The Brauer–Manin obstruction for curves having split Jacobians
Journal de théorie des nombres de Bordeaux, Volume 16 (2004) no. 3, p. 773-777

Let X𝒜 be a non-constant morphism from a curve X to an abelian variety 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 𝒜(k) and Ш(𝒜/k) are finite.

Soit X𝒜 un morphisme (qui n’est pas constant) d’une courbe X vers une variété abélienne 𝒜, tous définis sur un corps de nombres k. Supposons que X ne satisfait pas le principe de Hasse. Nous donnons des conditions suffisantes pour que l’obstruction de Brauer-Manin soit la seule obstruction au principe de Hasse. Ces conditions suffisantes sont légèrement plus fortes que de supposer que 𝒜(k) et Ш(𝒜/k) sont finis.

     author = {Siksek, Samir},
     title = {The Brauer--Manin obstruction for curves having split Jacobians},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {16},
     number = {3},
     year = {2004},
     pages = {773-777},
     doi = {10.5802/jtnb.469},
     mrnumber = {2144966},
     zbl = {1076.14033},
     language = {en},
     url = {}
Siksek, Samir. The Brauer–Manin obstruction for curves having split Jacobians. Journal de théorie des nombres de Bordeaux, Volume 16 (2004) no. 3, pp. 773-777. doi : 10.5802/jtnb.469.

[1] 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. | MR 1406090 | Zbl 0857.14018

[2] J.W.S. Cassels, A. Fröhlich, Algebraic number theory. Academic press, New York, 1967. | MR 215665 | Zbl 0153.07403

[3] D. Coray, C. Manoil, On large Picard groups and the Hasse Principle for curves and K3 surfaces. Acta Arith. LXXVI.2 (1996), 165–189. | MR 1393513 | Zbl 0877.14005

[4] J.E. Cremona, Algorithms for modular elliptic curves. second edition, Cambridge University Press, 1996. | MR 1628193 | Zbl 0758.14042

[5] V.A. Kolyvagin, Euler systems. I The Grothendieck Festschrift, Vol. II, 435–483, Progr. Math. 87, Birkhäuser, Boston, 1990. | MR 1106906 | Zbl 0742.14017

[6] V. Scharaschkin, The Brauer-Manin obstruction for curves. To appear.

[7] A.N. Skorobogatov, Torsors and rational points. Cambridge Tracts in Mathematics 144, Cambridge University Press, 2001. | MR 1845760 | Zbl 0972.14015