15 June 2008 Growth of Selmer rank in nonabelian extensions of number fields
Barry Mazur, Karl Rubin
Author Affiliations +
Duke Math. J. 143(3): 437-461 (15 June 2008). DOI: 10.1215/00127094-2008-025


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


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Barry Mazur. Karl Rubin. "Growth of Selmer rank in nonabelian extensions of number fields." Duke Math. J. 143 (3) 437 - 461, 15 June 2008. https://doi.org/10.1215/00127094-2008-025


Published: 15 June 2008
First available in Project Euclid: 3 June 2008

zbMATH: 1151.11023
MathSciNet: MR2423759
Digital Object Identifier: 10.1215/00127094-2008-025

Primary: 11G05
Secondary: 11R23 , 14G05 , 20C15

Rights: Copyright © 2008 Duke University Press


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Vol.143 • No. 3 • 15 June 2008
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