Journal of the Ramanujan Mathematical Society

Open Access Open Access  Restricted Access Subscription Access
Open Access Open Access Open Access  Restricted Access Restricted Access Subscription Access

Sufficient Conditions for the Existence of Rational Points on Diagonal Cubic Surfaces


Affiliations
1 Bannnai laboratory at Keio University in Japan, Japan
     

   Subscribe/Renew Journal


In this paper, we construct diagonal cubic surfaces over ℚ which have a ℚ-rational point under the assumption that the Tate-Shafarevich group of elliptic curve X3 + Y3 = AZ3 is finite. We can also check that there is no Brauer-Manin obstruction for these surfaces without the finiteness assumption of the Tate-Shafarevich group.
User
Subscription Login to verify subscription
Notifications
Font Size

  • [BF] C. L. Basile and T. A. Fisher, Diagonal cubic equations in four variables with prime coefficients, Rational points on algebraic varieties, Progr. Math., Birkh¨auser, Basel, 199 (2001) 1–12.

  • [Coh] H. Cohen, Number theory, Vol. I. Tools and Diophantine equations, Graduate Texts in Mathematics, Springer, New York, 239 (2007).

  • [Cora] D. F. Coray, Algebraic points on cubic hypersurfaces, Acta Arith., 30 no. 3, (1976) 267–296.

  • [Corn] Patrick Corn, Del Pezzo surfaces and the Brauer-Manin obstruction, PhD thesis, University of California, Berkeley (2005).

  • [CKS] Jean-Louis Colliot-Th´el`ene, Dimitri Kanevsky and Jean-Jacques Sansuc, Arithm´etique des surfaces cubiques diagonales, Diophantine Approximation and Transcendence Theory (Bonn 1985), Lecture Notes in Math., Springer, Berlin, 1290 (1987) 1–108.

  • [PV] Bjorn Poonen and Jos´e Felipe Voloch, Random Diophantine equations, Arithmetic of higher dimensional algebraic varieties (Palo Alto, CA, 2002), 2004, 175–184. With appendices by Jean-Louis Colliot-Th´el´ene and Nicholas M. Katz.

  • [Sat] Kazuki Sato, Rational points on diagonal cubic surfaces, J. Ramanujan Math. Soc., 30 no. 3, (2015) 295–308.

  • [SD] P. Swinnerton-Dyer, The solubility of diagonal cubic surfaces, Ann. Sci. ´Ecole Norm. Sup. (4), 34 no. 6, (2001) 891–912.

  • [Sko] A. N. Skorobogatov, Torsors and rational points, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 144 (2001).


Abstract Views: 259

PDF Views: 0




  • Sufficient Conditions for the Existence of Rational Points on Diagonal Cubic Surfaces

Abstract Views: 259  |  PDF Views: 0

Authors

Yosuke Shimizu
Bannnai laboratory at Keio University in Japan, Japan

Abstract


In this paper, we construct diagonal cubic surfaces over ℚ which have a ℚ-rational point under the assumption that the Tate-Shafarevich group of elliptic curve X3 + Y3 = AZ3 is finite. We can also check that there is no Brauer-Manin obstruction for these surfaces without the finiteness assumption of the Tate-Shafarevich group.

References