On Iwasawa $\mu$-invariants for Selmer groups over $\mathbb{Z}_2$-extensions

Abstract: Let $E$ be an elliptic curve over $\mathbb{Q}$ with good ordinary reduction at $p$, $E[p]$ the $p$-torsion points of $E$ and $\mathbb{Q}_\infty/\mathbb{Q}$ the cyclotomic $\mathbb{Z}_p$-extension. Further assume $E[p]$ is reducible as a $G_\mathbb{Q}$-representation over $\mathbb{F}_p$. In 1999, Greenberg offered sufficient conditions for the Pontryagin dual of the Selmer group over $\mathbb{Q}_\infty$ to have $\mu$-invariant 0, and obtained a lower bound when the $\mu$-invariant is positive. In this talk, we will focus on the case $p=2$ case and provide an upper bound for the $\mu$-invariants. Combining with Greenberg’s result, we are able to classify all the elliptic curves with $\mu$-invariant 0 or 1 under the additional assumptions that $E[4]$ is reducible. This is a joint work with Mulun Yin.

number theory

Audience: researchers in the topic

Pioneer Valley Number Theory Seminar