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 $K$, and we write $mathfrak{p}$ for either of the primes $K$ above $2$. Let $K_infty$ be the unique $mathbb{Z}_2$-extension of $K$ unramified outside $mathfrak{p}$ with $n$-th layer $K_n$. For certain quadratic extensions $F/K$, we prove a simple exact formula for the $lambda$-invariant of the Galois group of the maximal abelian 2-extension unramified outside $mathfrak{p}$ of the field $F_infty = FK_infty$. Equivalently, our result determines the exact $mathbb{Z}_2$-corank of certain Selmer groups over $F_infty$ of a large family of quadratic twists of the higher dimensional abelian variety with complex multiplication, which is the restriction of scalars to $K$ of the Gross curve with complex multiplication defined over the Hilbert class field of $K$. We also discuss computations of the associated Selmer groups over $K_n$ in the case when the $lambda$-invariant is equal to $1$.