Abstract
Kolyvagin has shown how to study the Shafarevich-Tate group of elliptic curves over imaginary quadratic fields via Kolyvagin classes constructed from Heegner points. One way to produce explicit non-trivial elements of the Shafarevich-Tate group is by proving that a locally trivial Kolyvagin class is globally non-trivial, which is difficult in practice. We provide a method for testing whether an explicit element of the Shafarevich-Tate group represented by a Kolyvagin class is globally non-trivial by determining whether the Cassels pairing between the class and another locally trivial Kolyvagin class is non-zero. Our algorithm explicitly computes Heegner points over ring class fields to produce the Kolyvagin classes and uses the efficiently computable cryptographic Tate pairing.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Barreto, P.S.L.M., Kim, H.Y., Lynn, B., Scott, M.: Efficient algorithms for pairing-based cryptosystems. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 354–368. Springer, Heidelberg (2002)
Breuil, C., Conrad, B., Diamond, F., Taylor, R.: On the modularity of elliptic curves over \(\bold Q\): wild 3-adic exercises. J. Amer. Math. Soc. 14(4), 843–939 (2001)
Cohen, H.: Number Theory Volume I: Tools and Diophantine equations. Graduate Texts in Mathematics, vol. 239. Springer, New York (2007), http://math.arizona.edu/~swc/notes/files/06CohenExtract.pdf
Cohen, H., Frey, G.: Handbook of Elliptic and Hyperelliptic Curve Cryptography. Chapman and Hall, CRC (2006)
Cremona, J.E.: Elliptic curves of conductor ≤ 25000, http://www.warwick.ac.uk/~masgaj/ftp/data/INDEX.html
Elkies, N.: Heegner point computations. In: Huang, M.-D.A., Adleman, L.M. (eds.) ANTS 1994. LNCS, vol. 877, pp. 122–133. Springer, Heidelberg (1994)
Eisenträger, K., Lauter, K., Montgomery, P.L.: Improved Weil and Tate pairings for elliptic and hyperelliptic curves. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 169–183. Springer, Heidelberg (2004)
Frey, G.: Applications of arithmetical geometry to cryptographic constructions. In: Finite fields and applications (Augsburg, 1999), pp. 128–161. Springer, Berlin (2001)
Frey, G., Lange, T.: Mathematical background of public key cryptography. In: Frey, G., Lange, T. (eds.) Arithmetic, geometry and coding theory (AGCT 2003), Soc. Math. France, Paris. Sémin. Congr., vol. 11, pp. 41–73 (2005)
Grigorov, G., Jorza, A., Patrikis, S., Stein, W., Tarniţǎ-Patraşcu, C.: Verification of the Birch and Swinnerton-Dyer conjecture for specific elliptic curves (preprint)
Gross, B.: Kolyvagin’s work on modular elliptic curves. In: L-functions and arithmetic (Durham, 1989), pp. 235–256. Cambridge Univ. Press, Cambridge (1991)
Gross, B., Zagier, D.: Heegner points and derivatives of L-series. Invent. Math. 84(2), 225–320 (1986)
Jetchev, D., Lauter, K., Stein, W.: Explicit Heegner Points: Kolyvagin’s Conjecture and Non-trivial Elements in the Shafarevich-Tate group (preprint, 2007)
Jetchev, D., Stein, W.: Visualizing elements of the Shafarevich-Tate group at higher level (preprint)
Kolyvagin, V.A.: Euler systems. In: The Grothendieck Festschrift, vol. II, pp. 435–483. Birkhäuser, Boston (1990)
Lang, S.: Algebraic groups over finite fields. Amer. J. Math. 78, 555–563 (1956)
Lichtenbaum, S.: Duality theorems for curves over p-adic fields. Inv. Math. 7, 120–136 (1969)
Mazur, B., Rubin, K.: private communication
McCallum, W.G.: Kolyvagin’s work on Shafarevich-Tate groups. In: L-functions and arithmetic (Durham, 1989), pp. 295–316. Cambridge Univ. Press, Cambridge (1991)
Milne, J.S.: Arithmetic duality theorems. Academic Press Inc., Boston (1986)
Nekovar, J., Schappacher, N.: On the asymptotic behaviour of Heegner points. Turkish J. of Math. 23(4), 549–556 (1999)
Serre, J.-P.: Local fields, GTM, vol. 67. Springer, New York (1979)
Serre, J.-P.: Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15(4), 259–331 (1972)
Silverman, J.H.: The arithmetic of elliptic curves. Springer, New York (1992)
Silverman, J.H.: Advanced topics in the arithmetic of elliptic curves. Springer, New York (1994)
Stein, W.: SAGE, http://modular.math.washington.edu/sage/
Watkins, M.: Some remarks on Heegner point computations (preprint, 2004)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Eisenträger, K., Jetchev, D., Lauter, K. (2008). Computing the Cassels Pairing on Kolyvagin Classes in the Shafarevich-Tate Group. In: Galbraith, S.D., Paterson, K.G. (eds) Pairing-Based Cryptography – Pairing 2008. Pairing 2008. Lecture Notes in Computer Science, vol 5209. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85538-5_8
Download citation
DOI: https://doi.org/10.1007/978-3-540-85538-5_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-85503-3
Online ISBN: 978-3-540-85538-5
eBook Packages: Computer ScienceComputer Science (R0)