Statistics for Iwasawa invariants of elliptic curves, $\rm{III}$

Abstract

Given a prime $p\geq 5$, a conjecture of Greenberg predicts that the $\mu$-invariant of the $p$-primary Selmer group should vanish for most elliptic curves with good ordinary reduction at $p$. In support of this conjecture, I show that the $5$-primary Iwasawa $\mu$- and $\lambda$-invariants simultaneously vanish for an explicit positive density of elliptic curves $E_{/\mathbb{Q}}$. The elliptic curves in question have good ordinary reduction at $5$, and are ordered by their height. The results are proven by leveraging work of Bhargava and Shankar on the distribution of $5$-Selmer groups of elliptic curves defined over $\mathbb{Q}$.