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On the Vanishing of μ-Invariants of Elliptic Curves over ℚ
Published online by Cambridge University Press: 20 November 2018
Abstract
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Let 
${{E}_{/\mathbb{Q}}}$ be an elliptic curve with good ordinary reduction at a prime 
$p\,>\,2$. It has a welldefined Iwasawa 
$\mu $-invariant 
$\mu {{\left( E \right)}_{p}}$ which encodes part of the information about the growth of the Selmer group 
$\text{Se}{{\text{l}}_{{{p}^{\infty }}}}\left( {{E}_{/{{K}_{n}}}} \right)$ as 
${{K}_{n}}$ ranges over the subfields of the cyclotomic 
${{\mathbb{Z}}_{p}}$-extension 
${{K}_{\infty }}/\mathbb{Q}$. Ralph Greenberg has conjectured that any such 
$E$ is isogenous to a curve 
${E}'$ with 
$\mu {{\left( {{E}'} \right)}_{p}}\,=\,0$. In this paper we prove Greenberg's conjecture for infinitely many curves $E$ with a rational 
$p$-torsion point, $p$ = 3 or 5, no two of our examples having isomorphic $p$-torsion. The core of our strategy is a partial explicit evaluation of the global duality pairing for finite flat group schemes over rings of integers.
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- Copyright © Canadian Mathematical Society 2005
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