manuscripta mathematica

, Volume 115, Issue 4, pp 437–466

The Hasse principle and the Brauer-Manin obstruction for curves

Article

Abstract.

We discuss a range of ways, extending existing methods, to demonstrate violations of the Hasse principle on curves. Of particular interest are curves which contain a rational divisor class of degree 1, even though they contain no rational point. For such curves we construct new types of examples of violations of the Hasse principle which are due to the Brauer-Manin obstruction, subject to the conjecture that the Tate-Shafarevich group of the Jacobian is finite.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bost, J.-B., Mestre, J.-F.: Moyenne arithmético-géométrique et périodes des courbes de genre 1 et 2. Gaz. Math. Soc. France 38, 36–64 (1988)MATHGoogle Scholar
  2. 2.
    Bremner, A.: Some interesting curves of genus 2 to 7. J. Number Theory 67, 277–290 (1997)CrossRefMATHGoogle Scholar
  3. 3.
    Bremner, A., Lewis, D.J., Morton, P.: Some varieties with points only in a field extension. Arch. Math. 43, 344–350 (1984)MATHGoogle Scholar
  4. 4.
    Bright, M.J., Swinnerton-Dyer, H.P.F.: Computing the Brauer-Manin obstructions. Math. Proc. Cam. Phil. Soc. 137, 1–16 (2004)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Bruin, N.: Chabauty methods using covers on curves of genus 2. http://www.math.leidenuniv.nl/reports/1999-15.shtml
  6. 6.
    Bruin, N.: KASH-based program for performing 2-descent on elliptic curves over number fields. Available from: http://www.math.uu.nl/people/bruin/ell.shar
  7. 7.
    Bruin, N., Flynn, E.V.: Towers of 2-covers of hyperelliptic curves. To appear in Trans. Am. Math. Soc.Google Scholar
  8. 8.
    Bruin, N.: Routines to accompany the article The primitive solutions to the equation x3 + y9 = z2. Available from: http://www.cecm.sfu.ca/ nbruin/eq239/routines.m
  9. 9.
    Cassels, J.W.S.: The arithmetic of certain quartic curves. Proc. Royal Soc. Edinburgh 100A, 201–218 (1985)MATHGoogle Scholar
  10. 10.
    Cassels, J.W.S., Flynn, E.V.: Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2. LMS–LNS 230, Cambridge University Press, Cambridge, 1996Google Scholar
  11. 11.
    Chabauty, C.: Sur les points rationnels des variétés algébriques dont l’irrégularité est supérieure à la dimension. C. R. Acad. Sci. Paris 212, 1022–1024 (1941)MATHGoogle Scholar
  12. 12.
    Coleman, R.F. Effective Chabauty. Duke Math. J. 52, 765–780 (1985)Google Scholar
  13. 13.
    Coray, D., Manoil, C.: On large Picard groups and the Hasse principle for curves and K3 surfaces. Acta. Arith. LXXVI.2, 165–189 (1996)Google Scholar
  14. 14.
    Flynn, E.V.: An explicit theory of heights. Trans. Am. Math. Soc. 347, 3003–3015 (1995)MATHGoogle Scholar
  15. 15.
    Flynn, E.V.: A flexible method for applying Chabauty’s theorem. Compositio Mathematica 105, 79–94 (1997)CrossRefMATHGoogle Scholar
  16. 16.
    Flynn, E.V., Smart, N.P.: Canonical heights on the Jacobians of curves of genus 2 and the infinite descent. Acta Arith. 79, 333–352 (1997)MATHGoogle Scholar
  17. 17.
    Flynn, E.V., Wetherell, J.L.: Finding rational points on bielliptic genus 2 curves. Manuscripta Math. 100, 519–533 (1999)CrossRefMATHGoogle Scholar
  18. 18.
    Flynn, E.V.: On Q-derived polynomials. Proc. Edinburgh Math. Soc. 44, 103–110 (2001)CrossRefMATHGoogle Scholar
  19. 19.
    Flynn, E.V., Wetherell, J.L.: Covering collections and a challenge problem of Serre. Acta Arithmetica XCVIII.2, 197–205 (2001)Google Scholar
  20. 20.
    The Magma Computational Algebra System. Available from http://magma.maths.usyd.edu.au/magma/
  21. 21.
    McCallum, W.G.: On the method of Coleman and Chabauty. Math. Ann. 299 (3), 565–596 (1994)MATHGoogle Scholar
  22. 22.
    Poonen, B., Stoll, M.: The Cassels-Tate pairing on polarized abelian varieties. Ann. Math. (2) 150 (3), 1109–1149 (1999)Google Scholar
  23. 23.
    Scharaschkin, V.: Local Global Problems and the Brauer-Manin Obstruction. PhD Thesis, University of Michigan, 1999Google Scholar
  24. 24.
    Scharaschkin, V.: The Brauer-Manin obstruction for 0-cycles. Manuscript, 2003Google Scholar
  25. 25.
    Siksek, S.: On the Brauer-Manin obstruction to the Hasse principle for curves of split Jacobians. Preprint, 2003Google Scholar
  26. 26.
    Siksek, S., Skorobogatov, A.N.: On a Shimura curve that is a counterexample to the Hasse principle. Bull. London Math. Soc. 35 (3), 409–414 (2003)CrossRefMATHGoogle Scholar
  27. 27.
    Skorobogatov, A.N.: Torsors and rational points. CTM 144, Cambridge Univ. Press, 2001Google Scholar
  28. 28.
    Stoll, M.: Two simple 2-dimensional abelian varieties defined over Open image in new window with Mordell-Weil rank at least 19. C. R. Acad. Sci. Paris, Série I, 321, 1341–1344 (1995)Google Scholar
  29. 29.
    Stoll, M.: On the height constant for curves of genus two. Acta Arith. 90, 183–201 (1999)MATHGoogle Scholar
  30. 30.
    Stoll, M.: Implementing 2-descent for Jacobians of hyperelliptic curves. Acta Arith. 98, 245–277 (2001)MATHGoogle Scholar
  31. 31.
    Stoll, M.: On the height constant for curves of genus two, II. Acta Arith. 104, 165–182 (2002)MATHGoogle Scholar
  32. 32.
    Swinnerton-Dyer, H.P.F.: Arithmetic of diagonal quartic surfaces, II. Proc. London Math. Soc. 80, 513–544 (2000)CrossRefMATHGoogle Scholar
  33. 33.
    Weil, A.: Variétés abeliennes et courbes algébriques. Hermann & Cie, Paris, 1948Google Scholar
  34. 34.
    Wetherell, J.L.: Bounding the number of rational points on certain curves of high rank. PhD Dissertation, University of California at Berkeley, 1997Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department Mathematical SciencesUniversity of LiverpoolLiverpoolEngland

Personalised recommendations