Growth of Selmer Rank in Nonabelian Extensions of Number Fields

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Growth of Selmer Rank in Nonabelian Extensions of Number Fields

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Title: Growth of Selmer Rank in Nonabelian Extensions of Number Fields
Author: Mazur, Barry C.; Rubin, Karl

Note: Order does not necessarily reflect citation order of authors.

Citation: Mazur, Barry C., Karl Rubin. 2008. Growth of Selmer rank in nonabelian extensions of number fields. Duke Mathematical Journal 143(3): 437-461.
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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.

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 \(n\).
Published Version: doi:10.1215/00127094-2008-025
Terms of Use: This article is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#OAP
Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:10355855
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