dc.contributor.author Mazur, Barry C. dc.contributor.author Rubin, Karl dc.date.accessioned 2013-02-28T19:36:33Z dc.date.issued 2008 dc.identifier.citation Mazur, 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.issn 0012-7094 en_US dc.identifier.uri http://nrs.harvard.edu/urn-3:HUL.InstRepos:10355855 dc.description.abstract 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 $Z_p$-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. en_US 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 $n$. dc.description.sponsorship Mathematics en_US dc.language.iso en_US en_US dc.publisher Duke University Press en_US dc.relation.isversionof doi:10.1215/00127094-2008-025 en_US dash.license OAP dc.title Growth of Selmer Rank in Nonabelian Extensions of Number Fields en_US dc.type Journal Article en_US dc.description.version Accepted Manuscript en_US dc.relation.journal Duke Mathematical Journal en_US dash.depositing.author Mazur, Barry C. dc.date.available 2013-02-28T19:36:33Z dc.identifier.doi 10.1215/00127094-2008-025 * dash.contributor.affiliated Mazur, Barry
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