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More references

Open problems

- 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.

- 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).

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

- 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)

- 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)

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.

- 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)

- 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
y
^{2}=x^{3}+ t^{5}- 1/t^{5}-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).

- 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)

- 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.

- 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.

- 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.

- 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.