptotic growth of the number of rational points of bounded height on the variety as a function of the height bound. This growth is strongly related to the geometry of the variety, in particular to its Picard number. We will see some heuristics and experiments that suggest an analogue of the conjecture for K3 surfaces.
Papers:

K3 surfaces with Picard number one and infinitely many rational points, Algebra and Number Theory, Vol. 1, No. 1 (2007), 1-15.
Quartic K3 surfaces without nontrivial automorphisms, Mathematical Research Letters (MRL), Volume 13 (2006), Issue 3, 423-439.
D. McKinnon, Counting Rational Points on K3 Surfaces, J. Number Theory 84 (2000), no. 1, 49--62.
More references to papers on the arithmetic of K3 surfaces.

Olivier Wittenberg
Title: Brauer groups and rational points of K3 surfaces.
Abstract: This will be an expository talk. Rational points of K3 surfaces over number fields do not always satisfy weak approximation or even the Hasse principle. Understanding which K3 surfaces enjoy these properties and which do not is one of the basic problems in the study of the arithmetic of K3 surfaces. I will discuss examples and known results in this direction, most of which involve Brauer groups, and mention a few open questions as well.
Papers:

Transcendental Brauer-Manin obstruction on a pencil of elliptic curves, in Arithmetic of higher-dimensional varieties (Palo Alto, CA, 2002; edited by B. Poonen and Yu. Tschinkel), 259--267, Progress in Mathematics 226, Birkhäuser Boston, Boston, MA, 2004.
Chapter 1 of Intersections de deux quadriques et pinceaux de courbes de genre 1, LNM 1901, Springer-Verlag, 2007.
D. Harari and A. Skorobogatov, Nonabelian descent and the arithmetic of Enriques surfaces, Int. Math. Res. Not. 2005, no. 52, 3203--3228.
P. Swinnerton-Dyer, Arithmetic of diagonal quartic surfaces. II. Proc. London Math. Soc. (3) 80 (2000), no. 3, 513--544.
A. Skorobogatov and P. Swinnerton-Dyer, 2-descent on elliptic curves and rational points on certain Kummer surfaces. Adv. Math. 198 (2005), no. 2, 448--483.

Noriko Yui
Title: On the modularity of certain K3 surfaces with non-symplectic group actions.
Abstract: This is a joint work with Ron Livné (Jerusalem) and Matthias Schütt (Copenhagen). We consider complex K3 surfaces with a non-symplectic group acting trivially on the algebraic cycles. Vorontsov and Kondo classified those K3 surfaces with transcendental lattice of minimal rank. The purpose of this talk is to study the Galois representations associated to these K3 surfaces. The rank of transcendental lattices is even and varies from 2 to 20, excluding 8 and 14. We show that these K3 surfaces are dominated by Fermat surfaces and hence they are all of CM type. We establish the modularity of the Galois representations associated to the transcendental parts of these K3 surfaces. Time permitting, we briefly discuss mirror symmetry for these K3 surfaces.

Yuri Zarhin
Title: Variants of the Tate conjecture with finite coefficients and their applications.
Abstract: We'll discuss interactions between various conjectures of Tate that deal with the algebraicity of Galois-invariant cohomology classes and homomorphisms of abelian varieties and their Tate modules over finitely generated fields. We also consider an analogue of Tate's conjecture on homomorphisms that deals with points of sufficiently large order (instead of Tate modules) . Applications to finiteness results for Brauer groups will be given (joint work with Alexei Skorobogatov).
Papers:

A finiteness theorem for the Brauer group of abelian varieties and K3 surfaces, J. Algebraic Geometry 17 (2008) 481-502, (joint with A. Skorobogatov), also on arXiv.
The Brauer group of an Abelian variety over a finite field, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 211-243; English translation: Math. USSR Izv. 20 (1983), 203-234.
Hodge groups of K3 surfaces, Journal für die reine und angewandte Mathematik, Vol. 341 (1983), 193-220.
A finiteness theorem for unpolarized Abelian varieties over number fields with prescribed places of bad reduction, Inventiones mathematicae, Vol. 79 (1985), 309-321.