Duke Mathematical Journal

Growth of Selmer rank in nonabelian extensions of number fields

Barry Mazur and Karl Rubin

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let $p$ be an odd prime number, let $E$ be an elliptic curve over a number field $k$, and let $F/k$ be a Galois extension of degree twice a power of $p$. We study the $\mathbf{Z}_p$-corank $\mathrm{rk}_p(E/F)$ of the $p$-power Selmer group of $E$ over $F$. We obtain lower bounds for $\mathrm{rk}_p(E/F)$, generalizing the results in [MR], which applied to dihedral extensions.

If $K$ is the (unique) quadratic extension of $k$ in $F$, if $G = \mathrm{Gal}(F/K)$, if $G^+$ is the subgroup of elements of $G$ commuting with a choice of involution of $F$ over $k$, and if $\mathrm{rk}_p(E/K)$ is odd, then we show that (under mild hypotheses) $\mathrm{rk}_p(E/F) \ge [G:G^+]$.

As a very specific example of this, suppose that $A$ is an elliptic curve over $\mathbf{Q}$ with a rational torsion point of order $p$ and without complex multiplication. If $E$ is an elliptic curve over $\mathbf{Q}$ with good ordinary reduction at $p$ such that every prime where both $E$ and $A$ have bad reduction has odd order in $\mathbf{F}_p^\times$ and such that the negative of the conductor of $E$ is not a square modulo $p$, then there is a positive constant $B$ depending on $A$ but not on $E$ or $n$ such that $\mathrm{rk}_p(E/\mathbf{Q}(A[p^n])) \ge B p^{2n}$ for every $n$

Article information

Duke Math. J. Volume 143, Number 3 (2008), 437-461.

First available in Project Euclid: 3 June 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G05: Elliptic curves over global fields [See also 14H52]
Secondary: 14G05: Rational points 11R23: Iwasawa theory 20C15: Ordinary representations and characters


Mazur, Barry; Rubin, Karl. Growth of Selmer rank in nonabelian extensions of number fields. Duke Math. J. 143 (2008), no. 3, 437--461. doi:10.1215/00127094-2008-025. http://projecteuclid.org/euclid.dmj/1212500463.

Export citation


  • J. Coates, T. Fukaya, K. Kato, and R. Sujatha, Root numbers, Selmer groups, and noncommutative Iwasawa theory, preprint, 2007.
  • J. E. Cremona, Algorithms for Modular Elliptic Curves, Cambridge Univ. Press, Cambridge, 1992.
  • T. Dokchitser and V. Dokchitser, On the Birch-Swinnerton-Dyer quotients modulo squares, preprint, \arxivmath/0610290v2[math.NT]
  • —, Regulator constants and the parity conjecture, preprint,\hfill\arxiv0709.2852[math.NT]
  • D. S. Dummit and R. M. Foote, Abstract Algebra, 3rd ed., Wiley, Hoboken, N.J., 2004..
  • R. Greenberg, Galois theory for the Selmer group of an abelian variety, Compositio Math. 136 (2003), 255--297.
  • —, Iwasawa theory, projective modules, and modular representations, in preparation.
  • M. Harris, Systematic growth of Mordell-Weil groups of abelian varieties in towers of number fields, Invent. Math. 51 (1979), 123--141.
  • L. Howe, Twisted Hasse-Weil $L$-functions and the rank of Mordell-Weil groups, Canad. J. Math. 49 (1997), 749--771.
  • B. D. Kim, The parity conjecture for elliptic curves at supersingular reduction primes, Compos. Math. 143 (2007), 47--72.
  • H. Koch and B. B. Venkov, ``Über den $p$-Klassenkörperturm eines imaginär-quadratischen Zahlkörpers'' in Journées Arithmétiques de Bordeaux (Bordeaux, France, 1974), Astérisque 24 --.25, Soc. Math. France, Montrouge, 1975, 57--67.
  • M. Lazard, Groupes analytiques $p$-adiques, Inst. Hautes Études Sci. Publ. Math. 26 (1965), 389--603.
  • B. Mazur and K. Rubin, Finding large Selmer rank via an arithmetic theory of local constants, Ann. of Math. (2) 166 (2007), 579--612.
  • P. Monsky, Generalizing the Birch-Stephens theorem, I: Modular curves, Math. Z. 221 (1996), 415--420.
  • A. Movahhedi and T. NguyễN-Quang-\Garyỗ, ``Sur l'arithmétique des corps de nombres $p$-rationnels'' in Séminaire de Théorie des Nombres, Paris, 1987--88. (Paris, 1987--88.), Progr. Math. 81, Birkhäuser, Boston, 1990, 155--200.
  • J. Neková\UR, On the parity of ranks of Selmer groups, II, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), 99--104.
  • —, Selmer Complexes, Astérisque 310, Soc. Math. France, Montrouge, 2006.
  • —, On the parity of ranks of Selmer groups, IV, preprint, 2007.
  • D. E. Rohrlich, Galois theory, elliptic curves, and root numbers, Compositio Math. 100 (1996), 311--349.
  • —, Scarcity and abundance of trivial zeros in division towers, to appear in J. Algebraic Geom.
  • R. Schoof, Infinite class field towers of quadratic fields, J. Reine Angew. Math. 372 (1986), 209--220.
  • J.-P. Serre, Abelian $\ell$-adic Representations and Elliptic Curves, Benjamin, New York, 1968.
  • —, Représentations linéaires des groupes finis, 2ème éd., Hermann, Paris, 1971.
  • —, ``Résumé des cours de 1985--1986.'' in Qeuvres: Collected Papers, IV, Springer, Berlin, 2000, 33--37.