# The Brauer–Manin obstruction for curves having split Jacobians

Samir Siksek^{[1]}

- [1] Department of Mathematics and Statistics Faculty of Science Sultan Qaboos University P.O. Box 36 Al-Khod 123, Oman

Journal de Théorie des Nombres de Bordeaux (2004)

- Volume: 16, Issue: 3, page 773-777
- ISSN: 1246-7405

## Access Full Article

top## Abstract

top## How to cite

topSiksek, Samir. "The Brauer–Manin obstruction for curves having split Jacobians." Journal de Théorie des Nombres de Bordeaux 16.3 (2004): 773-777. <http://eudml.org/doc/249265>.

@article{Siksek2004,

abstract = {Let $X \rightarrow \{\mathcal\{A\}\}$ be a non-constant morphism from a curve $X$ to an abelian variety $\{\mathcal\{A\}\}$, all defined over a number field $k$. Suppose that $X$ is a counterexample to the Hasse principle. We give sufficient conditions for the failure of the Hasse principle on $X$ to be accounted for by the Brauer–Manin obstruction. These sufficiency conditions are slightly stronger than assuming that $\{\mathcal\{A\}\}(k)$ and $\{\cyrX\}(\{\mathcal\{A\}\}/k)$ are finite.},

affiliation = {Department of Mathematics and Statistics Faculty of Science Sultan Qaboos University P.O. Box 36 Al-Khod 123, Oman},

author = {Siksek, Samir},

journal = {Journal de Théorie des Nombres de Bordeaux},

language = {eng},

number = {3},

pages = {773-777},

publisher = {Université Bordeaux 1},

title = {The Brauer–Manin obstruction for curves having split Jacobians},

url = {http://eudml.org/doc/249265},

volume = {16},

year = {2004},

}

TY - JOUR

AU - Siksek, Samir

TI - The Brauer–Manin obstruction for curves having split Jacobians

JO - Journal de Théorie des Nombres de Bordeaux

PY - 2004

PB - Université Bordeaux 1

VL - 16

IS - 3

SP - 773

EP - 777

AB - Let $X \rightarrow {\mathcal{A}}$ be a non-constant morphism from a curve $X$ to an abelian variety ${\mathcal{A}}$, all defined over a number field $k$. Suppose that $X$ is a counterexample to the Hasse principle. We give sufficient conditions for the failure of the Hasse principle on $X$ to be accounted for by the Brauer–Manin obstruction. These sufficiency conditions are slightly stronger than assuming that ${\mathcal{A}}(k)$ and ${\cyrX}({\mathcal{A}}/k)$ are finite.

LA - eng

UR - http://eudml.org/doc/249265

ER -

## References

top- J.W.S. Cassels, E.V. Flynn, Prolegomena to a middlebrow arithmetic of curves of genus $2$. L.M.S. lecture notes series 230, Cambridge University Press, 1996. Zbl0857.14018MR1406090
- J.W.S. Cassels, A. Fröhlich, Algebraic number theory. Academic press, New York, 1967. Zbl0153.07403MR215665
- D. Coray, C. Manoil, On large Picard groups and the Hasse Principle for curves and K3 surfaces. Acta Arith. LXXVI.2 (1996), 165–189. Zbl0877.14005MR1393513
- J.E. Cremona, Algorithms for modular elliptic curves. second edition, Cambridge University Press, 1996. Zbl0758.14042MR1628193
- V.A. Kolyvagin, Euler systems. I The Grothendieck Festschrift, Vol. II, 435–483, Progr. Math. 87, Birkhäuser, Boston, 1990. Zbl0742.14017MR1106906
- V. Scharaschkin, The Brauer-Manin obstruction for curves. To appear. Zbl1169.14021
- A.N. Skorobogatov, Torsors and rational points. Cambridge Tracts in Mathematics 144, Cambridge University Press, 2001. Zbl0972.14015MR1845760

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.