Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Nontrivial elements of Sha explained through K3 surfaces

Authors: Adam Logan and Ronald van Luijk
Journal: Math. Comp. 78 (2009), 441-483
MSC (2000): Primary 14H40, 11G10, 14J27, 14J28
Published electronically: May 2, 2008
MathSciNet review: 2448716
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We present a new method to show that a principal homogeneous space of the Jacobian of a curve of genus two is nontrivial. The idea is to exhibit a Brauer-Manin obstruction to the existence of rational points on a quotient of this principal homogeneous space. In an explicit example we apply the method to show that a specific curve has infinitely many quadratic twists whose Jacobians have nontrivial Tate-Shafarevich group.

References [Enhancements On Off] (What's this?)

  • 1. W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. MR 749574
  • 2. E. Bombieri and D. Mumford, Enriques’ classification of surfaces in char. 𝑝. II, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 23–42. MR 0491719
  • 3. Martin Bright, Brauer groups of diagonal quartic surfaces, J. Symbolic Comput. 41 (2006), no. 5, 544–558. MR 2209163, 10.1016/j.jsc.2005.10.001
  • 4. M. Bright, Computations on Diagonal Quartic Surfaces, Unpublished Ph.D. dissertation, Cambridge University, 2002, available at
  • 5. Martin Bright and Peter Swinnerton-Dyer, Computing the Brauer-Manin obstructions, Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 1, 1–16. MR 2075039, 10.1017/S0305004104007571
  • 6. N. Bruin and E. V. Flynn, Exhibiting SHA[2] on hyperelliptic Jacobians, J. Number Theory 118 (2006), no. 2, 266–291. MR 2225283, 10.1016/j.jnt.2005.10.007
  • 7. J. W. S. Cassels and E. V. Flynn, Prolegomena to a middlebrow arithmetic of curves of genus 2, London Mathematical Society Lecture Note Series, vol. 230, Cambridge University Press, Cambridge, 1996. MR 1406090
  • 8. P. Deligne, Cohomologie des intersections complètes, Exposé XI in Groupes de monodromie en géométrie algébrique (SGA 7 II), Lecture Notes in Math. 340, Springer, Berlin, 1973.
  • 9. Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
  • 10. Robin Hartshorne, Equivalence relations on algebraic cycles and subvarieties of small codimension, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974) Amer. Math. Soc., Providence, R.I., 1973, pp. 129–164. MR 0369359
  • 11. A. Logan, The Brauer-Manin obstruction on del Pezzo surfaces of degree 2 branched along a plane section of a Kummer surface, Math. Proc. Camb. Phil. Soc., to appear.
  • 12. Yu. I. Manin, Cubic forms, 2nd ed., North-Holland Mathematical Library, vol. 4, North-Holland Publishing Co., Amsterdam, 1986. Algebra, geometry, arithmetic; Translated from the Russian by M. Hazewinkel. MR 833513
  • 13. James S. Milne, Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
  • 14. D. R. Morrison, On 𝐾3 surfaces with large Picard number, Invent. Math. 75 (1984), no. 1, 105–121. MR 728142, 10.1007/BF01403093
  • 15. Bjorn Poonen and Michael Stoll, The Cassels-Tate pairing on polarized abelian varieties, Ann. of Math. (2) 150 (1999), no. 3, 1109–1149. MR 1740984, 10.2307/121064
  • 16. Edward F. Schaefer, Computing a Selmer group of a Jacobian using functions on the curve, Math. Ann. 310 (1998), no. 3, 447–471. MR 1612262, 10.1007/s002080050156
  • 17. Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237
  • 18. Tetsuji Shioda, Algebraic cycles on certain 𝐾3 surfaces in characteristic𝑝, Manifolds–Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973) Univ. Tokyo Press, Tokyo, 1975, pp. 357–364. MR 0435084
  • 19. Alexei Skorobogatov, Torsors and rational points, Cambridge Tracts in Mathematics, vol. 144, Cambridge University Press, Cambridge, 2001. MR 1845760
  • 20. Michael Stoll, Implementing 2-descent for Jacobians of hyperelliptic curves, Acta Arith. 98 (2001), no. 3, 245–277. MR 1829626, 10.4064/aa98-3-4
  • 21. Ronald van Luijk, An elliptic 𝐾3 surface associated to Heron triangles, J. Number Theory 123 (2007), no. 1, 92–119. MR 2295433, 10.1016/j.jnt.2006.06.006
  • 22. Ronald van Luijk, K3 surfaces with Picard number one and infinitely many rational points, Algebra Number Theory 1 (2007), no. 1, 1–15. MR 2322921, 10.2140/ant.2007.1.1
  • 23. Electronic resources, available at

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 14H40, 11G10, 14J27, 14J28

Retrieve articles in all journals with MSC (2000): 14H40, 11G10, 14J27, 14J28

Additional Information

Adam Logan
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, ON, Canada, N2L 3G1

Ronald van Luijk
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada, V5A 1S6

Received by editor(s): June 16, 2007
Received by editor(s) in revised form: November 19, 2007
Published electronically: May 2, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.