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Program

Abstracts, relevant papers, and slides

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)

- Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points. Inventiones math. 134 (1998) 579-650 (joint with A. N. Skorobogatov and Sir Peter Swinnerton-Dyer).

- Nonabelian descent and the arithmetic of Enriques surfaces, Int. Math. Res. Not. 2005, no. 52, 3203--3228 (joint with A. Skorobogatov).

- Computational Investigation of Wehler's K3 surfaces.
- Dynatomic Cycles for Morphisms of Projective Varieties.

- Finite groups of symplectic automorphisms of K3 surfaces in positive characteristic, to appear in Annals of Mathematics (joint with I. Dolgachev).
- K3 surfaces with a symplectic automorphism of order 11, to appear in J. of the European Mathematical Society (joint with I. Dolgachev).

- Elliptic K3 surfaces with geometric Mordell-Weil rank 15
- Explicit sections on Kuwata's elliptic surfaces

- K3 surfaces associated with curves of genus 2, Int. Math. Res. Not., Vol. 2008, Article ID rnm165.

- Vanishing and Non-Vanishing Dirichlet Twists of L-Functions of Elliptic Curves (joint with J. Fearnley and H. Kisilevsky).
- Elliptic parameters and defining equations for elliptic fibrations on a Kummer surface, Advanced Studies in Pure Mathematics 50, 2008, Algebraic Geometry in East Asia - Hanoi 2005, 177-215 (joint with T. Shioda)

- Counting Rational Points on K3 Surfaces, J. Number Theory 84 (2000), no. 1, 49--62.

- Correspondences of a K3 surface with itself via moduli of sheaves. I
- Finiteness of the number of arithmetic groups generated by reflections in Lobachevsky spaces
- On ground fields of arithmetic hyperbolic reflection groups
- On ground fields of arithmetic hyperbolic reflection groups. II
- On ground fields of arithmetic hyperbolic reflection groups. III
- Self-correspondences of K3 surfaces via moduli of sheaves and arithmetic hyperbolic reflection groups

- 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)

- 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)

- Nonabelian descent and the arithmetic of Enriques surfaces, Int. Math. Res. Not. 2005, no. 52, 3203--3228 (joint with D. Harari).
- 2-descent on elliptic curves and rational points on certain Kummer surfaces. Adv. Math. 198 (2005), no. 2, 448--483 (joint with P. Swinnerton-Dyer).
- A finiteness theorem for the Brauer group of abelian varieties and K3 surfaces J. Algebraic Geometry 17 (2008) 481-502, (joint with Yu. Zarhin), also on arXiv.
- Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points. Inventiones math. 134 (1998) 579-650 (joint with J.-L. Colliot-Thélène and Sir Peter Swinnerton-Dyer).

- Arithmetic of diagonal quartic surfaces. II. Proc. London Math. Soc. (3) 80 (2000), no. 3, 513--544.
- 2-descent on elliptic curves and rational points on certain Kummer surfaces. Adv. Math. 198 (2005), no. 2, 448--483 (joint with A. Skorobogatov).
- Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points. Inventiones math. 134 (1998) 579-650 (joint with J.-L. Colliot-Thélène and A. Skorobogatov).

- An isogeny of K3 surfaces (with B. van Geemen, Bull. London Math. Soc. 38 (2006), 209--223)
- The Hasse zeta function of a K3 surface related to the number of words of weight 5 in the Melas codes (with C. Peters and M. van der Vlugt, J. reine angew. Math. 432 (1992), 151--176)
- Selfdual and non-selfdual 3-dimensional Galois representations (with B. van Geemen, Compos. Math 97 (1995), 51--70)
- A singular K3 surface related to sums of consecutive cubes (with M. Kuwata, Indag. Math. 11 (2000), 419--435)
- Sections on certain j=0 elliptic surfaces (with J. Chahal and M. Meijer, Comment. Math. Univ. St. Pauli 49 (2000), 79--89)
- Arithmetic of the [19,1,1,1,1,1] fibration (with M. Schuett), Comment. Math. Univ. St. Pauli 55 (2006), 9--16)
- Explicit equations of some elliptic modular surfaces (with N. Yui, Rocky Mountain J. of Math. 37 (2007), 663--687)
- Congruent number problems and their variants (with N. Yui, 613--639 in Algorithmic Number Theory (J.P. Buhler and P. Stevenhagen, eds.), MSRI Publications, ISBN 978-0-521-80854-5, 2008.)
- Explicit elliptic K3 surfaces with rank 15 (with F. de Zeeuw, to appear in Rocky Mountain J. of Math., 2009.)

- 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, also available at arXiv:math.AG/0511542).
- Nontrivial elements of Sha explained through K3 surfaces and accompanying files, Math. Comp. 78 (2009), 441-483. (joint work with Adam Logan, also available at arXiv:0706.0541).
- K3 surfaces with Picard number three and canonical vector heights, Mathematics of Computation, Volume 76 (2007), 1493-1498 (joint work with Arthur Baragar, different version available at arXiv:math.AG/0602166 ).
- An elliptic K3 surface associated to Heron triangles, Journal of Number Theory, Volume 123 (2007), 92-119, also available at arXiv:math.AG/0411606).
- A K3 surface associated to certain integral matrices with integral eigenvalues, Canadian Mathematical Bulletin, Volume 49 (4), 2006, 560-577, also available at arXiv:math.AG/0411600).
- Cubics points on cubic curves and the Brauer-Manin obstruction for K3 surfaces and accompanying files, also available at arXiv:0708.2752.
- Density of rational points on diagonal quartic surfaces (joint with A. Logan and D. McKinnon).

- 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.

- 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.