On the $\mu$-invariant of two-variable $2$-adic $\boldsymbol{L}$-functions

Abstract Let $K={\mathbb {Q}}(\sqrt {-7})$ and $\mathcal {O}$ the ring of integers in $K$ . The prime $2$ splits in $K$ , say $2{\mathcal {O}}={\mathfrak {p}}\cdot {\mathfrak {p}}^*$ . Let $A$ be an elliptic curve defined over $K$ with complex multiplication by $\mathcal {O}$ . Assume that $A$ has good ordinary reduction at both $\mathfrak {p}$ and ${\mathfrak {p}}^*$ . Write $K_\infty$ for the field generated by the $2^\infty$ –division points of $A$ over $K$ and let ${\mathcal {G}}={\mathrm {Gal}}(K_\infty /K)$ . In this paper, by adopting a congruence formula of Yager and De Shalit, we construct the two-variable $2$ -adic $L$ -function on $\mathcal {G}$ . Then by generalizing De Shalit’s local structure theorem to the two-variable setting, we prove a two-variable elliptic analogue of Iwasawa’s theorem on cyclotomic fields. As an application, we prove that every branch of the two-variable measure has Iwasawa $\mu$ invariant zero.