On the Vanishing of μ-Invariants of Elliptic Curves over Q

Trifković, Mak

Canadian journal of mathematics, Tome 57 (2005) no. 4, pp. 812-843

Résumé

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.

Détail

DOI : 10.4153/CJM-2005-032-9

Mots-clés : 11R23

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