|
Nontrivial elements of Sha explained through K3 surfaces
Author(s):
Adam
Logan;
Ronald
van Luijk.
Journal:
Math. Comp.
78
(2009),
441-483.
MSC (2000):
Primary 14H40, 11G10, 14J27, 14J28
Posted:
May 2, 2008
Retrieve article in:
PDF
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:
-
- 1.
- W. Barth, C. Peters, and A. van de Ven, Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 4, Springer-Verlag, 1984. MR 749574 (86c:32026)
- 2.
- E. Bombieri and D. Mumford, Enriques' classification of surfaces in char.
 , II. Complex Analysis and Algebraic Geometry - Collection of papers dedicated to K. Kodaira, W.L. Baily and T. Shioda, eds. Iwanami and Cambridge Univ. Press (1977), 23-42. MR 0491719 (58:10922a) - 3.
- M. Bright, Brauer Groups of Diagonal Quartic Surfaces, J. Symbolic Computation 41 (2006), 544-558. MR 2209163 (2007h:14022)
- 4.
- M. Bright, Computations on Diagonal Quartic Surfaces, Unpublished Ph.D. dissertation, Cambridge University, 2002, available at www.boojum.org.uk.
- 5.
- M. Bright and H. P. F. Swinnerton-Dyer, Computing the Brauer-Manin obstructions, Math. Proc. Camb. Phil. Soc. 137 (2004), 1-16. MR 2075039 (2005c:11081)
- 6.
- N. Bruin and E. V. Flynn, Exhibiting SHA[2] on hyperelliptic Jacobians, J. Number Theory 118 (2006), 266-291. MR 2225283 (2006m:11091)
- 7.
- J. W. S. Cassels and E. V. Flynn, Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2, LMS Lecture Notes 230, Cambridge University Press (1996). MR 1406090 (97i:11071)
- 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.
- R. Hartshorne, Algebraic Geometry, GTM 52, Springer-Verlag, New York, 1977. MR 0463157 (57:3116)
- 10.
- R. Hartshorne, Equivalence relations of algebraic cycles and subvarieties of small codimension, Algebraic Geometry, Arcata 1974, Amer. Math. Soc. Proc. Symp. Pure Math. 29 (1975), 129-164. MR 0369359 (51:5592)
- 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. Manin, Cubic forms: algebra, geometry, arithmetic, North-Holland Mathematical Library 4, North-Holland, Amsterdam, 1986 (2nd edition, translated by M. Hazewinkel). MR 833513 (87d:11037)
- 13.
- J. S. Milne, Étale Cohomology, Princeton Mathematical Series 33, Princeton University Press, Princeton, 1980. MR 559531 (81j:14002)
- 14.
- D.R. Morrison, On K3 surfaces with large Picard number, Invent. Math. 75 (1984), 105-121. MR 728142 (85j:14071)
- 15.
- B. Poonen and M. Stoll, The Cassels-Tate pairing on polarized abelian varieties, Ann. of Math. (2) 150 (1999), no. 3, 1109-1149. MR 1740984 (2000m:11048)
- 16.
- E.F. Schaefer, Computing a Selmer group of a Jacobian using functions on the curve, Math. Ann. 310 (1998), 447-471. MR 1612262 (99h:11063)
- 17.
- J.-P. Serre, Local Fields, GTM 67, Springer-Verlag, New-York, 1979. MR 554237 (82e:12016)
- 18.
- T. Shioda, Algebraic cycles on certain K3 surfaces in characteristic
 , Manifolds-Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973), Univ. Tokyo Press, Tokyo (1975), 357-364. MR 0435084 (55:8046) - 19.
- A. Skorobogatov, Torsors and rational points, Cambridge Tracts in Mathematics 144. Cambridge University Press, Cambridge, 2001. MR 1845760 (2002d:14032)
- 20.
- M. Stoll, Implementing 2-descent for Jacobians of hyperelliptic curves, Acta Arith. 98 (2001), no. 3, 245-277. MR 1829626 (2002b:11089)
- 21.
- R. van Luijk, An elliptic K3 surface associated to Heron triangles, Journal of Number Theory, 123 (2007), 92-119. MR 2295433 (2007k:14077)
- 22.
- R. van Luijk, K3 surfaces with Picard number one and infinitely many rational points, J. Algebra Number Theory 1 (2007), no. 1, 1-15. MR 2322921
- 23.
- Electronic resources, available at http://oldweb.cecm.sfu.ca/~rluijk/ShaK3/.
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
Email:
a5logan@math.uwaterloo.ca
Ronald
van Luijk
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada, V5A 1S6
Email:
rmluijk@gmail.com
DOI:
10.1090/S0025-5718-08-02105-4
PII:
S 0025-5718(08)02105-4
Received by editor(s):
June 16, 2007
Received by editor(s) in revised form:
November 19, 2007
Posted:
May 2, 2008
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org