On the Vanishing of $mu$-Invariants of Elliptic Curves over $qq$

[摘要] Let $E_{/qq}$ be an elliptic curve with good ordinary reduction at aprime $p>2$. It has a well-defined Iwasawa $mu$-invariant $mu(E)_p$which encodes part of the information about the growth of the Selmergroup $sel E{K_n}$ as $K_n$ ranges over the subfields of thecyclotomic $zzp$-extension $K_infty/qq$. Ralph Greenberg hasconjectured that any such $E$ is isogenous to a curve $E'$ with$mu(E')_p=0$. In this paper we prove Greenberg's conjecture forinfinitely many curves $E$ with a rational $p$-torsion point, $p=3$ or$5$, no two of our examples having isomorphic $p$-torsion. The coreof our strategy is a partial explicit evaluation of the global dualitypairing for finite flat group schemes over rings of integers.

[学科分类] 数学（综合）