Algebra & Number Theory

Hasse principle for Kummer varieties

Yonatan Harpaz and Alexei Skorobogatov

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


The existence of rational points on the Kummer variety associated to a 2-covering of an abelian variety A over a number field can sometimes be established through the variation of the 2-Selmer group of quadratic twists of A. In the case when the Galois action on the 2-torsion of A has a large image, we prove, under mild additional hypotheses and assuming the finiteness of relevant Shafarevich–Tate groups, that the Hasse principle holds for the associated Kummer varieties. This provides further evidence for the conjecture that the Brauer–Manin obstruction controls rational points on K3 surfaces.

Article information

Algebra Number Theory, Volume 10, Number 4 (2016), 813-841.

Received: 8 May 2015
Revised: 8 February 2016
Accepted: 12 March 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14G05: Rational points
Secondary: 11J95: Results involving abelian varieties

Kummer varieties Hasse principle


Harpaz, Yonatan; Skorobogatov, Alexei. Hasse principle for Kummer varieties. Algebra Number Theory 10 (2016), no. 4, 813--841. doi:10.2140/ant.2016.10.813.

Export citation


  • M. Bhargava, C. Skinner, and W. Zhang, “A majority of elliptic curves over $\Q$ satisfy the Birch and Swinnerton-Dyer conjecture”, preprint, 2014.
  • S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models, Results in Mathematics and Related Areas (3) 21, Springer, Berlin, 1990.
  • M. Bright, T. D. Browning, and D. Loughran, “Failures of weak approximation in families”, Compos. Math. (online publication April 2016), 1–41.
  • W. Burnside, Theory of groups of finite order, 2nd ed., Cambridge University Press, 1911.
  • S. D. Cohen, “The distribution of the Galois groups of integral polynomials”, Illinois J. Math. 23:1 (1979), 135–152.
  • J.-L. Colliot-Thélène and A. N. Skorobogatov, “Descent on fibrations over ${\bf P}\sp 1\sb k$ revisited”, Math. Proc. Cambridge Philos. Soc. 128:3 (2000), 383–393.
  • J.-L. Colliot-Thélène, A. N. Skorobogatov, and P. Swinnerton-Dyer, “Hasse principle for pencils of curves of genus one whose Jacobians have rational $2$-division points”, Invent. Math. 134:3 (1998), 579–650.
  • T. Dokchitser and V. Dokchitser, “Root numbers and parity of ranks of elliptic curves”, J. Reine Angew. Math. 658 (2011), 39–64.
  • I. V. Dolgachev, Classical algebraic geometry: A modern view, Cambridge University Press, 2012.
  • M. R. Gonzalez-Dorrego, “$(16,6)$ configurations and geometry of Kummer surfaces in ${\bf P}\sp 3$”, Mem. Amer. Math. Soc. 107:512 (1994), vi+101.
  • C. Hall, “An open-image theorem for a general class of abelian varieties”, Bull. Lond. Math. Soc. 43:4 (2011), 703–711. With an appendix by E. Kowalski.
  • Y. Harpaz and O. Wittenberg, “On the fibration method for zero-cycles and rational points”, Ann. of Math. $(2)$ 183:1 (2016), 229–295.
  • D. Holmes and R. Pannekoek, “The Brauer–Manin obstruction on Kummer varieties and ranks of twists of abelian varieties”, Bull. Lond. Math. Soc. 47:4 (2015), 565–574.
  • Z. Klagsbrun, “Elliptic curves with a lower bound on 2-Selmer ranks of quadratic twists”, Math. Res. Lett. 19:5 (2012), 1137–1143.
  • Z. Klagsbrun, “Selmer ranks of quadratic twists of elliptic curves with partial rational two-torsion”, preprint, 2012.
