Hostname: page-component-784d4fb959-jtjl6 Total loading time: 0 Render date: 2025-07-15T08:37:59.841Z Has data issue: false hasContentIssue false
  • English
  • Français

On the parity of ranks of Selmer groups IV. With an appendix by Jean-Pierre Wintenberger

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  03 December 2009

Jan Nekovář
Jan Nekovář*
Affiliation:
Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie (Paris 6), Théorie des Nombres (Case 247), 4, place Jussieu, F-75252 Paris cedex 05, France (email: nekovar@math.jussieu.fr)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove the parity conjecture for the ranks of p-power Selmer groups (p⁄=2) of a large class of elliptic curves defined over totally real number fields.

Keywords

Selmer groupselliptic curvesparity conjecturemodularity

MSC classification

Secondary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11G05: Elliptic curves over global fields 14G10: Zeta-functions and related questions (Birch-Swinnerton-Dyer conjecture)

Information

Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 6 , November 2009 , pp. 1351 - 1359
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Bertolini, M. and Darmon, H., Kolyvagin’s descent and Mordell–Weil groups over ring class fields, J. Reine Angew. Math. 412 (1990), 63–74.Google Scholar
[2]Coates, J., Fukaya, T., Kato, K. and Sujatha, R., Root numbers, Selmer groups and non-commutative Iwasawa theory, J. Algebraic Geom., to appear.Google Scholar
[3]Cornut, C. and Vatsal, V., Nontriviality of Rankin–Selberg L-functions and CM points, in L-functions and Galois representations, Durham, July 2004, LMS Lecture Note Series, vol. 320 (Cambridge University Press, Cambridge, 2007), 121–186.Google Scholar
[4]Curtis, C. W. and Reiner, I., Methods of representation theory, Vol. I (Wiley, New York, 1981).Google Scholar
[5]Deligne, P. and Pappas, G., Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant, Compositio Math. 90 (1994), 59–79.Google Scholar
[6]Dokchitser, T. and Dokchitser, V., Parity of ranks for elliptic curves with a cyclic isogeny, J. Number Theory 128 (2008), 662–679.Google Scholar
[7]Dokchitser, T. and Dokchitser, V., On the Birch–Swinnerton–Dyer quotients modulo squares, Ann. of Math. (2), to appear, http://arxiv.org/abs/0610290.Google Scholar
[8]Dokchitser, T. and Dokchitser, V., Self-duality of Selmer groups, Proc. Cam. Phil. Soc. 146 (2009), 257–267, http://arxiv.org/abs/0705.1899.Google Scholar
[9]Dokchitser, T. and Dokchitser, V., Regulator constants and the parity conjecture, Invent. Math., to appear, http://fr.arxiv.org/abs/0709.2852.Google Scholar
[10]Greenberg, R., Trivial zeros of p-adic L-functions, Contemp. Math. 165 (1994), 149–173.Google Scholar
[11]Greenberg, R., Iwasawa theory, projective modules, and modular representations, Preprint.Google Scholar
[12]Harris, M., Shepherd-Barron, N. and Taylor, R., A family of Calabi–Yau varieties and potential automorphy, Preprint (2006). Available at http://www.math.harvard.edu/∼rtaylor/, Ann. of Math. (2), to appear.Google Scholar
[13]Jacquet, H. and Langlands, R. P., Automorphic forms on GL(2), Lecture Notes in Mathematics, vol. 114 (Springer, Berlin, 1970).Google Scholar
[14]Kim, B. D., The parity theorem of elliptic curves at primes with supersingular reduction, Compositio Math. 143 (2007), 47–72.CrossRefGoogle Scholar
[15]Kim, B. D., The parity conjecture over totally real fields for elliptic curves at supersingular reduction primes, Preprint.Google Scholar
[16]Mazur, B. and Rubin, K., Finding large Selmer rank via an arithmetic theory of local constants, Ann. of Math. (2) 166 (2007), 581–614.Google Scholar
[17]Mazur, B. and Rubin, K., Growth of Selmer rank in nonabelian extensions of number fields, Duke Math. J. 143 (2008), 437–461.CrossRefGoogle Scholar
[18]Monsky, P., Generalizing the Birch–Stephens theorem. I. Modular curves, Math. Z. 221 (1996), 415–420.Google Scholar
[19]Moret-Bailly, L., Groupes de Picard et problèmes de Skolem I, II, Ann. Sci. École Norm. Sup. (4) 22 (1989), 161–179, 181–194Google Scholar
[20]Nekovář, J., On the parity of ranks of Selmer groups II, C. R. Acad. Sci. Paris, Sér. I Math. 332 (2001), 99–104.CrossRefGoogle Scholar
[21]Nekovář, J., Selmer complexes, Astérisque, vol. 310 (Societé Mathématique de France, Paris, 2006).Google Scholar
[22]Nekovář, J., The Euler system method for CM points on Shimura curves, in L-functions and Galois representations, Durham, July 2004, LMS Lecture Note Series, vol. 320 (Cambridge Univesity Press, Cambridge, 2007), 471–547.Google Scholar
[23]Nekovář, J., Growth of Selmer groups of Hilbert modular forms over ring class fields, Ann. Sci. École Norm. Sup. (4) 41 (2008), 1003–1022.Google Scholar
[24]Rapoport, M., Compactifications de l’espace de modules de Hilbert–Blumenthal, Compositio Math. 36 (1978), 255–335.Google Scholar
[25]Serre, J.-P., Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259–331.Google Scholar
[26]Serre, J.-P., Quelques applications du théorème de densité de Chebotarev, Publ. Math. Inst. Hautes Études Sci. 54 (1981), 323–401.Google Scholar
[27]Skinner, C. M., Modularity of Galois representations, Les XXIIèmes Journées Arithmetiques (Lille, 2001), J. Théor. Nombres Bordeaux 15 (2003), 367–381.Google Scholar
[28]Skinner, C. M. and Wiles, A., Nearly ordinary deformations of irreducible residual representations, Ann. Fac. Sci. Toulouse Math. (6) 10 (2001), 185–215.Google Scholar
[29]Taylor, R., On Galois representations associated to Hilbert modular forms, Invent. Math. 98 (1989), 265–280.Google Scholar
[30]Taylor, R., Remarks on a conjecture of Fontaine and Mazur, J. Inst. Math. Jussieu 1 (2002), 1–19.CrossRefGoogle Scholar
[31]Taylor, R., On icosahedral Artin representations II, Amer. J. Math. 125 (2003), 549–566.Google Scholar
[32]Wiles, A., On p-adic representations for totally real fields, Ann. of Math. (2) 123 (1986), 407–456.Google Scholar
[33]Wiles, A., Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), 443–551.Google Scholar
[34]Zhang, S.-W., Heights of Heegner points on Shimura curves, Ann. of Math. (2) 153 (2001), 27–147.Google Scholar