Duke Mathematical Journal

Growth of Selmer rank in nonabelian extensions of number fields

Barry Mazur and Karl Rubin

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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 Zp-corank rkp(E/F) of the p-power Selmer group of E over F. We obtain lower bounds for rkp(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=Gal(F/K), if G+ is the subgroup of elements of G commuting with a choice of involution of F over k, and if rkp(E/K) is odd, then we show that (under mild hypotheses) rkp(E/F)[G:G+].

As a very specific example of this, suppose that A is an elliptic curve over Q with a rational torsion point of order p and without complex multiplication. If E is an elliptic curve over Q with good ordinary reduction at p such that every prime where both E and A have bad reduction has odd order in Fp× 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 rkp(E/Q(A[pn]))Bp2n for every n

Article information

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

First available in Project Euclid: 3 June 2008

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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. https://projecteuclid.org/euclid.dmj/1212500463

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  • 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.