On the $\lambda$-invariant of Selmer Groups Arising from Certain Quadratic Twists of Gross Curves

Let $q$ be a prime with $q \equiv 7 \mod 8$, and let $K=\mathbb{Q}(\sqrt{-q})$. Then $2$ splits in $K$, we write $\mathfrak{p}$ for either of the primes $K$ above $2$. $K_\infty$ unique $\mathbb{Z}_2$-extension unramified outside $\mathfrak{p}$. For certain quadratic biquadratic extensions $\mathfrak{F}/K$, prove simple exact formula $\lambda$-invariant Galois group maximal abelian 2-extension field $\mathfrak{F}_\infty = \mathfrak{F} K_\infty$. Equivalently, our result determines $\mathbb{Z}_2$-corank Selmer groups over $\mathfrak{F}_\infty$ large family twists higher dimensional variety complex multiplication, which is restriction scalars to Gross curve multiplication defined Hilbert class $K$. We also exhibit computations associated $K_n$ case when equal $1$; here denotes $n$-th layer $K_\infty/K$.