  • K. Kramer, “Arithmetic of elliptic curves upon quadratic extension”, Trans. Amer. Math. Soc. 264:1 (1981), 121–135.
  • M. Kuwata and L. Wang, “Topology of rational points on isotrivial elliptic surfaces”, Internat. Math. Res. Notices 4 (1993), 113–123.
  • Q. Liu, “Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète”, Trans. Amer. Math. Soc. 348:11 (1996), 4577–4610.
  • B. Mazur, “Rational points of abelian varieties with values in towers of number fields”, Invent. Math. 18 (1972), 183–266.
  • B. Mazur, “The topology of rational points”, Experiment. Math. 1:1 (1992), 35–45.
  • B. Mazur, “Speculations about the topology of rational points: an update”, pp. 165–182 in Columbia University Number Theory Seminar (New York, 1992), Astérisque 228, Société Mathématique de France, Paris, 1995.
  • B. Mazur and K. Rubin, “Finding large Selmer rank via an arithmetic theory of local constants”, Ann. of Math. $(2)$ 166:2 (2007), 579–612.
  • B. Mazur and K. Rubin, “Ranks of twists of elliptic curves and Hilbert's tenth problem”, Invent. Math. 181:3 (2010), 541–575.
  • J. S. Milne, Arithmetic duality theorems, Perspectives in Mathematics 1, Academic Press, Boston, 1986.
  • A. Polishchuk, Abelian varieties, theta functions and the Fourier transform, Cambridge Tracts in Mathematics 153, Cambridge University Press, 2003.
  • B. Poonen and E. Rains, “Self cup products and the theta characteristic torsor”, Math. Res. Lett. 18:6 (2011), 1305–1318.
  • B. Poonen and E. Rains, “Random maximal isotropic subspaces and Selmer groups”, J. Amer. Math. Soc. 25:1 (2012), 245–269.
  • B. Poonen and M. Stoll, “The Cassels–Tate pairing on polarized abelian varieties”, Ann. of Math. $(2)$ 150:3 (1999), 1109–1149.
  • C. H. Sah, “Cohomology of split group extensions, II”, J. Algebra 45:1 (1977), 17–68.
  • A. Skorobogatov, Torsors and rational points, Cambridge Tracts in Mathematics 144, Cambridge University Press, 2001.
  • A. Skorobogatov, “Diagonal quartic surfaces”, (2009). Abstract from Belabas et al., “Explicit methods in number theory”, Oberwolfach Rep. 6:3 (2009), 1843–1920.
  • A. Skorobogatov, “del Pezzo surfaces of degree 4 and their relation to Kummer surfaces”, Enseign. Math. $(2)$ 56:1-2 (2010), 73–85.
  • A. Skorobogatov and P. Swinnerton-Dyer, “2-descent on elliptic curves and rational points on certain Kummer surfaces”, Adv. Math. 198:2 (2005), 448–483.
  • A. N. Skorobogatov and Y. G. Zarhin, “A finiteness theorem for the Brauer group of abelian varieties and $K3$ surfaces”, J. Algebraic Geom. 17:3 (2008), 481–502.
  • A. N. Skorobogatov and Y. G. Zarhin, “The Brauer group of Kummer surfaces and torsion of elliptic curves”, J. Reine Angew. Math. 666 (2012), 115–140.
  • P. Swinnerton-Dyer, “Arithmetic of diagonal quartic surfaces, II”, Proc. London Math. Soc. $(3)$ 80:3 (2000), 513–544.
  • P. Swinnerton-Dyer, “The solubility of diagonal cubic surfaces”, Ann. Sci. École Norm. Sup. $(4)$ 34:6 (2001), 891–912.
  • R. Wisbauer, Foundations of module and ring theory, Algebra, Logic and Applications: A handbook for study and research 3, Gordon and Breach Science Publishers, Philadelphia, PA, 1991.
  • O. Wittenberg, Intersections de deux quadriques et pinceaux de courbes de genre 1/Intersections of two quadrics and pencils of curves of genus 1, Lecture Notes in Mathematics 1901, Springer, Berlin, 2007.