Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Tate-Shafarevich groups and $ K3$ surfaces

Author: Patrick Corn
Journal: Math. Comp. 79 (2010), 563-581
MSC (2000): Primary 14H40; Secondary 11G10
Published electronically: June 5, 2009
MathSciNet review: 2552241
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper explores a topic taken up recently by Logan and van Luijk, finding nontrivial $ 2$-torsion elements of the Tate-Shafarevich group of the Jacobian of a genus-$ 2$ curve by exhibiting Brauer-Manin obstructions to rational points on certain quotients of principal homogeneous spaces of the Jacobian, whose desingularizations are explicit $ K3$ surfaces. The main difference between the methods used in this paper and those of Logan and van Luijk is that the obstructions are obtained here from explicitly constructed quaternion algebras, rather than elliptic fibrations.

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

  • 1. Paola Argentin.
    Sur certaines surfaces de Kummer.
    Ph.D. thesis, Université de Genève, 2006.
  • 2. Wieb Bosma, John Cannon, and Catherine Playoust.
    The Magma algebra system. I. The user language.
    J. Symbolic Comput., 24(3-4):235-265, 1997.
    Computational algebra and number theory (London, 1993), homepage at . MR 1484478
  • 3. M. J. Bright, N. Bruin, E. V. Flynn, and A. Logan.
    The Brauer-Manin obstruction and Sh [2].
    LMS J. Comp. Math., 10:354-377, 2007. MR 2342713 (2008i:11087)
  • 4. Martin Bright.
    Computations on diagonal quartic surfaces.
    Ph.D. thesis, Cambridge University, 2002.
  • 5. Martin Bright.
    Efficient evaluation of the Brauer-Manin obstruction.
    Math. Proc. Camb. Phil. Soc., 142(1):13-23, 2007. MR 2296387 (2007k:14026)
  • 6. Nils Bruin and E. V. Flynn.
    Exhibiting SHA[2] on hyperelliptic Jacobians.
    J. Number Theory, 118:266-291, 2006. MR 2225283 (2006m:11091)
  • 7. J. W. S. Cassels and E. V. Flynn.
    Prolegomena to a middlebrow arithmetic of curves of genus $ 2$.
    Cambridge University Press, 1996.
    London Math. Soc. Lecture Note Series 230. MR 1406090 (97i:11071)
  • 8. Patrick Corn.
    Del Pezzo surfaces and the Brauer-Manin obstruction.
    Ph.D. thesis, University of California, Berkeley, 2005.
  • 9. Patrick Corn.
    The Brauer-Manin obstruction on Del Pezzo surfaces of degree $ 2$.
    Proc. London Math. Soc., 95(3):735-777, 2007. MR 2368282 (2009a:14027)
  • 10. Andrew Kresch and Yuri Tschinkel.
    On the arithmetic of del Pezzo surfaces of degree $ 2$. Proc. London Math. Soc. (3) 89(3):545-569, 2004. MR 2107007 (2005h:14060)
  • 11. Adam Logan.
    MAGMA algorithm for computing the Brauer-Manin obstruction on a Del Pezzo surface of degree $ 4$.$ \sim$adaml/math/index.html, 2004.
  • 12. Adam Logan and Ronald van Luijk.
    Nontrivial elements of Sha explained through $ K3$ surfaces.
    Math. Comp., 78:441-483, 2009. MR 2448716
  • 13. James Milne.
    Class field theory.
    Online lecture notes,, 1997.
  • 14. B. Poonen and M. Stoll.
    The Cassels-Tate pairing on polarized abelian varieties.
    Ann. Math., 150:1109-1149, 1999. MR 1740984 (2000m:11048)
  • 15. Bjorn Poonen and E. F. Schaefer.
    Explicit descent for Jacobians of cyclic covers of the projective line.
    J. Reine Angew. Math., 488:141-188, 1997. MR 1465369 (98k:11087)
  • 16. Alexei Skorobogatov.
    Torsors and rational points.
    Cambridge University Press, Cambridge, 2001.
    Cambridge Tracts in Mathematics, 144. MR 1845760 (2002d:14032)
  • 17. Michael Stoll.
    Implementing $ 2$-descent on Jacobians of hyperelliptic curves.
    Acta Arith., 98:245-277, 2001. MR 1829626 (2002b:11089)

Similar Articles

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

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

Additional Information

Patrick Corn
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602-7403
Address at time of publication: Department of Mathematics and Computer Science, St. Mary’s College of Maryland, 18952 E. Fisher Road, St. Mary’s City, Maryland 20686-3001

Received by editor(s): March 27, 2008
Received by editor(s) in revised form: February 4, 2009
Published electronically: June 5, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society