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ISSN 1088-6842(e) ISSN 0025-5718(p)

Tate-Shafarevich groups and surfaces

Author(s): Patrick Corn.
Journal: Math. Comp. 79 (2010), 563-581.
MSC (2000): Primary 14H40; Secondary 11G10
Posted: June 5, 2009
MathSciNet review: 2552241
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Abstract: This paper explores a topic taken up recently by Logan and van Luijk, finding nontrivial -torsion elements of the Tate-Shafarevich group of the Jacobian of a genus- curve by exhibiting Brauer-Manin obstructions to rational points on certain quotients of principal homogeneous spaces of the Jacobian, whose desingularizations are explicit 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:

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 http://magma.maths.usyd.edu.au/magma/ . 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 .
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 .
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 . 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 .
http://www.liv.ac.uk/adaml/math/index.html, 2004.
12.: Adam Logan and Ronald van Luijk.
Nontrivial elements of Sha explained through surfaces.
Math. Comp., 78:441-483, 2009. MR 2448716
13.: James Milne.
Class field theory.
Online lecture notes, http://www.jmilne.org/math/CourseNotes/math776.pdf, 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 -descent on Jacobians of hyperelliptic curves.
Acta Arith., 98:245-277, 2001. MR 1829626 (2002b:11089)

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