Tate-Shafarevich groups and $K3$ surfaces
HTML articles powered by AMS MathViewer
- by Patrick Corn PDF
- Math. Comp. 79 (2010), 563-581 Request permission
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
- Paola Argentin. Sur certaines surfaces de Kummer. Ph.D. thesis, Université de Genève, 2006.
- Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
- M. J. Bright, N. Bruin, E. V. Flynn, and A. Logan, The Brauer-Manin obstruction and Sh[2], LMS J. Comput. Math. 10 (2007), 354–377. MR 2342713, DOI 10.1112/S1461157000001455
- Martin Bright. Computations on diagonal quartic surfaces. Ph.D. thesis, Cambridge University, 2002.
- Martin Bright, Efficient evaluation of the Brauer-Manin obstruction, Math. Proc. Cambridge Philos. Soc. 142 (2007), no. 1, 13–23. MR 2296387, DOI 10.1017/S0305004106009844
- N. Bruin and E. V. Flynn, Exhibiting SHA[2] on hyperelliptic Jacobians, J. Number Theory 118 (2006), no. 2, 266–291. MR 2225283, DOI 10.1016/j.jnt.2005.10.007
- 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, DOI 10.1017/CBO9780511526084
- Patrick Corn. Del Pezzo surfaces and the Brauer-Manin obstruction. Ph.D. thesis, University of California, Berkeley, 2005.
- Patrick Corn, The Brauer-Manin obstruction on del Pezzo surfaces of degree 2, Proc. Lond. Math. Soc. (3) 95 (2007), no. 3, 735–777. MR 2368282, DOI 10.1112/plms/pdm015
- Andrew Kresch and Yuri Tschinkel, On the arithmetic of del Pezzo surfaces of degree 2, Proc. London Math. Soc. (3) 89 (2004), no. 3, 545–569. MR 2107007, DOI 10.1112/S002461150401490X
- Adam Logan. MAGMA algorithm for computing the Brauer-Manin obstruction on a Del Pezzo surface of degree $4$. http://www.liv.ac.uk/$\sim$adaml/math/index.html, 2004.
- Adam Logan and Ronald van Luijk, Nontrivial elements of Sha explained through $K3$ surfaces, Math. Comp. 78 (2009), no. 265, 441–483. MR 2448716, DOI 10.1090/S0025-5718-08-02105-4
- James Milne. Class field theory. Online lecture notes, http://www.jmilne.org/math/CourseNotes/math776.pdf, 1997.
- 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, DOI 10.2307/121064
- Bjorn Poonen and Edward F. Schaefer, Explicit descent for Jacobians of cyclic covers of the projective line, J. Reine Angew. Math. 488 (1997), 141–188. MR 1465369
- Alexei Skorobogatov, Torsors and rational points, Cambridge Tracts in Mathematics, vol. 144, Cambridge University Press, Cambridge, 2001. MR 1845760, DOI 10.1017/CBO9780511549588
- Michael Stoll, Implementing 2-descent for Jacobians of hyperelliptic curves, Acta Arith. 98 (2001), no. 3, 245–277. MR 1829626, DOI 10.4064/aa98-3-4
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
- Email: corn@math.uga.edu, pkcorn@smcm.edu
- Received by editor(s): March 27, 2008
- Received by editor(s) in revised form: February 4, 2009
- Published electronically: June 5, 2009
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 79 (2010), 563-581
- MSC (2000): Primary 14H40; Secondary 11G10
- DOI: https://doi.org/10.1090/S0025-5718-09-02264-9
- MathSciNet review: 2552241