Duke Mathematical Journal

Growth of Selmer rank in nonabelian extensions of number fields

Barry Mazur and Karl Rubin

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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 $\mathbf{Z}_p$-corank $\mathrm{rk}_p(E/F)$ of the $p$-power Selmer group of $E$ over $F$. We obtain lower bounds for $\mathrm{rk}_p(E/F)$, generalizing the results in [MR], which applied to dihedral extensions.

If $K$ is the (unique) quadratic extension of $k$ in $F$, if $G = \mathrm{Gal}(F/K)$, if $G^+$ is the subgroup of elements of $G$ commuting with a choice of involution of $F$ over $k$, and if $\mathrm{rk}_p(E/K)$ is odd, then we show that (under mild hypotheses) $\mathrm{rk}_p(E/F) \ge [G:G^+]$.

As a very specific example of this, suppose that $A$ is an elliptic curve over $\mathbf{Q}$ with a rational torsion point of order $p$ and without complex multiplication. If $E$ is an elliptic curve over $\mathbf{Q}$ with good ordinary reduction at $p$ such that every prime where both $E$ and $A$ have bad reduction has odd order in $\mathbf{F}_p^\times$ 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 $\mathrm{rk}_p(E/\mathbf{Q}(A[p^n])) \ge B p^{2n}$ for every $n$

Article information

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

Dates
First available in Project Euclid: 3 June 2008

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1212500463

Digital Object Identifier
doi:10.1215/00127094-2008-025

Mathematical Reviews number (MathSciNet)
MR2423759

Zentralblatt MATH identifier
1151.11023

Subjects
Primary: 11G05: Elliptic curves over global fields [See also 14H52]
Secondary: 14G05: Rational points 11R23: Iwasawa theory 20C15: Ordinary representations and characters

Citation

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


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