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dc.contributor.authorMazur, Barry C.
dc.contributor.authorRubin, Karl
dc.date.accessioned2013-02-28T19:36:33Z
dc.date.issued2008
dc.identifier.citationMazur, Barry C., Karl Rubin. 2008. Growth of Selmer rank in nonabelian extensions of number fields. Duke Mathematical Journal 143(3): 437-461.en_US
dc.identifier.issn0012-7094en_US
dc.identifier.urihttp://nrs.harvard.edu/urn-3:HUL.InstRepos:10355855
dc.description.abstractLet \(p\) 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 \(Z_p\)-corank \(rk_p(E/F)\) of the \(p\)-power Selmer group of \(E\) over \(F\). We obtain lower bounds for \(rk_p(E/F)\), generalizing the results in [MR], which applied to dihedral extensions. If \(K\) is the (unique) quadratic extension of \(k\) in \(F\), if \(G = Gal(F/K)\), if \(G+\) is the subgroup of elements of \(G\) commuting with a choice of involution of \(F\) over \(k\), and if \(rk_p(E/K)\) is odd, then we show that (under mild hypotheses) \(rkp(E/F)\ge[G:G+]\). 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 \(F\frac{x}{p}\) 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 \(rk_p(E/Q(A[p^n]))/geBp^{2n}\) for every \(n\).en_US
dc.description.sponsorshipMathematicsen_US
dc.language.isoen_USen_US
dc.publisherDuke University Pressen_US
dc.relation.isversionofdoi:10.1215/00127094-2008-025en_US
dash.licenseOAP
dc.titleGrowth of Selmer Rank in Nonabelian Extensions of Number Fieldsen_US
dc.typeJournal Articleen_US
dc.description.versionAccepted Manuscripten_US
dc.relation.journalDuke Mathematical Journalen_US
dash.depositing.authorMazur, Barry C.
dc.date.available2013-02-28T19:36:33Z
dc.identifier.doi10.1215/00127094-2008-025*
dash.contributor.affiliatedMazur, Barry


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